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MATHEMATICAL SCIENCESDIVISION

1992 Programs

0 rAV,,. . AUG 20 19930

OFFICE OF NAVAL RESEARCH800 NORTH QUINCY STREETARLINGTON, VA 22217-5660

This document is issued for the information of U.S. Government scientificpersonnel and contractors. It is not considered part of the scientific literatureand should not be cited as such.

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Mathematical Sciences Division 1992 Programs (U) 61153N

6. AUTHOR(S)

Edited by Dr. Neil L. Gerr

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION

Office of Naval Research REPORT NUMBER

Mathematical Sciences DivisionCode 1111 OCNR 111192-12Arlington, VA 22217- 5660

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Mathematical Sciences Division OCNR 111192-12Code 1111Arlington, VA 22217-5660

"11. SUPPLEMENTARY NOTES

This is a compilation of abstracts describing research projects sponsored bythe ONR Mathematical Sciences Division in FY92.

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This report documents research performed under the sponsorship of the ONRMathematical Sciences Division during fiscal year 1992.

14. SUBJECT TERMS 15. NUMBER OF PAGES214 + ivDiscrete Mathematics Numerical Analysis 24+i

Applied Analysis Operations Research - 16. PRICE CODE

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MATHEMATICAL SCIENCESDIVISION

1992 Programs

OFFICE OF NAVAL RESEARCH800 NORTH QUINCY STREETARLINGTON, VA 22217-5660

This document is issued for the information of U.S. Government scientificpersonnel and contractors. It is not considered part of the scientific literatureand should not be cited as such.

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

FOREWORD

This book describes research projects carried out undersponsorship of the Mathematical Sciences Division of the Office ofNaval Research (ONR) during fiscal year 1992. The Division'sresearch is organized in six programs:

- Applied Analysis

- Discrete Mathematics

- Numerical Analysis

- Operations Research

- Probability and Statistics

- Signal Analysis

These programs are described in the Introduction. The individualresearch projects supported by each are described in the chaptersthat follow.

This book is but one of several means by which wecommunicate our interests to the scientific community. If you wishfurther information, please do not hesitate to contact us; ournames and phone numbers are listed in the Introduction. We welcomeyour interest in our programs.

Neil L. GerrDirectorMathematical Sciences Division

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CONTENTS

FOREWORD ................. .......................... i

INTRODUCTION ........................... iii

APPLIED ANALYSIS .................................... 1

DISCRETE MATHEMATICS ............. .................. 33

NUMERICAL ANALYSIS ........................ 69

OPERATIONS RESEARCH ....................... 97

PROBABILITY AND STATISTICS ......... ............... 123

SIGNAL ANALYSIS ................ ..................... 165

DIVISION PROJECTS .............. .................... 201

INDEX ...................... .......................... 209

ii

INTRODUCTION

The Mathematical Sciences Division supports basic research that iscarried out under grants and contracts that are awarded on acompetitive basis. Proposals received in response to the ONR BroadAgency Announcement published in the Commerce Business Daily areevaluated according to th2 following criteria:

1. Scientific Merit.

2. Navy Relevance.

3. Potential for success.

4. Cost.

In general, proposals are evaluated in-house by the appropriateScientific Officer. All of the Division Scientific Officers havePh.D.s and are well versed in current Navy needs.

As noted in the Foreword, the Mathematical Sciences Division isorganized in six programs. These are described below and theresponsible Scientific Officer is indicated. Dr. Neil L. Gerr isthe Director of the Mathematical Sciences Division. Dr. Bobby R.Junker is the Director of the Mathematical and Physical SciencesDirectorate, of which the Mathematical Sciences Division is a part.

I. Applied AnalysisScientific Officer: Dr. Reza Malek-Madani (Acting)

(703) 696-4314

The program develops and exploits methods in modern and classicalmathematical analysis, with emphasis on mathematical models ofphysical phenomena. Naval interest in waves and flows motivatesresearch in convection-diffusion, with emphasis in the areas ofshock capturing, inertial manifolds, vortex-free surfaceinteraction, and control of fluids. Acoustics, electromagnetics,modeling the ocean bottom and the ocean medium, and nondestructiveevaluation motivate research in inverse scattering and mathematicalinverse theory.

II. Discrete MathematicsScientific Officer: Dr. Marc J. Lipman

(703) 696-4310

The goal is to develop areas of discrete mathematics which haveapplication to engineering and information sciences. This includesspecific structural parameters and invariants for combinatorialconfigurations such as graphs, networks, sequences, and partiallyordered sets. The use of discrete structures for modeling,consensus formation, and classification is of particular interest.

iii

III. Numerical AnalysisScientific officer: Dr. Richard L. Lau

(703) 696-4316

Research is conducted on the numerical solution of partialdifferential equations. Areas of particular interest at presentinclude the problems of engineering design, acoustic propagation,geophysical fluid mechanics, and electromagnetic scattering.Emphasis is placed on methods to estimate, control, and reducecomputational error, and on the analysis of error induced bymathematical approximations, such as dimensional reduction, and bydata errors.

IV. Operations ResearchScientific Officer: Dr. Donald K. Wagner

(703) 696-4313

The program focuses on mathematical programming and its applicationto problems in the areas of logistics, resource allocation,production, and command and control. This includes research inlinear, nonlinear, integer and combinatorial optimization, and insequencing and scheduling. Theoretical development, algorithmdesign and analysis, computational techniques for large-scaleproblems, and innovative and relevant applications are all ofinterest. Interior-point methods and related algebraic andgeometric concepts are an area of particular emphasis.

V. Probability and StatisticsScientific Officer: Dr. Julia Abrahams

(703) 696-4320

The focus is on probabilistic analysis and statistical inferencefor random fields and spatial processes which model the temporaland spatial evolution of physical phenomena, with particularemphasis on oceanographic applications. Stochastic geometry andgeometric probability in conjunction with image analysis and themodeling of shape are of interest. Several efforts developingstochastic models for the neural sciences are also supported.

VI. Signal AnalysisScientific Officer: Dr. Neil L. Gerr (Acting)

(703) 696-4321

This program aims to develop the mathematical and statisticaltheory that underlies modern signal analysis. There is a strongemphasis on stochastic processes and time series, particularlythose which provide realistic models for random signals and noise,and their characterization in the spectral domain. A primary goalis to develop new techniques for detecting and classifyingbroadband and transient acoustic signals received in noise.

iv

Applied Analysis

1

TITLE: Fast Algorithms for Three-Dimensional InverseScattering Problems

PI: Andrew E. YagleUniversity of MichiganDepartment of EECS(313) 763-9810

FUNDING AGENCY: University Research Initiative

R&T PROJECT CODE: 400x068yip03

CONTRACT NO: N0001490J1897

CURRENT END DATE: 31 AUG 1993

SCIENTIFIC OFFICER: Reza Malek-Madani

Technical Objective:Layer-stripping algorithms that reconstruct the scatteringpotential at a wavefront in a three-dimensional medium and use theinformation obtained to recursively propagate the wavefield furtherinto the medium will be developed. The algorithms to be developedwill be fast on serial computers and will be parallelizable.

Approach:The scattering problems will be expressed by the Schroedinger waveequation with time-invariant potential and by acoustic waveequations with variable velocity and density. Three-dimensionalone-dimensional layer-stripping algorithms developed by theproposer and others. Computations will be carried out to verifythe theoretical results.

Progress:Recent work addressed the problem of inverting spherical means, thegoal being to recover a function from its averages over all spherespassing through the origin. Geometric inversion was used totransform the problem into an inverse Radon transform problem. Byreflecting functions about the sphere marking the outer limit ofknown data, averages over spheres are transformed into averagesover planes. In addition, it was shown that measurements at asingle sensor are sufficient to reconstruct the scatteringpotential in either the Born or Rytov approximations providedeither 1) wideband data are available, or 2) the average value ofthe potential is known.

3

TITLE: Acoustics, Waves and Flows

PI: Joseph B. KellerLeland Stanford Junior UniversityDepartment of Mathematics(415) 723-0851

FUNDING AGENCY: Office of Naval Research

R&T PROJECT CODE: 4112549---07


CURRENT END DATE: 30 SEP 1993

SCIENTIFIC OFFICER: Reza Malek-Marani

Technical Objective:The first objective is to treat the two-dimensional case of aninfinitely long straight vortex parallel to the surface and obtaina complete description of the motion of the surface and the vortex.Extension will then be made to the three-dimensional linear caseand the two-dimensional nonlinear case. Finally, thethree-dimensional nonlinear case will be addressed. Both short andlong-time solutions will be derived.

Approach:The surface will be represented by a Green's function integral orsome equivalent expression. To evaluate the integral, asymptoticmethods will be used. Nonlinearity will be taken care of byperturbation expansions and numerical methods. These methodsdepend on the use of complex variables and identification of all ofthe singularities of the flow. A hybrid Galerkin-perturbationmethod, an adaption of the perturbation-expansion method that isexpected to be more powerful for the cases under consideration,will also be used.

Progress:Simplified equations governing the potential flow and shape ofslender jets and thin sheets of liquid were derived, taking accountof surface tension. Families of s-imilarity solutions of theseequations were introduced. For jets and symmetrical sheets theysatisfy ordinary differential equations. The properties of thesesimilarity solutions were examined analytically and numerically.They can be used to describe the motion of a liquid sheet on asolid, the thickening and flow following the breaking of jets, andthe merging of two jets.

4

TITLE: Inverse and Optimization Problems for Low FrequencyData and Low Frequency Electromagnetic Inverse Problems

PI: Joyce R. McLaughlin and Margaret CheneyRensseAaer Polytechnic InstituteDepartment of Mathematical Sciences(518) 276-6349




CURRENT END DATE: 31 DEC 1992


Technical Objective:The wave speed in a medium of finite extent will be determined fromthe interior eigenvalues and scattered field data at frequenciesthat are not eigenvalues. The information content of eacheigenvalue will be determined and stability of the algorithm willbe shown. Under the expansion task, improved techniques forelectromagnetic inverse problems will be developed.

Approach:The theory of Riesz bases and non-self-adjoint operators will beused. Algorithms will be based on spectral methods. Research ondetermining wave speed will be carried out first for thespherically symmetric problem and then for fully multi-dimensionalproblems. For electromagnetic inversion, multidimensionallayer-stripping methods will be developed.

Progress:A connection between inverse scattering and inverse spectral theorywas established in the context of spherically symmetric threedimensional inverse acoustic scattering. It was shown thataverages of the scattering amplitude determine the eigenvalues foran eigenvalue problem. Using techniques of inverse eigenvalueproblems, a uniqueness theorem to determine the sound speed wasobtained. New properties of this nonlinear inverse problem are 1)the "size" of inhomogeneities in the sound speed can be determinedby the largest eigenvalues, 2) the sound speed is uniquelydetermined by the eigenvalues if the "size" is small, and 3) nolinear approximation is required.

5

Title: Energy Stability and Bifurcations inLangmuir Circulation

PI: Isom H. HerronHoward UniversityMathematics Department(202) 806-6830




CURR-NT EVID DATE: 30 SEP 1993


Technical Objective:The work aims to develop f or flows on unbounded domains therelationship between spectral theory on one hand and weaklynonlinear stability and bifurcation theory on the other. Flows inboundary layers, jets and shear layers will be addressed. Theeffect of curvature on the nonlinear stability of various flowswill be derived. Incompressible flows in two and three dimensionswill be considered.

Approach:Weighted inner-product spaces and computational experiments will beused to establish the relationship between spectral theory andnonlinear stability theory and bifurcation theory. Nonlinearstability of flows with curvature will be investigated by themethod of energy, which has proven to be effective in explainingTaylor-Couette flow between rotating cylinders.

Progress:The linearized equations for the evolution of disturbances to fourwall bounded flows were treated. The flows are plane Couette flowand plane Poiseuille flow, Hagen-Poiseuille pipe flow, and theasymptotic suction profile. By looking at the vorticity it wasproved that plane Couette and Hagen-Poiseuille flow are linearlystable.

6

Title: Mathematical Analysis of Strong FluidMechanical Effects in- Turbulence andDetonation Physics

PI: Andrew J. MajdaPrinceton UniversityDepartment of Mathematics(609) 258-5114






Technical Objective:Fully developed turbulence involves velocity fields which have acontinuous range of excited space and time scales. These inducesubstantial additional dissipation at high Reynolds numbers due tothe development of energetic small scales. A primary goal here isto assess the effect of the motion at arbitrarily small lengthscales on the large length scales without resolving the smallscales in detail.

Approach:Classes of model problems that retain some of the subtle featuresof turbulent transport and for which renormalization theory can besolved with mathematical rigor will be developed for turbulentadvection-diffusion. These will provide an unambiguous check onthe variety of ad hoc methods for renormalizing turbulenttransport. In addition, behaviors observed for various regimes ofvelocity statistics will suggest improved theories for eddydiffusivity in more general contexts.

Progress:Developed a rigorous renormalization theory for a model problem ofadvection-diffusion of a passive scalar by an incompressiblevelocity field. In particular, explicit formulas for therenormalized anomalous time scaling as well as both the explicitGreen's function for the eddy diffusivity equation and higher orderstatistics were derived.

7

Title: Wave Propagation and Inverse Problems inHeterogeneous Elastic and AnelasticLayered Media

PI: William W. SymesRice UniversityDepartment of Mathematical Sciences(713) 527-4805






Technical Objective:The resolution of wave velocity from reflected waves, and itsdependence on model completeness, will be quantified in the contextof layered acoustic and elastic media. The resolution of theparameters of viscoelestic and poroelastic layered solids from pre-and post-critical reflected wave data will be quantified underrealistic assumptions for interwell and shallow seismology andseabed sounding.

Approach:Perturbation theory, high-frequency asymptotics, and energyestimates will be used to determine the sensitivity of reflectiondata to small perturbations in various classes of model parameters.

Progress:A new formulation was introduced for the velocity inversion problemvia reparameterization by travel-time. The new formulationrequires the solution of a large scale optimization problem. Theinversion is regularized (the optimization made non-pathological)by incorporating a penalty term that enforces the appropriatemathematical constraint. Initial results applying the approach tothe problem of plane wave reconstruction are very encouraging.

8

Title: Rotational Inviscid Flows with Free Boundaries

PI: J. R. OckendonOxford UniversityMathematical Institute(0865) 448-6527



CONTRACT NO: N0001491JI163



Technical Objective:The objective in vortex/free surface interaction is to developanalysis explaining the bouncing phenomenon for line vorticesbeneath a free surface, the movement of curved vortices near a freesurface and the singularities that occur when vortices are near afree surface. The objectives in marine meteorology are to determinethe smoothness of the solution of a meteorological variationalproblem over a modern measure space and to use of properties ofthis solution to design practical numerical methods for weatherfronts.

Approach:The research on vortex/free surface interaction will be carried outusing ordinary-differential-equation models, generalized-functionrepresentations of the vortices, asymptotic techniques, transformtechniques, point-vortex methods, vortex-blob methods andring-vortex methods. The research in meteorology will be carriedout using geometrical and linear programming methods for solvingproblems in Lp spaces, generalized solutions of the Monge-Ampereequation that preserve local potential vorticity and numericalsolution procedures.

Progress:The mathematical theory of total internal reflection of light raysemanating from a real source above an interface between twohomogeneous media was analyzed using the method of multiple scales.This allows geometrical optics to be applied away from the criticalray, and ray optics to be matched into this framework near theinterface. Most recently, models for phase changes in isothermalsuperconductors were identified with vector versions of previouslyanalyzed free boundary models of supercooled solidification. Thismay allow knowledge from solidification theory to be applied tosuperconductor design.

9

Title: Mathematical Stability Analyses ofBenard Convection

PI: David J. WollkindWashington State UniversityPure and Applied Mathematics Department(509) 332-2162






Technical Objective:Linear stability of the conduction solution for aerosol one-layerRayleigh-Benard models and weakly nonlinear stability of theconduction solution for a two-layer Benard-Marangoni system will bedetermined. The resulting theoretical predictions will be comparedto experimental results on buoyancy-driven columnar instability inaerosol-air mixtures. The discrepancy between experimental resultsand currently available theoretical results will be at leastpartially resolved.

Approach:Asymptotic methods will be used throughout. First, the analysiswill be carried out in an unbounded region. Then a layer of finitedepth will be considered. Normal-mode expansions will be used tocreate systems of ordinary differential equations, which will besolved by Galerkin and other numerical methods. In the research ontwo-layer cases, a certain interfacial condition will be assumed.

Progress:The weak nonlinear stability of the pure conduction solution for anaerosol one-layer Rayleigh-Benard model of a Boussinesqparticle-gas system in thermal equilibrium was addressed. Areduction of the threshold temperature gradient associated with theoccurrence of the supercritically equilibrated state was predictedfor a clean gas, that reduction increasing with decreasing layerdepth.

10

Title: An Analytical Study of Some Problems in PartialDifferential Equations with Applications toFluid Dynamics and Wave Propagation

PI: George H. KnightlyUniversity of MassachusettsDepartment of Mathematics(413) 545-0823

FUNDING AGENCY: office of Naval Research



CURRENT END DATE: 31 OCT 1992


Technical Objective:Research will be performed on (a) stability and bifurcation forviscous, incompressible Couette flow under various driving forces;(b) well-posedness of the equations governing two-phase phenomenain the presence of convection and properties of the regionseparating the solid and liquid phases; (c) stability and accuracyof marching methods for predicting wave propagation ininhomogeneous media (Helmholtz equation of underwater acoustics).

Approach:In area (a), methods developed recently by others for a class ofdifferential equations in Banach space will be adapted. In area(b), methods of nonlinear functional analysis will be utilized. Inarea (c), the methods of stability analysis for numerical schemeswill be utilized to estimate accuracy and computational stability.

Progress:Periodic waves were shown to exist and their expansions derived fora variety of Couette-Poiseuille problems in the narrow gap limit ofviscous flow between rotating sliding cylinders under applied axialand circumferential pressure gradients. For rotating Couette flow,a continuum of periodic waves bifurcating from the basic spiralflow was found. A connection between such spiral flows andLangmuir circulation in upper ocean mixing was established.Langmuir circulation in the form of periodic waves was generatedfor certain problems in which the Stokes drift is not aligned withthe wind direction.

11

Title: The Application of Inverse Methods tothe Ocean Environment

PI: Norman BleisteinColorado School of MinesDepartment of Mathematical & Computer Sciences(303) 273-3255

FUNDING AGENCY: Office of-Naval Research





Technical Objective:An inverse-scattering theory for complex background structures andrealistic (separated) source/receiver configurations will bedeveloped. Inversion procedures for both isotropic and anisotropicelastic media will be created. Methods for separating signals andfor correlation of reflector locations will be developed. Computercodes to implement these procedures will be written. Extension ofboth modeling and codes to three dimensions will be carried out.In addition, enhanced methods for direct propagation predictionwill be developed.

Approach:The Born approximation and backprojection are the basis of theanalysis and the computational procedure. The backprojectionmethod will be extended to allow for complex background structuresand separated source/receiver configurations. Time windows will beused to separate signals. The postprocessing analysis forcorrelation of reflector locations will be based on processing withdifferent travel time inversion operators. In addition, the ideasof wavefield splitting, invariant embedding, and phase spaceanalysis will be used.

Progress:Conventional seismic inversion methods for imaging structure insidethe earth require reasonable background velocities. A newtechnique was introduced in which velocity analysis and imaging areperformed at the same time. The medium is assumed to consist ofconstant-velocity layers separated by arbitrary smooth interfaces.The goal of the inversion is to determine layer velocities andinterface depths. The approach is based on the fact that velocityerror yields image distortion. The sensitivity to velocity errorwas derived for some special cases. The technique is now beingtested on synthetic and experimental data.

12

Title: Uniqueness Theorems and Numerical Solutionof 3-D Inverse Scattering Problems

PI: Alexander G. RammKansas State UniversityDepartment of Mathematics(913) 532-6750






Technical Objective:New uniqueness theorems for three-dimensional electromagneticinverse scattering problems will be proved. A constructiveanalytical method for inversion of the scattering data at a fixedfrequency will be obtained. Stability of this method will beestablished. Convergence of the method for exact data will beproved. Error estimates as the error of the measurements tends tozero will be found.

Approach:Techniques of functional analysis, including Sobolev space theory,will be used to establish inversion formulas and prove uniqueness.Techniques of numerical error analysis will be used to establishthe properties of the computational method.

Progress:Results were obtained in a number of areas:- A method was developed for determining the stability of the exactinversion problem for 3D fixed-energy scattering. Error estimatesfor the inversion of noisy 3D fixed-energy scattering were derived.- An algorithmically verifiable characterization of the class ofscattering amplitudes corresponding to small local potentials wasobtained.- Necessary and sufficient conditions on the scattering data werefound for the scatterer to be spherically symmetric.

13

Title: Mathematical Methods in Applied WavePropagation

PI: Gregory A. KriegsmanNew Jersey Institute of TechnologyDepartment of Mathematics(201) 596-5785






Technical Objective:The method of normal modes, which has been successfully applied torange-independent oceans, will be applied to range-dependent oceanenvironments. An "effective-medium" wave equation that describesscattering of acoustic waves by the ocean bottom will be developed.The acoustic fields both near and far from a target will bedescribed. Computational experiments will be carried out.

Approach:Analytical, numerical and computational methods will be usedthroughout the research. Slowly varying range dependence will beassumed. Asymptotic procedures will be used to determine reflectedand transmitted-wave components at the ocean bottom.

Progress:An algorithm was developed which uses phase information from thegeometrical optics limit for acoustic object reconstruction.Essentially, the phase in the back scattered direction determinesthe tangent plane at the unknown specular point. This planedepends on the two spherical angles which describe the incidentwave direction. The observation that the tangent plane envelopesthe object as these angles are varied allows the derivation of anexplicit formula for the surface in terms of the measured scatteredphase.

14

Title: Stability, Bifurcation and Fracture in Ice Mechanics

PI: Reza Malek-MadaniU.S. Naval AcademyDepartment of Mathematics(301) 267-3603



CONTRACT NO: N0001492WR24006


SCIENTIFIC OFFICER: Neil L. Gerr

Technical Objective:Global weak solutions for the nonlinear equations that modelcompressible, visco-plastic deformation of quasi-two-dimensionalsheets of ice will be obtained. Localization of flow near coastalboundaries will be determined for circular regions. For singlethree-dimensional blocks of ice, the formation of wing cracks byprimary-secondary bifurcation will be studied. The stability ofwing cracks as equilibrium solutions of the governing equationswill be proved.

Approach:Recently developed functional-analytic results on minimization ofnonlinear functionals will be used to prove existence of globalweak solutions of the two-dimensional equations. Steady-statesolutions for a circular domain will be used to quantify thelocalization of the flow near coastal boundaries. The ice blockwill be modelled as an anisotropic thermally conducting materialwith a nonlinear constitutive law. Solution of the nonlinearequation will be accomplished by the methods of functionalanalysis.

Progress:Recent work addresses the system of partial differential equationsthat model simple shearing of a slab of thermo-plastic material.The class of constitutive laws that gives rise to a variationalformulation was identified, and a phase-plane argument was used toconstruct time-independent solutions that may be interpreted assteady-state shear bands. Techniques from bifurcation theory wereused to classify stable and unstable solutions in terms of theshape of the solution branch that arises when strain-rate isplotted against shearing force.

15

Title: Phenomenological Behavior of Multipolar ViscousFluids

PI: Hamid BelloutNorthern Illinois UniversityDepartment of Mathematical Sciences(815) 753-6763


R&T PROJECT CODE: 4112699----02




Technical Objective:A new constitutive "multipolar" theory to describe the flow ofviscous fluids will be formulated. This theory assumes nonlinearrelations between the stress tensors and the higher spatialderivatives of the velocity. For various steady flows, thesolutions of the new equations will be compared to the Prandtlboundary-layer solutions of the standard Navier-Stokes equations.Computations will be carried out for an isothermal, incompressible,dipolar fluid. Continuous dependence and existence results will beproved.

Approach:The physics of fluid flow will be used to create the nonlinearconstitutive relations. Existence and continuous dependence of thesolutions of the new equations with the nonlinear constitutiverelations will be established by methods of functional analysis(Sobolev spaces). The comparison of the new equations with thestandard Navier-Stokes equations will be carried out using Prandtlboundary-layer theory and computations.

Progress:Recent work addressed the existence, uniqueness, and stability ofsolutions to the full three-dimensional initial-boundary valueproblem for an isothermal, incompressible, dipolar viscous fluid.The boundary conditions considered were more general than thoseused previously in that they allow for time-dependentnonhomogeneous boundary data. This is equivalent to the case wherefluid is flowing into and/or out of the region of interest, thusallowing more possible flows.

16

Title: Nonlinear Aspects of MultidimensionalInverse Scattering Problems

PI: Adrian I. NachmanUniversity of RochesterDepartment of Mathematics(716) 275-4416






Technical Objective:The d-bar method for multidimensional acoustic inverse scatteringwill be further developed to take into account practicallimitations. The integral equations in several complex variablesthat arise in the d-bar method will be extended to allow for thepresence of singularities. The exceptional points (singularities)that arise in the complex frequency range will be studied. Amethod to calculate the exceptional points from the scattering dataand to use them in the inversion procedure will be developed.

Approach:Functional-analytic methods will be used to extend the integralequations in several complex variables that arise in the d-barmethod to allow for the presence of singularities. Thesingularities will be studied using the spectral theory ofnon-self -adjoint operators, weighted inner product spaces, harmonicanalysis estimates, perturbation methods and numericalcalculations.

Progress:The problem of recovering the quantum mechanical two-bodyHamiltonian -D+V in L2(Rn), n>=3, from knowledge of thecorresponding scattering matrix at one energy was addressed. Itwas shown that if the potential V has compact support and is inLn/2 then it is uniquely determined by the scattering matrix; theproof gives a method to reconstruct the potential from thescattering matrix.

17

Title: Functional Regularity of EntropySolutions to Nonlinear Conservation Laws

PI: Ronald A. DevoreUniversity of South CarolinaDepartment of Mathematics(803) 777-2632


R&T PROJECT CODE:- 4112701---02




Technical Objective:Functional analysis for calculating entropy solutions of nonlinearscalar conservation laws in several space dimensions will bedeveloped. Regularity (that is, smoothness that is inherited ormaintained as time passes) of the solution of a multidimensionalscalar conservation law will be established. Stability andconvergence of leading numerical methods for calculating thesolutions of conservation laws will be determined in a Besov-spacesetting.

Approach:To prove regularity for conservation laws, the theory of Besovspaces will be used. To characterize the solutions of conservationlaws, a model based on phase averaging of the kinetic solutions oftransport equations related to the conservation law will be used.The theory of nonlinear approximation of (discontinuous) functionsin Besov spaces will be used to establish stability and convergenceof numerical methods.

Progress:Questions of the behavior of solutions to the heat equation innonconvex regularity spaces were addressed. In particular, it wasshown that convolution with the heat kernel is bounded on B(n,p),the space of Lp functions that have n derivatives, if 1/p - n < 1.In other work, stability, total variation, and error properties ofspectral viscosity approximations were derived.

18

Title: Reduction of Partial DifferentialEquations to Ordinary DifferentialEquations

PI: Ciprian FoiasIndiana UniversityDepartment of Mathematics(812) 855-1448






Technical Objective:Algebraic procedures for the reduction of partial differentialequations governing fluid flows, in particular, Navier-Stokesequations, to systems of ordinary differential equations will bedeveloped. To this end, a family of analytical sets approximatingthe global attractors of the partial differential equation, afamily that describes the interaction between small and largeeddies, will be studied. The theory of a canonical normal form ofthese equations will be developed. This theory will be used tosolve the Hopf bifurcation equation in turbulence theory.

Approach:The partial differential equations will be treated as vector fieldsin infinite-dimensional phase spaces. The zeros of polynomial mapswill be used to approximate the attractor of the equations. Theflow on the attractor will be studied through the algebraicproperties of the sets of zeros. When the attractor is a singlefixed point, the normal form of the equation will be used tointegrate explicitly the partial differential equation.

Progress:In recent work it was proved that the nonlinear operator associatedwith the normal form of the Navier-Stokes equations with potentialbody forces is unique. A method for computing this operator interms of a sequence of defining polynomials was derived. Thesepolynomials can be used to compute the flow of the Navier-Stokessystem. The approach is also applicable to other nonlinearparabolic equations.

19

Title: Regularity and Approximation for HyperbolicConservation Laws in Several Space Dimensions

PI: Bradley J. LucierPurdue UniversityDepartment of Mathematics(317) 494-1979






Technical Objective:Two types of regularity for conservation laws will be established:1) If the initial condition is in a certain Besov space, then thesolution remains in that space at later times; and 2) If theinitial condition is in a certain space rougher than the Besovspace, the solution is at later times in the Besov space (that is,it gains regularity or smoothness). It will be shown that thesolution of the phase average of the so-called kineticapproximation to the conservation law converges to the solution ofthe conservation law.

Approach:Results from approximation theory and operator theory thatcharacterize the multidimensional functions that can be wellapproximated by wavelet-type decompositions will be used to provethe regularity results. Fourier multiplier theory, waveletdecompositions and comparison theorems of Kuznetsov will be used toestablish regularity and convergence of solutions of the kineticequation.

Progress:A mathematical theory for image compression based on nonlinearapproximation theory was developed which describes of suchoperations as quantization, threshold coding, and clipping. Theresults show how to optimally use the wavelets for compression byminimizing the error, defined in terms of an Lp metric, between theoriginal image and the compressed image. In particular, it wasshown how to quantize in order to achieve optimal compression undera given metric. The issue of the choice of metric was alsoaddressed, and it was demonstrated that the Ll metric, rather thanthe more commonly used L2 metric, leads to quantization whichbetter matches the Threshold Sensitivity Curve used bypsychologists to model the human visual system.

20

Title: Nonlinear Systems of Conservation Laws

PI: Gui-Qiang ChenUniversity of ChicagoDepartment of Mathematics(312) 702-7386






Technical Objective:New, more efficient nonlinear methods of solving certain classes ofmultidimensional nonlinear conservation laws for compressible andincompressible flow will be formulated. A rigorous theoreticalbasis, including stability, convergence and error-estimate results,for these methods will be developed. The methods will be used incomputations for more complicated classes of multidimensionalconservation laws to guide further theoretical development.

Approach:Nonlinear versions of various methods of functional analysis andnumerical analysis will be used. These methods include compactnessmethods, estimate methods, asymptotic expansions, geometric-measuremethods and topological methods. Well-posedness and qualitativebehavior of solutions in measure spaces such as L-infinty and BVwill be used as the basis for analyzing numerical methods andproposing improvements.

Progress:The convergence of a pure second-order shock-capturing scheme forthe system of isentropic gas dynamics with L-infinity initial datawas established. The scheme is a version of the classical MUSCL(monotonic upstream scheme for conservation laws) modified in orderthat points near the vacuum not take negative density and toprevent the local entropy from dramatically increasing near shockwaves. Convergence of the scheme for the piston problem, which iscomplicated by the reflection of shock waves at the rigid wall, wasalso addressed.

21

Title: Discontinuous Solutions for MultidimensionalConservation Laws

PI: Dening LiWest Virginia UniversityDepartment of Mathematics(304) 293-2013






Technical Objective:The stability of a contact discontinuity for compressible flow willbe determined. The Cauchy problem with discontinuous initial datacontaining shocks, rarefaction waves or vortex sheets will beaddressed. A mathematical theory for the existence and stabilityof Mach node formation will be developed. The structure of thesolution of the two-dimensional Euler equations with piecewisesmooth initial data will be determined.

Approach:Existence and stability of compressible-flow solutions will bedetermined using energy estimates and iterative techniques. Thestructure of solutions of the Cauchy problem will be derived usinglinearization and Nash-Moser iteration. Mach node formation willbe examined using asymptotic analysis and techniques forfree-boundary problems. The two-dimensional Euler equations willbe addressed by explicit construction of piecewise smoothsolutions. Computational experiments will be used occasionally toguide the analysis.

Progress:For a one-dimensional two-phase degenerate Stefan problem, it wasproved that the boundary and the solutions up to the boundary areC-infinity smooth. The proof is based on performing thehodographic transformation to fix the free boundary andestablishing a nonlinear a priori estimate for the solution.

22

Title: Analysis, Approximation, and Computationof Flow Control Problems

PI: Max GunzburgerVirginia Polytechnic InstituteDepartment of Mathematics(703) 231-5945






Technical objective:The goal is to solve optimal-control problems for incompressiblefluid flows. Functional-analytical, numerical-analytical andcomputational research will be carried out on drag minimization,avoiding hot spots, maximization of mixing, minimization of mixingand achieving an a priori specified flow field. Control mechanismsto be used in the models will be physically implementable controlssuch as velocity at the boundary (injection, sucking), temperatureat the boundary, body forces (magnetism) and shape of the boundary.

Approach:The mathematical analysis of the models will be carried out usingnonlinear functional analysis for Navier-Stokes equations instandard and novel Sobolev spaces and using Lagrange-multipliertechniques for the constraints. Discretization will be carried outby finite elements. Rigorous numerical analysis of the discreteproblems will be carried out by extending finite-element analysisof discretizations of the uncontrolled Navier-Stokes equations.Computer codes for solving the discrete problems will be written.

Progress:A number of results were extended from steady-state flows totime-dependent flows. In particular, necessary conditions fromwhich optimal states and controls can be determined were given inthe form of systems of partial differential equations. Analysis ofthese optimality systems shows that practical computationalsolutions can be derived.

23

Title: Control Theory of Nonlinear PartialDifferential Equations with Applicationsto Fluid Mechanics

PI: Thomas P. SvobodnyWright State UniversityDepartment of Mathematics and Statistics(513) 873-2379






Technical Objective:The grantee will address optimal-control problems forincompressible fluid flows. Functional-analytic andnumerical-analytic research will be carried out on stabilization,drag reduction, flutter reduction and boundary stress minimizationfor laminar and turbulent flows by boundary shape control, boundaryvelocity control and hybrid shape/velocity control.

Approach:Throughout the research, the techniques of Lagrangian analysis ofoptimization problems will be applied. Generalizations of thenotion of weak solution of inhomogeneous boundary-value problemswill be used. In deriving and manipulating a maximum principle,boundary controls of bounded variation will be used. Estimateswill be derived using the concept of monotonicity of solutions ofelliptic equations. Invariant manifold theory will be used toprove the stabilization results. Discretization will be carriedout by finite elements.

Progress:Necessary conditions were derived for the existence of optimalcontrols for the boundary control problem associated with thestationary Navier-Stokes equations. The control is the velocity onpart or all of the boundary of the flow domain. The use of theLagrange multiplier rule was justified as a means to derive firstorder necessary conditions for optimality, which were expressed asa system of partial differential equations.

24

Title: Inverse Scattering and Nonlinear Waves

PI: Mark J. AblowitzUniversity of ColoradoProgram in Applied Mathematics(303) 492-4668






Technical Objective:1) The existence of optimal controls and the validity of thePontryagin maximum principle will be proved. Suboptimal controlsproduced by discretizations will be derived. 2) The influence ofcurrents, periodic waves and change of wind on Langmuir circulation(upper-ocean mixing) will be determined. 3) Wavelet based methodsfor capturing shocks and inverting the Radon transform will bedeveloped. 4) Multidimensional inverse scattering methods will bedeveloped.

Approach:1) Dynamic programming, the Ekeland variational principle,invariant manifolds and viscosity solutions of Hamilton-Jacobiequations will be used to obtain results on control of fluids. 2)The system of partial differential equations governing Langmuircirculation will be rewritten as a singular evolution equation inan infinite-dimensional space and solved by techniques developedfor singular Navier-Stokes equations. 3) Wavelets will be used totransform smoothing operators into sparse matrices to solvefluid-flow equations and the generalized Radon transform. 4) TheDBAR method, which supplants the Riemann-Hilbert problem ofone-dimensional inversion by a linear equation for the DBARderivative of the eigenfunction whose coefficients are thescattering data, will be extended from continuous to discreteproblems.

Progress:New results were derived for optimal control for viscous fluids.In particular, the relationship between two notions of generalizedsolution for the Hamilton-Jacoby-Bellman equation was determined.A large class of viscous flow problems was then formulated in termsof infinite dimensional nonlinear control, with exteriorhydrodynamic flow an example. An existence theorem for optimalcontrol was produced.

25

Title: Mathematical Algorithms for Multidimensional InverseScattering Problems in Inhomogeneous Media

PI: Michael KlibanovUniversity of North Carolina, CharlotteDepartment of Mathematics(704) 547-4551


R&T PROJECT CODE: 4112714----01




Technical Objective:The technical objective is to develop improved mathematicalalgorithms for 3-D inverse scattering problems. Areas of specificinterest include inversion at low frequencies, time-dependent data,convergence rates and numerical efficiency.

Approach:Significant improvement in medium recovery is anticipated throughthe use of specially designed Newton-type iterative methods. Sincethese require a good initial approximation, alternating directionmethods, which are less dependent on initialization, will also beconsidered. Hybrid methods will be developed, implemented, andevaluated numerically.

26

Title: Transport, Mixing, and Dispersion in GeophysicalFlows: A Dynamical Systems Approach

PI: Stephen R. WigginsCalifornia Institute of TechnologyDepartment of Applied Mechanics(818) 356-4127




CURRENT END DATE: 14 JUN 1992


Technical Objective:The theory of nonlinear dynamics will be developed and applied toproblems of transport, mixing, and dispersion in geophysical flows.The strongly nonlinear deterministic problem of the motion oftracer particles under a specified velocity field is of particularinterest.

Approach:The mechanisms governing transport and mixing will be determined:1) in coherent structures of modon type (i.e., elliptical regionsof localized vorticity), and 2) for passive tracers and potentialvorticity associated with Rossby waves in eastward and westwardjets. A Hamiltonian formulation for the dynamics of an ellipticalvortex in time-varying strain rate and vorticity fields will beused. The size and geometry of the mixing region will becorrelated with those of the vortex distribution via invariantmanifold theory. Melnikov theory will be used to quantify thegeometry and size of the mixing region in terms of parameters suchas circulation and strain rate.

Progress:Recent work addressed the connection between chaos in dynamicalsystems and open shear flow turbulence. A simple dynamical modelbased on physical arguments relative to large and small eddymotions was developed for a broad class of flows. Using theMelnikov technique, it was shown that the system has chaoticsolutions under determinisitic forcing. The model exhibits manyfeatures of shear flow transition, including 1) there is aninherent cascade process, 2) the value of the control parameter(inverse Reynolds number) at onset of chaos is not unique butdepends on the forcing (disturbance) amplitude, and 3) chaospersists as the control parameter tends to zero, the forcingamplitude required to induce chaos simultaneously tending to zero.

27

Title: Problems in Compressible Wave Motion

PI: Jeffrey B. RauchUniversity of MichiganDepartment of Mathematics(313) 763-1184


R&T PROJECT CODE: 4112716--O01




Technical Objective:The objective is to develop new understanding of the role of wavefocusing in 2-D conservation laws, particularly in the context ofradial motions of a 2-D ideal gas and nonlinear geometric optics.In addition, the method of viscous perturbation will be applied tothe problem of shock capturing in mixing fluids. This involves theapplication of small viscous perturbations to nonconservedprimitive variables, making the algorithm conservative to the orderof numerical approximation.

Approach:Small radial and compactly supported perturbations of the constantstates for a 2-D ideal gas in the exterior of the unit disc will bemodeled and analyzed. The radial solution to the Euler equationwill be analyzed in the asymptotic regime of nonlinear geometricoptics. Computations of mixing fluids using a nonconservativescheme on primitive variables will be used to generate shocks withoscillation-free interfaces.

28

Title: Analysis and Computation of ViscousShock Layers and Related Problems

PI: Jonathan B. GoodmanNew York UniversityCourant Institute of Mathematical Sciences(212) 998-3001






Technical Objective:The goal is to establish the multidimensional stability of viscousshock fronts for the equations of gas dynamics and to produce errorestimates for adaptive finite elements when applied to flows withlocalized singularities. The error estimates will be obtainedwithout restriction on the shape of the finite elements.

Approach:The stability of a planar viscous shock profile will be establishedfor the equations of gas dynamics with perturbations. Stabilityestimates will be derived via the Kreiss symmetrizer as applied tothe linearized equations. Adaptive space-time finite elements willbe applied to flows with shock fronts and to the problem ofdetonation ignition by the collision of two shocks in a combustiblegas.

29

Title: Shock-capturing in Conservation Laws andApplications of Dynamical Systems

PI: Chris Jones and Constantin DafermosBrown UniversityDivision of Applied Mathematics(401) 863-1483


R&T PROJECT CODE: 4112718 --- 01




Technical Objective:The stability of viscous diffusion waves for systems ofconservation laws will be established. Then, a systematic analysisof various fan admissibility criteria will be carried out forstrictly and nonstrictly hyperbolic conservation laws. Finally, aclass of Petrov-Galerkin type finite element methods will bedeveloped and applied to multi-dimensional conservation laws.

Approach:An eigenvalue tracking technique, a topological invariant, andsingular perturbation will be used to obtain stable viscousdiffusion waves. Generalized characteristics will be developedwith the goal of understanding the global properties of solutionsdirectly and without appealing to Glimm's random choice method. Aclass of Petrov-Galerkin finite elements will be combined with aclass of total variation stable time discretization and theresulting algorithm will be applied to two-dimensional gasdynamics.

30

Title: Nonlinear Stability Analyses in Fluid Dynamics

PI: Yuriko RenardyVirginia Polytechnic InstituteDepartment of Mathematics(703) 231-8258






Technical Objective:The goal is to characterize the stability properties of all elevensolutions of the system of partial differential equations thatdescribe double-diffusive convection. Among these solutions arestanding rolls, travelling rolls, and travelling patchwork quilts.

Approach:Bifurcation analysis of double-diffusive convection will beperformed by applying the theory of pattern selection introduced byRoberts, Swift, and Wagner. This approach has been used by thePrincipal Investigators in their previous analysis of the two-layerBenard problem. The Center Manifold Theorem plays a crucial rolein establishing connections between the linear stability andspectral properties of such viscoelastic flows.

31

Title: Conference on Dynamical Systems and Applications

PI: I. E. BlockSociety for Industrial and Applied Mathematics(215) 382-9800






Technical Objective:The conference will be held in Salt Lake City on October 15-19,1992. The conference will bring mathematicians together withengineers, physicists and biologists. The conference program willinclude twelve invited speakers and thirty minisymposia. Thetopics of minisymposia range from computational complexity andchaos to low dimensional dynamics of partial differentialequations.

Approach:In each subtopic of nonlinear dynamical systems, speakers that areworld leaders in their area are invited to deliver a lecture. Theminisymposia will complement the concepts and ideas that areintroduced in the invited lectures.

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Discrete Mathematics

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33

TITLE: Studies in Traversability in Graphs

PI: Herbert S. WilfUniversity of PennsylvaniaDepartment of Mathematics(215) 898-8474





SCIENTIFIC OFFICER: Marc J. Lipman

Technical Objective:A deeper understanding of the relationship between combinatorialidentities and the rational functions which "prove" them will besought. New lower bounds for maximum sizes of codes will bederived.

Approach:The combinatorial identity associated with a particular rationalfunction will be determined. This will yield new relationshipsbetween families of combinatorial identities. Computer-usefulrepresentations of hypergeometric series will be built. This willpermit construction of a database for combinatorial identities.The spectrum of the distance graph of a code will be used to derivelower bounds on the independence number of that graph, and hence alower bound on the maximum size of the code.

Progress:The grantee has significantly extended his method whereby computerscan quickly find proofs of hypergeometric summation identities ina single variable. This extension allows for automation of proofsof summation-integration identites that involve several summationvariables or q-sums or multivariate q-sums. In particular, suchsums always satisfy recurrences with polynomial coefficients,computers can find these recurrences, and that bounds for theorders of the recurrences can be given in advance. Using theseresults, the grantee produced computer-generated proofs of a numberof important and famous identities.

35

TITLE: On-Line Algorithms and Polynomial Time Approximation

PI: Henry A. KiersteadArizona State UniversityDepartment of Mathematics(602) 965-6485




CURRENT END DATE: 30 NOV 1992


Technical Objective:Three general areas will be investigated. The first involveson-line algorithms and dynamic storage allocation. The secondconcerns the problem of computing the chromatic index of amultigraph. In particular, it is unknown if this can be done inpolynomial time. The third includes several classic problems inthe dimension theory of partially ordered sets.

Approach:Algorithms on partially ordered sets will be developed for findingimportant parameters and structures. The complexity of on-linealgorithms will be studied. The chromatic index of multigraphs andvarious aspects of the dimension theory of partially ordered setswill provide important examples to test the approach.

Progress:The grantee characterized those trees for which the first-fiton-line graph coloring algorithm has worst-case behavior that isbounded when restricted to acting on graphs which do not containthat tree as an induced subgraph. This means that first-fit yieldsa coloring requiring no more colors than a known function of theminimum number of colors necessary to color the graph. Some treeswere identified for which first-fit fails to have this property.

36

TITLE-- Matching and Covering in Graphs and Networks

PI: Michael D. PlummerVanderbilt UniversityDepartment of Mathematics(615) 322-6668



CONTRACT NO: NO0014w91J1142



Technical Objective:Improved lower and upper bounds for the number of perfect matchingsin a graph will be sought and a decomposition theory for bricksdeveloped. Well-covered graphs having girth less than five willalso be studied in the hopes of obtaining a completecharacterization of all well-covered graphs.

Approach:The approach to the structure of bricks and bicritical graphsinvolves the n-extendable graphs, i.e., graphs in which everymatching of size n lies in a perfect matching. Previous methodswill be generalized to characterize four and five regularwell-covered graphs.

Progress:Grantee produced a new degree sum condition and a new neighborhoodunion condition each of which guarantees that a graph isn-extendable for a suitable choice of n. In the particular case ofn equal to 1, the graph is guaranteed to be bicritical as well as1-extendable.

37

TITLE: Structure of Ordered Sets and Graphs

PI: Dwight DuffusEmory UniversityDepartment of Mathematics(404) 727-7580






Technical Objective:The goal is to develop structural properties of graphs and orderedsets. Random orders will be studied as will a variety of othercombinatorial problems involving familiar, highly structuredobjects such as Boolean lattices and subspace lattices of finitevector spaces.

Approach:Fibres and cutsets in both determined and random ordered sets willbe analyzed. Order preserving maps will be counted which will helpdetermine which ordered sets can be sorted when comparisons areunreliable. A search will be made for a polynomial algorithm forthe m-machine unit time preemptive scheduling problem. The topicof random maximal graphs for various properties will be explored.

Progress:The grantees produced an example of a finite poset with no one-element maximal antichain and containing no fibre of size at mosthalf the size of the poset. But they also showed that there is anabsolute constant, at most two-thirds, so that every finite posetcontains a fibre of size at most that constant times the size ofthe poset. In addition, they proved that the elements of anarbitrary poset can be partitioned into two sets so that everytwo-element maximal antichain spans both sets.

38

TITLE: Intersection Graphs and Geometric/CombinatorialOptimization

PI: Edward R. ScheinermanThe Johns Hopkins UniversityDepartment of Mathematical Sciences(301) 338-7210






Technical Objective:Three areas will be investigated: (i) random representions, leadingto a general theory of random intersection graphs, (ii) containmentrepresentions of partially ordered sets, particularly circle andsphere containment orders, and (iii) generalized chromatic numberson random graphs, leading to generalized perfect graphs.

Approach:The theory of random interval graphs, invented by the contractor,will continue to be developed. Results will be used to investigatethe dimension of random interval orders. Classes of partiallyordered sets which arise as the containment orders of variousgeometric sets such as circles and angles will be studied. Randommethods will be applied to fault tolerant analysis of non-arbitraryinterconnections between processors, such as the hypercubearchitecture.

Progress:Grantee proved that the vertex containment (edge containment,bounded-face containment) order of a 3-connected planar graph is acircle order. This has an important corollary about plane drawingsof planar graphs. Grantee also proved that the vertex containment(edge containment) order of any graph is a sphere order. Granteeestablished global growth rates for the numbers of graphs withhereditary properties. Hereditary properties can have constant,polynomial, exponential, factorial, and superfactorial rates ofgrowth as a function of the number of vertices in the graph.

39

TITLE: Tolerance Intersection Models

PI: Michael S. JacobsonUniversity of LouisvilleDepartment of Mathematics(502) 588-6826


R&T PROJECT CODE! 4111021---07



SCIENTIFIC OFFICER, Marc J. Lipman

Technical Objective:Tolerance intersection graphs and posets will be constructed for awide range of types of tolerance including- minimum, maximum, andsum tolerance. Comparisons with classic intersection models willdemonstrate the increased utility of these models.

Approach:For a given tolerance condition, such as minimum tolerance, thetolerance interval graphs will be characterized. Then the questionof unit intersection versus proper intersection will be extended totolerance interval structures. Properties of special classes oftolerance structures will be found, such as tolerance competitiongraphs. Finally, the concept of asymmetry in tolerance will bedeveloped, specifically tolerance intersection digraphs.

Progress:The grantee made fundamental discoveries about sphere-of-influencegraphs. Specifically, sphere-of-influence trees werecharacterized, and sufficient conditions determined for atriangle-free graph to be the sphere-of-influence graph of some setof points in the plane.

40

TITLE: Ordinal and Percentile Clustering

PI: Melvin JanowitzUniversity of MassachusettsDepartment of Mathematics and Statistics(413) 545-2871






Technical Objective:A new technique of cluster analysis will be developed that combinesthe ordinal theory of clustering with the theory of probabilisticmetric spaces. The goal is to improve classification in large datasets where there is variation within the objects to be clustered.

Approach:Various aspects of ordinal clustering and the related latticetheory will be developed. Some of the machinery from the theory ofprobabilistic metric spaces, notably properties of t-norms andtriangle functions, will be modified and extended to fit theapplications at hand. Algorithms necessary to implement the newtechniques will be developed and tested on the computer.

Progress:Software PCluster was created and is available on disk. This setof programs performs standard clustering as well as grantee's ownhierarchical method, percentile clustering, on PCs. Data can be ineither attribute or dissimilarity form. PCluster computesdissimilarities, clusters, and displays dendrograms in either textor graphic mode. More than a dozen standard clustering algorithmsare implemented. Grantee also characterized arbitrary quota ruleconsensus methods in a median semilattice. These methods allow forany fixed proportion of voters to control decisions.

41

TITLE: Distance and Extremal Subgraphs in Graphs and Digraphs

PI: Gary ChartrandWestern Michigan UniversityDepartment of Mathematics and Statistics(616) 387-4533

FUNDING AGENCY. Office of Naval Research





Technical Objective:The primary objectives are to fully develop, in the context ofgraph theory, the concept of common substructures of a given familyof structures, and construct families of metric structures inspaces of graphs.

Approach:Several definitions of the greatest common subgraph and smallestcommon supergraph of a collection of graphs will be explored. Newparameters involving generalized distance (e.g., n-eccentricity,n-radius, n-center) will be compared. Metrics based on edge-slidedistance, edge-rotation distance, and Steiner distances will bedeveloped and investigated.

Progress:Introduced two metrics defined on the subgraphs of a graph: Onefor induced subgraphs with the same number of vertices, one foredge-induced subgraphs with the same number of edges. Each is ageneralization of the classic graph distance. Produced anefficient algorithm for computing the vertex subgraph metric. Forany graph and specified order of subset, constructed a graph whichmodels this distance, so that the subgraph distance in the originalgraph is the ordinary graph distance in the construct. Constructedtwo metrics in digraphs based on the usual digraph "distance,"which is itself not a metric.

42

TITLE: Ordered Sets and Intervals

PI: Kenneth P. BogartDartmouth CollegeMathematics and Computer Science Department(603) 646-3178






Technical Objective:New types of ordered sets and graphs, and algorithms defined onthem, will be developed and applied in the context of schedulingproblems. The notion of tolerance will be introduced to the studyof interval orders and interval graphs.

Approach:New representations of interval orders and interval graphs thatemploy tolerance will be used to model scheduling situations inwhich there is uncertainly concerning the actual timing of theevents modeled. These representations give rise to "extendedinterval orders." The properties of these orders will bedeveloped.

Progress:Dowling lattices were characterized in terms of the existence ofspecial bases and their weight functions. Dowling lattices giverise to a large class of non-representable matroids, and they aremodularly complemented. These lattices are generalizations of thelattice of partitions of a set, and this characterization yields anew characterization of this lattice as an important corollary.

43

TITLE: Intersection Multigraphs and Related Constructs

PI: Terry A. McKeeWright State UniversityDepartment of Mathematics and Statistics(513) 873-2702






Technical Objective:A new theory of intersection multigraphs will be developed as ageneralization of the theory of intersection graphs.

Approach:Characterization theorems for interval, intersection, andbi-intersection multigraphs will be proved and exploited. Theextensive use of neighborhood union conditions will be integratedinto the grantee's intersection multigraph theory. The propertiesof generalized complete bipartite graphs will be developed as theanalogue of cliques in bi-intersection multigraphs.

Progress:The grantee generalized the concept of boxicity to chordality ingraphs. He showed that series-parallel graphs have chordality 2,planar graphs have chordality 3, and that cordality is boundedabove by the tree width. He further showed how chordal graphsmodel decomposable loglinear model statistical contingency tables.

44

TITLE: Seventh International Research Conference on GraphTheory, Combinatorics, Algorithms and Applications

PI: Yousef AlaviWestern Michigan UniversityMathematics and Statistics(616) 387-4510


R&T PROJECT CODE: 4111037---Q3


CURRENT END DATE: 31 MAR 1993


Technical Objective:The technical objective is to advance the state-of-the-art in themany areas of graph theory, combinatorics, algorithms, andapplications of these fields.

Approach:The conference will be held at Western Michigan University 1-5 June1992. All talks will be invited and selected for importance andrelevance to the conference's theme of applicability of themathematics presented. A refereed proceedings of the conferencewill be published in hardbound volumes by John Wiley and Sons.

45

TITLE: Discrete Structures in Classification and Other Topics inDiscrete Applied Mathematics

PI: F. R. McMorrisUniversity of LouisvilleMathematics Department(502) 588-5447

FUNDING AGENCY: Jffice of Naval Research





Technical Objective:The objective is to develop and extend the theory of aggregationfor discrete structured data. Since this theory encompasses thenotion of "common agreement," the results have application topattern recognition and the reconciliation of inconsistent data.Special attention will be given to waveform tree-coded data such asradar and sonar returns.

Approach:The theories of social choice and multicriteria decision makingwill be used as a general axiomatic model. Existing results innumerical taxonomy will be extended to more general cases. Newconsensus methods will be developed based on numerical taxonomy andpattern recognition.

Progress:The grantee gave several characterizations of the plurality rulefor constructing consensus sequences. Using the semilattice ofsubtrees of a rooted tree, he also characterized those digraphsthat are the acyclic intersection digraphs of subtrees of adirected tree. In addition, it was determined which graphs havep-vertex and p-edge clique cover numbers the same for all values ofp, and it was shown that trees are closed sphere-of-influencegraphs precisely when they have a perfect matching.

46

TITLE: Some Graph Theoretic Parameters Related to NetworkProblems

PI: Richard D. RingeisenClemson UniversityMathematical Sciences Department(803) 656-3436






Technical Objective:The objective of this research is the development of techniques fordesigning communication networks with enhanced reliability. Thesewill have applications in mobile packet radio and cellulartelephone systems and the design of computer networks.

Approach:Studies of cohesion and related graph theoretic parameters willcharacterize certain kinds of communication networks which are lessvulnerable to loss of links and nodes. Studies in graph drawingsshould lead to improvements in computer representations ofgraphical structures, and improved ability to model complexnetworks in the plane.

Progress:The grantee produced a new upper bound for the maximum number ofcrossings in a good drawing of the cube, 34, and a drawingachieving the bound. The grantee also determined exactly thebinding number of all cycle permutation graphs and bounds for thetoughness of such a graph. The crossing range of a graph yieldedinformation concerning whether or not a good drawing of a subgraphcan have more crossings than any such drawing for the originalgraph. Finally, the grantee produced many graphs which havedrawings that meet the theoretical upper bound for the number ofcrossings as well as classes of graphs which have no such drawing.

47

TITLE: Some Problems of Golomb Concerning Costas Arrays

PI: Oscar MorenoUniversity of Puerto RicoDepartment of Mathematics(809) 764-0000






Technical Objective:The technical ojective is to develop the theory of Costas arrays soas to show the existence of arrays with new orders and to createmethods for constructing families of these arrays.

Approach:Results concerning the primitive roots of finite fields will beused to limit the search for Costas arrays of certain orders.Necessary computer searching will be performed on a parallel systemconstructed by the research team.

Progress:Grantee constructed a family of p-phase sequences of complex pthroots of unity of length (p exp(n)) - 1 with excellent periodiccorrelation properties. In particular, the family isasymptotically optimal with respect to the correlation propertiesand requires only an alphabet of size three. The new sequences aresuitable for achieving code-division multiple-access and are easilyimplemented using shift registers.

48

TITLE: Mathematical Methods of Communication Signal Design

PI: Solomon W. GolombUniversity of Southern CaliforniaElectrical Engineering Department(213) 743-2990

FUNDING-AGENCY: Office of Naval Research




SCIENTTIC OFFICER: Marc J. Lipman.

Technical Objective:The objective is to develop the theory of Costas arrays, Florentinearrays, and Tuscan, Roman, and Vatican squares. Particularquestions of interest include the existence of classes of thesestructures, the development of general techniques for theirconstruction, and their properties. Also of interest is therelationship between these different types of combinatorialobjects. For example: Do there exist square Florentine arrayswhich are not Vatican squares?

Approach:Primitive roots in finite fields which sum to one guarantee theexistence of Costas arrays of specific orders. Permutationmatrices in which all vectors connecting pairs of ones havedifferent slopes are also Costas arrays. The grantee will usethese and similar results to construct Costas arrays of new orders,upper bound the number of such arrays, and construct the othercombinatorial objects of interest.

Progress:An infinite class of radar waveforms whose autocorrelation functionis unity in-phase and close to zero out-of-phase was produced.These include arbitrarily long binary sequences as the modulatingsignals in which the out-of-phase correlation is identically zero.The signals are derived from cyclic Hadamard difference sets andare also useful for range-doppler radar, since they have nearlyidentical two-dimensional autocorrelation functions in both timeand frequency. The grantee also showed that Legendre sequences ofperiod a prime congruent to 1 modulo 4 can be used to obtainsignals whose correlation is similar to those of period a prime twoless. The grantee generalized these results from signals based onthe Legendre symbol to some based on the Jacobi symbol modulo anysquarefree positive integer.

49

TITLE: Fault Tolerance in Graphs and Discrete Systems

PI: Frank HararyNew Mexico State UniversityDept. of Computer Sci. and Computer Res. Lab(505) 646-6238






Technical Objective:This project will develop new parameters to measure fault tolerancein families of graphs as well as develop the concept of faulttolerance in groups. The intention is to investigate the relationof fault tolerance to extremal graph theory.

Approach:The fault tolerance of a graph relative to a second, smaller graphwill be investigated by examining conditions on the complements ofthe two given graphs. Fault tolerance of a graph will be examinedrelative to a family of graphs. Graceful degradation will bestudied for fault tolerance in hypercubes. Fault tolerance ingroups with specified subsets will be investigated.

Progress:The grantees proved that any graphical structure guaranteed tocontain a specified graph G after the removal of a fixed number kof edges must in fact contain multiple edges whenever k is largeenough. They produced such multigraphs whenever the specified G isa cycle. They also produced optimal multigraphs for k = 1 in thecases of multi-dimensional meshes and hypercubes, two popularcomputer architectures. They developed a new technique forincremental fault-tolerant design of networks that reduces asystem's interconnection costs, and constructed a fast distributedalgorithm for reconfiguration around faults.

50

TITLE: Combinatorial Matrix Theory

PI: Charles R. JohnsonCollege of William and MaryDepartment of Mathematics(804) 221-2014


R&T PROJECT CODE: 4111050--03


CURRENT END DATE: 15 MAY 1993


Technical Objective:Combinatorial techniques will be applied in several areas of matrixtheory. One goal is to determine from the signs of the entrieswhen the eigenvalues of the matrix must be real. Another goal isto understand the properties of sparse matrices when information isgiven concerning the location of some of the zeros in the matrix.

Approach:Two types of conditions on eigenvalues of real matrices will beexamined with respect to properties of the entries. Matrixcompletion problems and determinantal inequalities will beaddressed, especially the minimum rank completion problem.Analysis of sparse matrices will follow from analysis of thelongest cycle in the associated non-zero digraph.

Progress:The grantee characterized the sign patterns of matrices that forcea matrix having that pattern to have its spectral radius among itseigenvalues, and identified a large number of patterns for whichsome matrices with that pattern have the same property. Thegrantee also gave simple bounds for the spectral radius of aproduct of matrices each of which has positive spectral radiuswhich is an eigenvalue. He constructed expected-value generalizedmatrix norms and showed their applicability to error matrices ininput-outpur modeling. He produced a complete classification forall square matrices for which zero is not in the field of valuesfor any power of the matrix.

51

TITLE: Twenty-Third Southeastern International Conference onCombinatorics, Graph Theory and Computing

PI: Frederick HoffmanFlorida Atlantic UniversityDepartment of Mathematics(407) 391-7745


R&T PROJICT CODE: 41110r5Tr---02




Technical Objective:The objective is to advance the state of knowledge and disseminatethe latest results in combinatorics, graph theory, and computing.

Approach:The conference will be in Boca Raton, Florida on 3-7 February 1992.It will bring together researchers in discrete mathematics todiscuss recent results in graph theory, combinatorics, computingand applications. The conference features a number ofinstructional lectures by leaders in these fields. This is now amajor annual international conference in its disciplines, and playsa significant role in disseminating new results and in fosteringcollaborative work.

52

TITLE: Towards a Theory of Generalized Degrees withApplications to Graphs and Network Problems

PI: Ralph J. FaudreeMemphis State UniversityDepartment of Mathematical Sciences(901) 678-2-482


R&T PROJECT CODE: 4111052- 02

CONTRACT NO: . N0001491JI085



Technical Objective:The primary objective is to determine the relationship between newneighborhood union conditions and various graph properties andparameters. Of particular interest is deriving routing andscheduling information from these parameters.

Approach:Neighborhood union properties yield bounds on the cardinalities ofindependent sets, arbitrary sets of determined cardinality, andcliques. These neighborhood union conditions will be used toobtain sufficient conditions for the existence of edge disjointhamilitonian paths or cycles, s-matchings, and multiple edgedisjoint matchings. The density of a graph is also bounded bydegree and neighborhood conditions. These bounds will bedetermined.

Progress:A generalization of the Ore condition for hamilton-connectedness ofgraphs was developed by utilizing both neighborhood union andintersection properties. A product lower bound was established forthe domination number of the cartesian product of graphs when atleast one graph has a special type of partition. Grantee provedthat any nonbipartite two-connected graph on n vertices, with nsufficiently large, and minimum degree at least 2n/(k+2), k odd,contains a k cycle. Neighborhood union conditions in claw-freegraphs that are sufficient to guarantee that the graph be traceableor pancyclic were derived.

53

TITLE: A Graph Theoretic Approach to Stability

PI: Margaret B. CozzensNortheastern UniversityDepartment of Mathematics(617) 437-2450






Technical Objective:The goal is to develop a theory of local node and arc stability,and in particular to determine optimal subgraphs relative to avariety of graphical stability parameters. Algorithms for findingoptimal subgraphs for the most important of these parameters willbe developed.

Approach:Cases of node failure, arc failure, and combination node-arcfailure will all be addressed. The first task is to determinewhich graphical parameters best preserve properties of interest.For those parameters, good algorithms will be written forconstructing parameter-preserving subgraphs. These algorithms willbe used to solve the problem of reconfiguring the network subjectto the type of failure being considered.

Progress:The grantees produced new bounds on the maximum number of crossingsin a drawing of a graph. They also constructed some graphs thatmeet the upper bound even though they contain many 4-cycles. Thegraph parameter "tenacity" was introduced by combining the notionsof graph integrity and graph toughness. They computed the tenacityof a large number of classes of graphs. Based on thosecomputations they have conjectured that the tenacity is alwaysdetermined by a maximum set of disjoint components in an inducedsubgraph of the graph. Preliminary work is complete on aMacintosh-based system to compute integrity and other vulnerabilityparameters of graphs.

54

TITLE: The Clemson Mini-Conference on Discrete Mathematics

PI: Renu LaskarClemson UniversityMathematical Sciences(803) 656-5218-


R&T PROJECT CODE: 4111054-02




Technical Objective:The objective is to advance the state of knowledge and disseminatethe latest results in discrete mathematics.

Approach:The conference will be held at Clemson University, Clemson SC, on3-4 OCT 1991. It will bring together researchers from all areas ofdiscrete mathematics to discuss recent results and attempt toextend them through sessions devoted to problem specific workinggroups.

Progress:Last year's Clemson Mini-Conference on Discrete Mathematics was thefifth in the series, and was held during October 11 and 12, 1990.The conference featured twelve major addresses as well as amplediscussion time for participants, and succeeded in its statedobjective. The proceedings are available.

55

TITLE: Graphs and Matrices: Combinatorial Analysis,Competitions, Covers, and Ranks

PI: Richard L. LundgrenUniversity of Colorado at DenverDepartment of Mathematics(303) 556-8442






Technical Objective:A mix of graph theoretic and matrix theoretic techniques will bedeveloped with an eye on applications to communication networks.Various clique covering methods will be extended.

Approach:The pattern of zeros in a matrix will be used to determine thecombinatorial properties of the matrix. Conditions will be foundunder which generalized competition graphs have special properties,such as being chordal graphs, or interval graphs. Furtherrelationships between biclique covers of digraphs and matrix rankswill be produced.

Progress:Grantees produced new results on minimum boolean and nonnegativeinteger rank of regular (0,1)-matrices giving equivalent results onminimum biclique covering numbers and partition numbers for regularbipartite graphs and digraphs. They produced bounds on the minimumnumber of complete bipartite subgraphs needed to cover andpartition the edges of a regular bigraph. The competition graph ofa digraph was generalized to the competition multigraph of adigraph and upper and lower bounds for the multicompetition numberfor several classes of multigraphs were computed. Grantees alsocharacterized those graphs that are the conflict graphs of stronglyconnected or hamiltonian digraphs.

56

TITLE: The Fifth Annual Cumberland Conference on Graph Theory andComputing

PI: Linda LawsonEast Tennesee State UniversityDepartment of Mathematics(615) 929-4349






Technical Objective:The technical objective is to advance the state of knowledge anddisseminate the latest results in discrete mathematics and computerscience.

Approach:The conference will be in Johnson City, Tennessee during 14-16 MAY1992. It will bring together researchers in discrete mathematicsand computer science. A specific focus is on the applications ofeach discipline to the problems of the other. Discrete topicsinclude graph models of computers, digraph scheduling models,graphical networks, and partially ordered data structures. Thecomputer science focus is primarily on algorithms for discretestructures. This is a growing conference with increasing influenceon reserach directions throughout the mid-south.

57

TITLE* Efficient Communication in Networks

PI: Roger C. Entringer-University of New MexicoDepartment of Mathematics and Statistics(505) 277-4613





SCIENTIFIC OFFICER: Marc•J. Lipman

Technical Objective:The objective of this research is to develop methods for measuringthe efficiency of communication in both static and dynamicnetworks. The further objective is to integrate aspects of bothvulnerability and efficiency into network design.

Approach:Measures of efficiency in networks will be developed and analyzed.These include delay in trees, optimal trees in networks, andbisection properties of subgraphs. Enumeration of trees underoptimality conditions will be used to extend results on trees togeneral networks.

Progress:The subdivision threshold for a graph G is the maximum number ofedges in a graph with specificed number of vertices so that nopiece of G is a proper subgraph of that graph. The granteesconstructed the subdivision threshold for a class of graphs andcharacterized the extremal graphs belonging to the class. Theyalso produced a list of all trees with eleven vertices and countedthe internal paths of each tree. This significant computationalresult extends the results known for smaller trees and validatestheir tree-counting methods. They also determined which trees havemaximum and minimum transmission, a quantity related to the numberand distribution of internal paths of the tree.

58

TITLE: Efficient Algorithms for Searching a Family of Trees

PI: Allen J. SchwenkWestern Michigan UniversityDepartment of Mathematics & Statistics(616) 387-4532






Technical Objective:The objective is to find optimal algorithms for searching a familyof trees or computing a parameter for all members of a family oftrees. Specific approaches will be sought for specialized familiesof trees, such as binary trees and trees with bounded degree.

Approach:Attention will focus on alternative orderings for families oftrees. Parameter specific orderings will be derived and comparedwith orderings known to be good for simple searching. Crudeasymptotic bounds on the computational cost will be replaced withprecise formal analysis. Partitionings for the family of treeswill be considered so as to determine the value of parallelizingthe search.

Progress:The grantee found bounds for the number of multicolored spanningtrees in the acyclic decompositions of a complete bipartite graph.It was also proved that the decompositions into n color classes,every one a star, are naturally in one-to-one correspondence withthe tournaments of order n - 1. Bounds for their number were alsofound. The grantee also produced a matrix whose determinant yieldsa lower bound on the full number of multicolored spanning trees.

59

TITLE: Diametral Path Graphs and Incremental Distance Sequences

PI: Jitender S. DeogunUniversity of Nebraska LincolnComputer Science and Engineering Department(402) 472-5033






Technical Objective:This research will produce structural information about the classof diametral path graphs similar to the type of information knownabout distances in interval graphs. Other classes of graphs withwell-behaved distance properties will be compared with this class.Efficient algorithms for computing distance properties for thisclass will be constructed.

Approach:Incremental distance sequences for diametral path graphs will beconstructed. Important parameters that are NP-hard to compute forgraphs in general will be examined in diametral path graphs. Theseinclude the size of minimal hamiltonian completions, and the orderof maximal cliques. It will be determined if everyco-comparability graph is a diametral path graph, and if the strong -

perfect graph conjecture holds for this class.

Progress:The problem of completing a hamiltonian path or cycle in acocomparability graph was proved to be of polynomial timecomplexity. Grantee used this theorem to construct a polynomialtime algorithm for determining when a cocomparability graph ishamiltonian. A jump number problem has been partially solved forthe interval order case by creating a polynomial time algorithm forconstructing a linear extension of an interval order that minimizesthe number of jumps from the original partial order.

60

TITLE: Flows and Cycle Covers - Research in Graph Theory

PI: Hong-Jian LaiWest Virginia UniversityDepartment of Mathematics(304) 293-2014






Technical Objective:The goal of this research is to prove several outstandingconjectures about the existence of specific-valued integer flows ongraphs and about the existence of cycle double covers in graphs.

Approach:The kernel-reduction technique will be applied to the cycle doublecover problem. Grantee's 4-cycle-reduction technique will beapplied to the k-flow problem. Grantee will develop and applyreduction techniques based on the set of graphs with a k-flow fora specific value of k.

Progress:Grantee proved Bondy's small cycle cover conjecture for the case ofplanar graphs, that is, that every planar graph with n vertices, nat least six, has a cycle cover with at most (2n - 1)/3 cycles.Those non-supereulerian graphs in which every two-edge-connectedinduced subgraph is supereulerian were characterized. The granteeproved that in any two-edge-connected graph not having a minor ofthe complete graph on five vertices every partial cycle coverextends to a full cycle cover.

61

TITLE: On the Optimality of Clustering

PI: Uriel G. RothblumRutgers UniversityRutgers Center for Operations Research(908) 932-5633






Technical Objective:The goal is to develop new methods for optimizing clusteringtechniques- and applying these methods to problems such aspartitioning and classification.

Approach:Both conic hulls and convex hulls of partitions of point sets willbe used to determine if a clustering method is optimal with respectto certain natural conditions. Methods used in operations researchand mathematical programming will be applied to find extreme pointsof partition polytopes.

62

TITLE: Cluster Analysis Algorithms

PI: Pierre HansenEcole des Hautes Commerciales, MontrealOperations Research(514) 340-6053






Technical Objective:This research will develop algorithms for a wide variety ofclustering problems. This includes methods for constructingdissimilarity indices, automated methods to select the bestpartitions among efficient ones, and new and informative ways torepresent partitions and hierarchies.

Approach:A wide variety of different families of clustering criteria will beexamined, with a view to finding a uniform methodology for devisingclustering algorithms. To test the methodology, algorithms will bewritten for many members of these families.

63

TITLE: International Conference on Random Mappings, Partitions,and Permutations

PI: Simon Tavare'University of Southern CaliforniaDepartment of Mathematics(213) 740-8766






Technical Objective:The objective is to advance the state of knowledge and disseminatethe latest results in random mappings, partitions and permutation.

Approach:A conference will be held including contributed papers as well asinvited addresses. The conference will be held at Los Angeles,California on 3-6 JAN 1992.

64

TITLE: Consensus by Voting; Tournaments; Centrality in Graphs

PI: Kenneth B. ReidCalifornia State University, San MarcosDepartment of Mathematics(619) 471-4164


R&T- PROJECT CODE: 4111Q64---01




Technical Objective:The objective of this research is to understand several methods ofconsensus building and clustering by preference, specifically theimpact of voting by agenda. Results will be applied to bothdesirable and non-desirable facility location problems.

Approach:Tournaments will be used to model a wide variety of restrictedpreference schemes. Different rules for selecting aggregatetournaments will be competitively analyzed with respect todesirable qualities. To apply these results to facility locationtheory, centrality measures in trees will be extended to moregeneral graphs. The notion of "centrality in a family" ofstructures will be analyzed as the basis for preference schemes.

65

TITLE: Image Representation, Clustering, and Search in ProximityGraphs and Pathfinder Networks

PI: Don DearholtMississippi State UniversityDepartment of Computer Science(601) 325-2756






Technical Objective:The objective of this research is to demonstrate the viability ofthe cluster-learning paradigm in pathfinder networks as a methodfor generating databases suitable for robotic vision systems.Pathfinder networks will be used to develop methods of representingimages as networks and the method will be tested on a suitable setof objects.

Approach:Research will be performed to improve the generation,representation, and presentation of pathfinder networks. They willbe applied to MSNET, a vision database for robotics applicationswhich supports sublinear search. Clustering in this databaseamounts to gathering similar images, and the representation ofimages as proximity graphs yields a new clustering methodology thatwill be investigated. Pathfinder networks will be built thatreflect the knowledge inherent in the new clusters.

66

TITLE: ONR Workshop on Discrete Structures in Classification

PI: Fred R. McMorrisUniversity of LouisvilleDepartment of Mathematics(502) 588-6826


R&T PROJECT CODE: 4111066 --- 01




Technical Objective:The objective of the workshop is to determine the state-of-the-artin Discrete Structured Classification, and chart new directions ofresearch relevant to Navy needs.

Approach:The workshop will be held in Chantilly, VA during 5-6 MAY 1992.There will be presentations by both Navy laboratory researchers andONR grantees in the area of Discrete Structured Classification(DSC). The laboratory presentations will focus on problem areas ofimportance to the Navy that may be solvable by the techniques ofDSC, so the grantees will know where the applications are.Extensive one-on-one interactions between grantees and Navypersonnel will be encouraged.

67

TITLE: Development of a Vector Extremal Model for the Navy'sAwaiting Instruction Problem

PI: A. CharnesThe University of Texas at AustinCenter for Cybernetic Studies(5121 471-1821

FUNDING AGENCY: Office of Naval Technology





Technical Objective:The grantee will develop a vector extremal model for the Navy'sAwaiting Instruction Problem that links all parts of the system andtakes into account the diverse goals and constraints of thedifferent schools. This model will permit computation of a dollarvalue for the delayed entry program based on instructor costs andawaiting instruction time.

Approach:Vector extremal problem formulations of new multipage constraineddirected networks will be essayed and computational methodsdeveloped to compute flows on these networks. Constraints on theflows will be used to model the effects of awaiting instructiontime and delayed entry. Operations research techniques will thenbe applied to determine minimum costs for the system.

68

Numerical Analysis

69

TITLE: Computational Studies of Stratocumulus Layers

PI: Joseph OligerLeland Stanford Junior UniversityDepartment of Computer Science(415) 723-0571


R&T PROJECT CODE: 411q001---04



SCIENTIFIC OFFICER: Richard L. Lau

Technical Objective:The aim is to use computation to explore the structure andmechanisms of the marine boundary layer and in the process todevelop new computational techniques which will have generalapplicability. A further objective is to provide predictivemodeling capability.

Approach:Initial research, aimed at basic understanding, will be carried outwith a simple Boussinesq model of the sort that has been used intheoretical studies of shear flows. When the basic physics,parameterization, and turbulence modeling considerations have beensorted out, a model using the Browning/Kreiss equations withsuitable adaptivity will be constructed for predictive use.

Progress:A stability analysis of the equations governing stratus layers islargely complete and scaling analysis is underway using datasupplied by meteorologists at Colorado State University. Prototypenumerical models based on both the Boussinesq and compressibleNavier-Stokes equations have been completed. All three of thesetools will be used in a mutually supprortive way to explore thestability behavior of the marine boundary layer and the appropriatemathematical model(s) for its description and prediction.

71

TITLE: Dynamics of and Characteristics ofNumerical Models of WeaklyNon-Linear Ocean Flows

PI: Leonard J. WalstadOregon State UniversityCollege of Oceanography(503) 737-2070


R&T PROJECT CODE: 411u001---01




Technical Objective:The aim is to characterize the performance of the reduced model anda primitive equations-based model (a surrogate for nearly allexisting ocean models) for solving boundary value problems inoceanography and for forecasting using data assimilation.

Approach:Six to ten different physical problems in the ocean will beconsidered. All are interconnected in ways which will shed lighton the performance of the two classes of models. Most problemswill be formulated in the context of the California Current so thatthe large scale ONR experimental efforts in the Current can be usedboth to supply data to the models and to check the accuracy/adequacy of the numerical results.

72

TITLE: Three Decades of Numerical Linear Algebra at Berkeley

PI: James DemmelUniversity of California, BerkeleyElectronics Research Laboratory(510) 643-5386


R&T PROJECT CODE: 411u002---01




Technical Objective:The aim is to bring together a distinguished group of researchersin the fields of numerical linear algebra and computer arithmeticso that the interface can be explored.

Approach:A conference will be held with 12 invited speakers. Participantswill be chosen both for expertise in their own fields and in lightof the degree to which their research suggests the potential forinteraction with the other topic of the meeting. The speakers willwrite review articles which will be puniished as a group in aspecial issue of the Journal of Numerical Linear Algebra withApplications, providing a valuable and lasting reference for otherresearchers in these fields and engineers in a broad range ofapplications.

73

TITLE: Research in Efficient Scientific Computation

PI: Beresford N. ParlettUniversity of California, BerkeleyCenter for Pure and Applied Mathematics(415) 642-6655






Technical Objective:This work aims to reduce the substantial computer time required todo the numerical linear algebra required for the computer solutionof partial differential equations. Among the equations addressedwill be those used in vehicle design and in weather and oceanprediction.

Approach:This research will concentrate on two distinct topics. The firstis promoting the sort of computer system regularity which leads toa uniform interface for performing scientific and engineeringcomputations. Here the work will build on previous research whichwas incorporated in the IEEE standard for computer arithmetic.Secondly, research on linear algerbra will emphasize matrixexponential and block and asymmetric eigenvalue algorithms.

Progress:A new and very efficient algorithm for the bidiagonal singularvalue problem was developed. The paper won the second Society forIndustrial and Applied Mathematics Linear Algebra Prize (1991).This exceptional work will find application in both signal/arrayprocessing and engineering computation.

74

TITLE: Adaptive Approaches and Error Estimations in FiniteElement Applications and Mathematical Modeling

PI: Ivo BabuskaUniversity of MarylandInstitute for Physical Science and Technology(301) 454-4469






Technical Objective:The objective is to develop algorithms for the computer solution ofpartial differential equations. This will make it possible tosolve new classes and categories of equations (by providing therequired speed) and, as a side benefit, speed up the solution ofequations which are presently used. An equally important objectiveis to provide error estimates and error control procedures toassure the user of the methods that the solutions accuratelyreflect physical reality. Both objectives are crucial for improvedvehicle design and for predictive capability in geophysical fluidmechanics.

Approach:New and improved adaptive solution methods will be developed.Approaches to a posteriori error estimation, first developed foradaptive algorithms, will be extended to more general errorestimation contexts. Other approaches, such as expert systems, forerror control will be explored.

Progress:Based on previous research under this grant, which developed theextraordinarily efficient HP finite element method, a veryefficient method for predicting the propagation of cracks instructures was developed. This new method appears to make itpossible for the first time to realistically implement thecurrently applicable milspec for cracks in aircraft structures.

75

TITLE: Adaptive Approaches and Error Estimates in FiniteElement Applications and Mathematical Modeling

PI: W. C. RheinboldtUniversity of PittsburghDepartment of Mathematics and Statistics(412) 624-8381






Technical Objective:The objective is to develop algorithms for the computer solution ofpartial differential equations. This will make it possible tosolve new classes and categories of equations (by providing therequired speed) and, as a side benefit, speed up the solution ofequations which are presently used. An equally important objectiveis to provide error estimates and error control procedures toassure the user of the methods that the solutions accuratelyreflect physical reality. Both objectives are crucial for improvedvehicle design and for predictive capabilities in geophysical fluidmechanics.

Approach:New and improved adaptive solution methods will be developed.Approaches to a posteriori error estimation, first developed foradaptive algorithms, will be extended to more general errorestimation contexts. Other approaches, such as expert systems, forerror control will be explored.

Progress:A general existence and uniqueness theory was developed fordifferential-algebraic equations which extends the well known ODEtheory. It was shown that differential-algebraic equations carrya geometric content in that they are locally equivalent to ODE's ona constraint manifold.

76

TITLE: Computing Methods for the Approximate Solutionof Time Dependent Problems

PI: Joseph 01igerLeland Stanford Junior UniversityDepartment of Computer Science(415) 723-0571






Technical Objective:The objective is to develop improved algorithms and programmingtechniques for the computer solution of problems in meteorology,oceanography, aerodynamics and hydrodynamics.

Approach:The goals will be approached via adaptive methods, meta-algorithms,improved open boundary conditions, parallel algorithms and improveddata structures.

Progress:Substantial progress was made on the long term goal of producing aset of software engineering tools suitable for use by scientistsimplementing methods for scientific and engineering computing. Akey part of this suite of tools which is nearing completion is apackage for creating adaptive methods for the solution of partialdifferential equations; advanced methods for creating andgraphically editing grids are included.

77

TITLE: Mathematical Treatment of Modeling in ComputationalMechanics

PI: Soren JensenUniversity of MarylandDepartment of Mathematics(301) 455-3294






Technical Objective:Most vehicles and platforms undergo structural analysis which isessential for performance and durability. Computer limitationsoften dictate that three dimensional structural problems be solvedvia a 2-D approximation. This project will develop techniqueswhich improve the computational gain from this approximation andtightly control the error involved.

Approach:The energy asymptotic method was recently applied to relativelysimple nonlinear monotone problems on simple domains. Under thisgrant more general nonlinear problems and more complex domains willbe investigated. Adaptive methods will be explored, methods forsystems of equations will be investigated and tight error estimateswill be derived. Analysis will be extended from just "brittle" to"ductile" and "plastic" materials and to more general constituativelaws appropriate to layered and inhomogeneous materials. Theprevious restriction to special deformations (anti-plane shear)will be removed.

Progress:A feedback method for dimensional reduction was developed for thenumerical solution of a class of nonlinear boundary value problemsassociated with anti-plane shear, Hencky's theory of plasticity orMaxwell's equations for the magnetic field potential. Underappropriate smoothness conditions the procedure was shown to beadaptive, i.e., optimal with respect to convergence rate.

78

TITLE: Adaptive Multi-Level Techniques for Solving PartialDifferential Equations

PI: Randolph E. BankUniversity of California, San DiegoDepartment of Mathematics(619) 534-4204






Technical Objective:The aim is to develop methods which will make possible the solutionof broad classes of partial differential equations which are notnow solvable either because operation counts are too large orbecause subtle nonlinearities preclude solution. Emphasis will beplaced on equations relevant to semiconductor simulations.

Approach:The approach will be via combinations of adaptive methods andmultigrid methods and a variety of geometric approaches tononlinear problems with parameters. Transformation techniques willbe applied to the circuit simulation problem. Many of the methodsdeveloped will be incorporated into transportable software.

Progress:A method was developed which is suitable for solving the linearequations of semiconductor device modeling in three dimensions --a first. The complexity of general sparse Gaussian elimination viabordering was analyzed and substantial storage savings weredocumented. Progress was made in understanding the discretizationof elliptic equations in three dimensions. Many improvements weremade in the PLTMG nonlinear partial differential equationssoftware.

79

TITLE: Numerical Solution of Radar Scattering Problemsvia Boundary Integral Equations

PI: Vladimir RokhlinYale UniversityDepartment of Computer Science(203) 432-1200






Technical Objective:The aim is to devise an algorithm 'hi. is capable of accuratelycalculating the 3D microwave scatt•:-• r Leld from most objects ofengineering interest in something Ikc one hour on an XMP-classcomputer.

Approach:The approach is to reduce the Helmholtz equation on an object to asecond kind integral equation on the boundary of the object. Theresulting system of dense linear equations can then be solved withvery high efficiency. The ultimate goal is a full threedimensional code, but as an interim measure, an axisymmetric codewill be developed. The axisymmetric code will be useful in itself(e.g., for radome calculations) and will allow many of the problemsof the 3D code to be anticipated and solved in a simpler context.

Progress:Several methods were developed for the direct solution of thelinear equations generated by the multipole method. For theseproblems and to the extent that the direct methods can be appliedin 3D, they largely eliminate the very troublesome problem ofill-conditioning for scattering applications. The mathematicsrequired to extend the fast multipole method to 3D scattering wascompleted; a simple demonstration of the ability to calculate thescattered field from an object of the order of one hundredwavelenths long will be forthcoming soon.

80

TITLE: Circuit and Semiconductor Simulation

PI: Donald J. RoseDuke UniversityDepartment of Computer Science(919) 660-6544






Technical Objective:The aim is to develop numerical methods which will apply to a rangeof partial differential equations but will be especially effectivefor semiconductor and circuit simulations. These will increase therange of semiconductor phenomena which can be modeled and greatlyincrease the size of circuits which can be simulated.

Approach:The most important idea to be pursued is that of operator theoreticmethods, an innovation of the PI. The concept is to view a set ofnonlinear equations as an operator in an appropriate Banach spaceand to apply the methods of functional analysis before proceedingto solution details. In this context it will likely prove possibleto use different time scales for different collections ofvariables.

Progress:The new norm decreasing approximate Newton method developed here,also known as waveform methods in the engineering literature, wasshown to provide a three to ten fold efficiency increase fortypical circuit simulation problems. For some applications such assmart power the advantage is likely to be much greater. Thetechnique, which can be characterized as space sensitive timeadaptivity, has been generalized beyond the circuit context.

81

TITLE: Fast PDE Solvers

PI: Bjorn JawerthUniversity of South CarolinaDepartment of Mathematics(803) 777-2632






Technical Objective:The aim is to explore both the range of applicability and thepotential efficiency of methods for the solution of partialdifferential equations based on the phi transform.

Approach:Four principal lines of research will be developed: 1) develop fastPDE solvers on nonsmooth domains, 2) incorporate adaptivity intothese fast solvers, 3) investigate the applicability of phitransform-based methods to time dependent problems, and 4)integrate geometric modeling with other developments.

Progress:Successful preliminary work was carried out to incorporate boundaryconditions into fast wavelet methods for solving partialdifferential equations, thus removing the restriction of suchmethods to integral equations. A new form of wavelet-based imagecompression was developed, and transition to the SLAM ImprovementProgram already seems assured.

82

TITLE: Numerical Solution of Partial Differential Equations

PI: Heinz 0. KreissUniversity of California, Los AngelesDepartment of Mathematics(213) 206-0048






Technical Objective:The aim is both to understand turbulence better and to use thisunderstanding to devise methods to compute a wider variety ofturbulent flows. For flows with different time scales the aim isto better understand the interactions of the different scales, todevise better numerical methods and to apply these insights andmethods to practical problems.

Approach:Turbulence investigations will include: smallest scales nearboundaries, maximal dissipativity, formation of inertial ranges,effect of compressibility, and stability of calculations. Formultiple time scales new numerical schemes will be developed,viscosity will be included, interactions between scales will beexplored, and applications will be made in oceanography andmeteorology.

Progress:A new approach to the computation of flow about a submarine,submarine wake calculations, and flow generated noise wasdeveloped. A new approach to the computation of high Reynoldsnumber compressible flow was developed. This approach was able topredict certain experimentally observed flow features previouslynot accessible to theory; it was awarded a NASA "SuperiorAchievement" prize.

83

TITLE: Numerical Solutions of Partial DifferentialEquations

PI: Tony F. ChanUniversity of California, Los AngelesDepartment of Mathematics(213) 825-2601




CURRENT END DATE: 31 JAN 1993


Technical Objective:The aim is to improve the performance of the so-called multipolemethod for the solution of the Helmholtz and similar equations.Performance enhancement will be pursued by examining improvedmethods for solving linear equations of the sort encountered inscattering. The scope of the method will be expanded by thecreation of boundary conditions appropriate to the physicalconditions of practical modern scattering problems. The virtues ofvarious ocean models and associated boundary and initial conditionswill be examined and characterized in a way useful for both oceanmodelers and experimenters.

Approach:Various forms of the preconditioned conjugate gradient algorithmwill be investigated. Implementation stategies which take accountof the opportunities and constraints of modern high-speed computerarchitectures will be devised and tested. Boundary conditionswhich correctly model layered scatterers, inhomogeneous bodies,internal baffels, etc. will be devised. Ocean research will bebased on constraints ("balances") applied to the Browning/Kreissand other models.

Progress:An alternative form of the mutipole method, which does not usemultipoles, was devised and is in the process of being implementedfor the scattering problem. Much enhanced efficiency isanticipated. New iterative techniques for the efficient solutiionof the integral equations generated by the multipole method weredevised and analysed. Moving Composite Overlapping gridtechniques were developed for free surface gravity wavecomputations.

84

TITLE: Fast Wavelet Transforms and the NumericalSolution of Hyperbolic Equations

PI: Stanley OsherUniversity of California, Los AngelesDepartment of Mathematics(213) 825-1758






Technical Objective:The aim is to extend numerical techniques under development for usein microwave scattering computations to the very different case ofhyperbolic equations, which typically describe fluid flow.

Approach:The plan is to begin with the Kreiss equation, and then to extendthis to an initial boundary value problem in one dimension. Theresearch will then proceed to higher dimensions, to Maxwell'sequation and finally perhaps to nonlinear equations, beginningpossibly with Berger's equation. The basic approach involves theuse of wavelets or other appropriate basis functions to "sparsify"the matrix representing the pde in a way that can be used togreatly reduce the computational complexity of the solutionprocedure.

Progress:The wavelet matrix compression technique has been applied toboundary value problems for constant coefficient hyperbolic partialdifferential equations. Both analysis and computationalexperiments have been successful and show great increases incomputational efficiency. The difficult issues involved in thenon-constant coefficient case are understood, but remain to beresolved.

85

TITLE: Efficient Solution Methods for Electromagneticand Acoustic Scattering

PI: Francis X. CanningRockwell International Corporation(805) 373-4598



CONTRACT NO: N0001491C0031



Technical Objective:The aim is to enhance the Impedance Localization Method byextending it in various ways so as to apply in three dimensions andto non-conducting and rough surfaces.

Approach:The approach to the three dimensional extension will be via two anda half dimensional problems--axisymmetric acoustic scattering andthe "wire grid" model of three dimensional microwave scattering.These projects will be pursued with NRL and NWC respectively. Byway of fundamental improvements in the method, two importantapproaches will be the consideration of multi-level numericalmethods and symbiotic relationships with the multipole method.

Progress:The impedance localization method has been successfully extendedfrom two dimensions to 3D axially symmetric problems--in theory, atleast. Prospects are excellent for a successful implementationnext year for the inelastic acoustic scattering problem. Resultshave been obtained which suggest that the full 3D submarine problemis within reach in the next few years.

86

TITLE: Acousto-Elastic Scattering and Radiation

PI: Luise S. SchuetzNaval Research LaboratoryStructural Acoustics Division(202) 767-3336



CONTRACT NO: N0001492WX24172



Technical Objective:The aim is to extend the Interaction Localization Method to the 3Daxisymmetric case to solve the acousto-elastic problem. Othergoals are to illuminate the basic method, to clearly drawimplementation issues, and to facilitate comparisons withwavelet-based methods.

Approach:Work will begin with the scattering problem and Neumann boundaryconditions. Then an algorithm will be developed and implementedfor the coupled acoustic-elastic problem. Analysis will be carriedout of the difficulties involved and improvements required forapplying these algorithms (codes) to problems of direct interest tothe development community.

Progress:The Interaction Matrix Localization(IML) method has been applied tothe 3D axisymetric submarine elastic radiation problem. Theexpected matrix sparsity was obtained. A storage problem remains,and a new approach has been devised, but not yet tested, for copingwith this problem.

87

TITLE: Fast Algorithms for Elliptic Partial DifferentialEquations

PI: Leslie GreengardNew York UniversityDepartment of Mathematics(212) 998-3315




CURRENT END DATE: 28 FEB 1993


Technical Objective:The aim is to characterize the best implementation techniques forapplying the fast multipole method to differential and integralequations. Issues include: 1. techniques for transformingdifferential equations to integral equations, 2. quadraturemethods, 3. linear algebra, 4. parallel implementation.

Approach:Work will begin with the Poisson equation in two and threedimensions. Then the far more difficult biharmonic and Helmholtzequations will be analyzed and appropriate methods will bedeveloped. The final stage will be to produce a general purposesolver for second order elliptic partial differential equations.

Progress:An implementation technique was developed for the solution ofPoisson's equation using the multipole idea, but in the case ofirregular domains and allowing the use of adaptivity. Roughdomains and adaptivity have not been compatible with pastimplementations of the multipole concept, and this implementationgreatly extends the applicability of the method.

88

TITLE: Approximate Models of Ocean Acoustics

PI: Martin H. SchultzYale UniversityDepartment of Computer Science(203) 432-1200






Technical Objective:The aim is to devise mathematical/computational models for oceanacoustic propagation such that the modeling and numerical errorsare well understood and minimal, given the practical constraints ofdata gathering and storage. A further constraint on the model isthat they must be capable of efficient execution on parallelcomputers, since such machines are required to realize acousticcomputations in real time.

Approach:The approach will be based on previous work by the PI on threedimensional parabolic equation approximations and theirimplementation in parallel (both unique research results). Withthese results goes a great deal of (still unimplemented) researchon crucial subproblems and related issues such the treatment ofinterfaces and the bottom and the inclusion of backscatter.

Progress:Great progress was made in understanding how to compute not onlyforward acoustic propagation but also the backscattered field.Within a year a complete numerical technology for the backscatterproblem should be available, including algorithms of practical useand an analysis of the errors induced by breaking the Helmholtzequation into outgoing and backscattered components. A potentiallyrevolutionary new algorithm for the parallel solution of partialdifferential equations (preconditioned conjugate gradient) wasdeveloped and used to solve a 3D problem on 10,000,000 grid pointsin a few minutes on a 1000 processor cube.

89

TITLE: Development of Overlapping Grid Techniques for theSolution of Ocean Models

PI: William D. HenshawInternational Business Machines Corporation(914) 945-3446






Technical Objective:The aim is to make composite grids, multigrids, and high ordermethods simultaneously available for ocean modeling computations.Whereas each technology separately might be implemented fairlyeasily, the combination requires substantial innovation.

Approach:High order methods combined with multigrids and composite gridswill be implemented for a new ocean model being developed under ONRsupport at NCAR. This model is being developed in closecollaboration with experimental oceanographers and will beexercised in forthcoming experiments in the California Current.

Progress:A successful two dimensional implementation of composite gridtechniques for use in the Browning/Kreiss ocean model was carriedout. This success points clearly to the way to achieve a goodthree dimensional implementation.

90

TITLE: Error Analysis in Numerical Solutionof Fluid-Structure Interaction Problems

PI: Ralph E. KleinmanUniversity of DelawareDepartment of Mathematics(302) 451-2000




CURRENT END DATE: 14 APR 1993


Technical Objective:To objective is to develop an analytical/numerical approach to theproblem of error estimation arising from the large scalecomputations associated with the solution of integral equations.The research has a very strong emphasis on error analysis forfluid-structure interaction problems.

Approach:The approach consists of analytic determination of the optimalerror norms associated with a particular fluid/structureformulation, via an appropriate choice of the underlying functionspace. Numerical implementation of the theory on model problemswill be performed.

Progress:Error analysis was successfully carried out for the hypersingularboundary integral equations of the first kind. Numericalexperiments were carried out to test the potential for usingresidual error as a measure of actual error in the case of Fredholmintegral equations of the second kind.

91

TITLE: Development of an Ocean Model Based Uponthe Reduced System of Equations

PI: Gerald BrowningColorado State UniversityCenter for Atmospheric Research(303) 491-8448






Technical Objective:The aim is to fully implement the Browning/Kreiss reduced system ofequations model and to test the model against standard ocean modelssuch the quasi-geostrophic based models.

Approach:The model will be implemented with bottom topography and for curveddomains. The implementation will incorporate modern numericalmethods such as a combination of multigrids and component grids.This combination will be developed at IBM and ported to the modelunder this effort. Testing and comparisons to other approacheswill use new "ocean-like" exact solutions of the Navier-Stokesequations developed under the predecessor to this effort.

92

TITLE: Numerical Methods for Flow Fields with Free Surfaces

PI: W. G. SzymczakNaval Surface Warfare CenterResearch Department(301) 394-2284






Technical Objective:The aim is to develop a computational method for simulating threedimensional free surface flows. The method will be applicable tothe prediction of cavitation damage, explosion bubble dynamics andwater exit and entry of projectiles.

Approach:A fixed domain method will be developed based on the GeneralizedHydrodynamic Formulation of Rogers which solves the conservationlaws subject to one sided constraints. Among the advantages ofthis formulation is that it can correctly model liquid on liquidcollisions.

93

TITLE: Analysis and Convergence of NumericalSchemes for an Anti-Plane Shear Problem

PI: Sonia M. GarciaU.S. Naval AcademyDepartment of Mathematics(301) 267-3703




CURRENT END DATE: 28 FEB 1993


Technical Objective:The aim is to develope a method to perform the computationsrequired to predict the formation and propagation of wing cracks inmaterials such as ice and steel. The method(s) will alsocontribute to advancing the state-of-the-art in solving nonlinearparabolic equations.

Approach:Several numerical schemes will be considered. First will be amodified Crank-Nicholson finite difference approximation in thetime variable together with a conbination finite element methods instress, temperature and displacement variables. An implementationwill be carried out.

94

TITLE: The 7th IMACS International Conferenceon Computer Methods for Partial Differential Equations

PI: Doyle KnightIMACSDepartment of Computer Science(908) 932-3998



CONTRACT NO: N000149231543



Technical Objective:The aim is to achieve communication and the sharing of ideas andtechniques among numerically intensive scientific disciplines whichdo not ordinarily communicate.

Approach:A conference will be organized which offers specialized sessions insuch subjects as ocean acoustics or lattice gases combined withmethodology sessions on subjects such as parallel computing oradaptive methods. This variety will attract the diverse group ofattendees required.

95

TITLE: The Numerical Analysis of EvolutionEquations Over Long Time Intervals

PI: Andrew M. StuartLeland Stanford Junior UniversityComputer Science Department(415) 723-2273



CONTRACT NO:



Technical Objective:The aim of this effort is to study the numerical approximation ofdifferential equations over a very long time interval. Becausethese equations either exhibit chaos or possess multiplesteady-state solutions, the standard techniques give rise tospurious solutions. In the context of two special classes ofdifferential equations, the main goal is to understand the class ofnonlinearities that allow the development of discretizations thatrelate the numerical approximation to the underlying continuousproblem.

Approach:Local error control mechanisms will be developed that will act tovary the time step in the discretization in such a way thatspurious solutions are avoided. For fixed time stepping schemes,the PI will generalize the theories of A, B, and G-stability todynamical systems with complex limit sets. Approximation ofhomoclinic and heteroclinic orbits of such dynamical systems willbe studied.

96

Operations Research

97

TITLE: Algorithm Design and Evaluation

Andrew V. GoldbergLeland Stanford Junior UniversityComputer Science Department(415) 723-4711


R&T PROJECT CODE: 400x081yip02



SCIENTIFIC OFFICER: Donald Wagner

Technical Objective:This research will provide new and efficient algorithms, bothsequential and parallel, for network flow problems and linearprogramming. Efficiency is measured in terms of time, memory andprocessors. Experimental evaluation of various implementations ofthe algorithms is intended.

Approach:The approach for finding more efficient algorithms for linearprogramming is to design an interior-point algorithm that does notuse transformations or projections since these operations arebottlenecks in current methods. For network flow problems, severaltechniques will be used in the design of better algorithms, such asimproved data structures, scaling and parallelization.

Progress:Progress has been made in the design of network optimizationalgorithms. Using a simple randomization strategy, the runningtime of a recent algorithm of Leighton et al. has been improved.Second, an algorithm for determining what is the minimum amount bywhich the capacities of an infeasible transportation problem mustbe increased to make it feasible has been found. Its running timesignificantly improves previous bounds. Work on a parallelimplementation of a maximum flow algorithm and a sequentialimplementation of a minimum-cost flow algorithm are also inprogress.

99

TITLE: Solving Integer Generalized Networks

PI: Michael A. TrickCarnegie-Mellon UniversityGraduate School of Industrial Administration(412) 268-3697


R&T PROJECT CODE:- 400xO96yipOl




Technical Objective:This research will provide new and efficient algorithms for integergeneralized network flow problems. The objective is to build onprevious research on cutting planes and linear generalized networkflow algorithms, and combine these techniques in a optimizationroutine for the integer version. Particular attention will begiven on how to properly exploit the underlying network structureof the problem.

Approach:The research will employ a three-part approach. First, anefficient code will be developed for the linear generalized networkflow problem. To be useful in the integer setting, the code willbe designed for efficient reoptimization after adding constraints.Second, useful valid inequalites will be found, and third,algorithms for solving the separation problem over the inequaliteswill be developed.

100

TITLE: New Approaches to Linear and Nonlinear Programming

PI: Walter MurrayLeland Stanford Junior UniversityDepartment of Operations Research(415) 723-1307


R&T PROJECT CODE: 411k003---07




Technical Objective:The objective of this research is to develop efficientinterior-point methods for linear and quadratic programming.Particular emphasis will be in the development of efficientprocedures for solving the linear least-squares problem, whichoccurs as a subproblem in interior-point methods.

Approach:This research will develop primal and dual interior-point methodsfor linear and quadratic programming-problems. The approach will beto view interior-point methods as barrier-function methods. Theassociated linear least-squares problem will be solved usingpreconditioned conjugate-gradient methods based on either LU orCholesky factorization techniques. Possible application of thesenew approaches to nonlinear problems will also be investigated.

Progress:A primal-dual interior point method for linear programming has beendeveloped that allows for an adaptive step length, which closes thegap between theoretically efficient short step methods andempirically efficient long step methods. An interior point methodfor nonlinear convex progamming problems has been developed. Thealgorithm uses the predictor-corrector strategy, and maintains thesparsity of the matrices assuming that the functions are separable.

101

TITLE: Mathematical Aspects of Discrete Event Dynamic Systems

PI: Yu-Chi HoHarvard CollegeDepartment of Applied Mathematics(617) 495-3992


R&T PROJECT CODE:. 4116651---07




Technical Objective:The objective is to develop a theory of discrete event dynamicsystems analogous to the control theory of continuous dynamicalsystems. Such a theory will provide a basis for the design ofoptimal controls for discrete systems and to increase our abilityto infer behavior of such systems from simulated results.

Approach:One important aspect involves deriving the capability toaccommodate simulations in which small perturbations cause theorder of events to change. This phenomenon makes the results ofreplications using different random number streams incomparable andthe corresponding mathematical analysis more difficult.Experiments will be conducted to test the theoretical results.

Progress:Computational experiments were conducted using the gradient surfacemethod for optimization of discrete event dynamical systems. Inthe gradient surface method, the gradient estimation is obtained byeither perturbation analysis or likelihood ratios, and theperformance gradient surface is obtained from observations atvarious points in a fashion similar to response surfacemethodology. This yields greatly increased efficiency. Severalpreviously computationally infeasible problems were solved withthis new method.

102

TITLE: New Algorithmic Developments in Nonlinear Optimization

PF: Anthony V. FiaccoGeorge Washington UniversityOperations Research Department(202) 994-7511






Technical Objective:The objective of this research is to develop and test barrier andpenalty methods for convex and non-convex programming problems.Emphasis will be on efficient implementations and provingconvergence of these methods. A new method for non-convexfractional programming will also be developed.

Approach:A theory for exterior-point methods will be developed that isparallel to that which exists for interior-point methods. Usingthe Courant quadratic-loss penalty function, exterior-pointalgorithms for both linear programming and convex programming willbe developed with the goal of establishing a polynomial-timecomplexity bound. Techniques for solving non-convex fractionalprogramming problems as a sequence of linear programming problemswill be investigated. By using techniques like those developedfor SUNT, the convergence rate of primal-dual methods fornon-convex programming problems will be examined.

Progress:A method that globally optimizes the sum of ratios of linear formshas been developed. It operates by transforming the decision spaceinto an image space in which the optimization is easily performedalong special search directions. In other work, methods forautomatically generating bounds on variables in a nonconvexprogramming problem were found. A primal-dual algorithm fornonconvex programming problems was devised and implemented. It wasshown to have a superlinear rate of convergence in the absence ofprimal and dual degeneracy.

103

TITLE: Intermodal Logistics

PI: H. D. RatliffGeorgia TechSchool of Industrial and Systems Engineering(404) 894-2307

FUNDING AGENCY: Office of-Naval Research





Technical Objective:The technical objective is to create models and solution algorithmsfor operational distribution and logiatics problems involvingintermodal networks, which are distribution networks that allow formore than one type of transportatior. vehicle. Of particularinterest are the problems of network design, routing and schedulingof vehicles and handling of material at the interface of differenttypes of vehicles.

Approach:The research will combine discrete optimization, optimization-basedheuristics and object-oriented interactive modeling for solving theproblems associated with intermodal networks. For example, newtechniques need to be developed for determining optimal location ofintermodal terminals, i.e. points where material is transferredform one mode of transportation to another. Standard networkoptimization techniques such as shortest path algorithms need to beextended to allow for quantity discounts and the capacity problemsencountered when switching from one mode to another.

Progress:Progress has been made in the area of guaranteed-time distributionnetworks, which are network problems in which the movement of itemsbetween any source-destination pair must be completed within aguaranteed period of time. Decisions have to be made regarding thelocation of accumulation points, which are where changes in themode of travel occur. Customers are allocated to these points insuch a way that travel time is minimized. For tree-structurednetworks, efficient algorithms have been developed that locateaccumulation points, allocate customers and provide thebest-possible time guarantee.

104

TITLE: Solution Procedures for Large-Scale CombinatorialOptimization Problems

PI: Karla L. HoffmanGeorge Mason UniversityOperations Research and Applied Statistics Department(703) 993-1670


R&T PROJECT C2)DE: 4116663---06




Technical Objective:The objective of this research is to develop new techniques forsolving large-scale integer optimization problems. Such problemsare extremely important in logistics, financial planning, andresource allocation.

Approach:Recently obtained results concerning the geometry of polyhedra willbe used to develop tight formulations of integer-programmingproblems. Good upper and lower bounds on the optimal solutionvalue will be obtained via numerical preprocessing. Abranch-and-cut strategy using variable fixing is employed as asolution procedure. Heuristics techniques are used to obtain theviolated inequalities needed in the cutting phase of thebranch-and-cut procedure.

Progress:Progress continues to be made on a branch-and-cut algorithm forsolving large-scale integer programming problems. Thebranch-and-cut approach combines automatic reformulationprocedures, heuristics for finding good feasible solutions andcutting-plane techniques in a tree-search format. It is currentlybeing used to solve set-partitioning problems arising from crewscheduling in the airline industry. To date, the largest suchproblem that has been solved to optimality has over one millionvariables. A similar polyhedral approach is being devised forsolving the set-covering problem with side constraints.

105

TITLE: Production Systems Analysis

PI: Russell E. KingNorth Carolina State UniversityIndustrial Engineering Department(919) 515-5186






Technical Objective:The objective of this effort is the analysis of fundamentalscheduling problems and related models. These include optimalresource planning and real-time scheduling of stochastic shop-floorprocesses. Scheduling systems will be developed that can adapt tothe highly stochastic nature of the shop floor and constantlychanging objectives of management.

Approach:The grantee will model and optimize real, complex schedulingsystems. This involves the development of specific models of theshop-floor process, real-time simulation of the inductionscheduling and rework prccess, and development of optimizationschemes for induction scheduling, resource planning and real-timeshop floor scheduling. The models will be tested and evaluated atCherry Point Naval Avaiation Depot.

Progress:Progress has been made in several areas, including scheduling ofmaterial handling systems and of Automated Guided Vehicles (AGVs).In a material handling system, the variability of products flowingthrough the system necessitates dynamic scheduling of the materialhandling devices. A branch-and-bound scheme was developed foreffective real-time scheduling of these devices, which resulted inreducing manufacturing lead time. In the scheduling AGVs, a Narkovdecision process model was developed to aid in optimal control ofthe AGV system. Combined with a heuristic scheme, the resultingscheduling system dominated standard industry techniques whentested via simulation.

106

TITLE: Combinatorial Optimization: Algorithms and Applications

PI: Egon BalasCarnegie-Mellon UniversityGraduate School of Industrial Administration(412) 268-2285


R&T PROJECT CODE:. 4116666---06




Technical Objective:The grantee will develop new computational methods for the solutionof NP-complete integer-programming problems, including thetraveling salesman problem and scheduling problems. This work willaddress the determination of facet-defining inequalities for theseinteger-programming problems, and will develop algorithms forparallel computers.

Approach:The grantee will use techniques from polyhedral combinatorics todevelop algorithms for certain integer-programming problem. TheCray supercomputer and the BBN Butterfly parallel computer will beused in computer testing of the algorithms.

Progress:New facets for the three index assignment polytope and theprecedence-constrained asymmetric TSP polytope have been found. Anew sequential lifting-projection procedure has been developed forsolving mixed-integer programming problems, and has been applied tosolve a hard TSP problem. The column-subtraction method designedfor set covering and packing problems was successfully adapted tosolve satisfiability problems. TSPs having as many as 10,000cities were solved using a two-matching-based branch-and-boundalgorithm. A parallel version of the simplex algorithm wasdeveloped and is being tested.

107

TITLE: Issues in Strong Polynomiality: Approximation Algorithmsand Graph Algorithms

PI: Dorit S. HochbaumUniversity of California, BerkeleySchool of Business(415) 642-4952






Technical Objective:The goal is to develop best-possible approximation algorithms forsome important NP-complete combinatorial optimization problems.The emphasis is on theoretical properties of algorithms developedand on their practical application. A second goal is to developstrongly polynomial algorithms for optimization problems anddetermine which problems admit a strongly polynomial algorithm.

Approach:The approach involves the use of results from complexity theory andgraph theory to develop approximation algorithms for various graphoptimization porblems. Complexity theory and linear-programmingtheory will be used in the development of strongly polynomialalgorithms for optimization problems, such as pre-Leontieflinear-programming problems and quadratic network flow problems.

Progress:An improved algorithm was developed for certain allocation problemsdefined on graphs. It improves the time bound of the bestalgorithm by a factor of log n, and it comes within a factor of logn of being optimal. In other work, the grantee is conducting anempirical analysis of a previously developed polynomial-timealgorithm for certain constrained non-linear optimization problems.The analysis shows that the algorithm, which uses an interior-pointalgorithm as a subroutine, dominates the widely used reducedgradient method.

108

TITLE: Computational and Mathematical Issues in optimization

PI: R. G. BlandCornell UniversitySchool of Operations Research and

Industrial Engineering(607) 255-9144



CONTRACT NO: N0001490F0058



Technical Objective:The objective is to develop new theoretical and computationaltechniques to solve large, complex optimization problems bothexactly and approximately. In particular, the emphasis will be onserial and parallel methods for discrete optimization, network flowand machine scheduling problems, and linear and nonlinearprogramming.

Approach:Among other approaches, investigations of interior-point methodswill extend to I sub p approximation and quadratic programming andalso to direct solution of combinatorial optimization problems,e.g. bipartite matching. In addition, interior-point methods willbe used as linear-programming solvers in polyhedral routines forcombinatorial optimization. The interplay between efficientinterior-point methods and methods of numerical linear algebra willbe exploited. Network flow research will attempt to extend thework of Tardos on strongly polynomial algorithms to multi-commodityand generalized-network flows.

Progress:This work has resulted in new and better algorithms for a varietyof important optimization problems. A non-exhaustive list includesa polynomial-time dual simplex method for the transshipmentproblem, a parallel algorithm, based on interior point methods, forbipartite matching, a combined phase 1-phase 2 interior pointmethod for linear programming, and interior and exterior pointmethods for convex quadratic programming.

109

TITLE: Analysis of Vehicle Routing and Inventory Problems

PI: David Simchi-LeviColumbia UniversityDepartment of Industrial Engineering &

Operations Research(212) 854-4334






Technical Objective:The objective is to develop worst case and probabilisticperformance analysis for a class of heuristic algorithms for thevehicle routing problem. The goal is to identify classes of therouting problem for which a specific heuristic is highly efficient.Issues relating the routing problem with related problems in anintegrated model such as inventory control will also be examined.

Approach:Worst case bounds on the performance of heuristic algorithms forvehicle routing problems are obtained by modifying techniques usedin related problems such as the traveling salesman problem on whichthe heuristics are based. Analytical results for the travelingsalesman problem can then be applied to the routing problem.

Progress:Progress has been in the development of heuristics for routing andinventory problems. For the bin-packing problem with a concavecost function, the so-called next-fit-increasing heuristic is shownto have good absolute and asymptotic worst-case bounds. For morespecialized cost functions, the grantee has found two heuristicsthat achieve the best possible performance ratio. For thedistance-constrained vehicle-routing problem, the grantee hasdeveloped a heuristic and has performed extensive computationaltesting with it. The testing shows that the heuristic is highlyeffective.

110

TITLE: Using Interior Point Methods in a Cutting PlaneApproach for Solving Integer Programming Problems

PI: John E. MitchellRensselaer Polytechnic InstituteMathematical Sciences Department(518) 276-6915






Technical Objective:The objective is to show how interior point methods can be used ina cutting plane algorithm to solve integer programming problems.A major issue to be addressed is that for a cutting plane procedureto work efficiently, it is necessary to be able to resolve a linearprogram efficiently when an extra constraint is added.

Approach:The approach is to apply the projective primal variant ofKarmarkar's algorithm to the dual of the current relaxation of aninteger programming problem. Cutting planes appear as extraconstraints in the dual problem and as extra variables in theprimal problem. A number of approaches for the earlyidentification of cutting planes will be examined.

Progress:The grantee has developed a column generation schmeme for linearprogramming based on Freund's shifted-barrier interior pointmethod. The method allows the solution of linear programmingproblems using only a selected subset of the variables, and socould be used in a cutting plane algorithm for integer programming.Under reasonable assumptions, the column generation schemeterminates in polynomial time.

111

TITLE: Mathematical Foundations of Combinatorial Search

PI: Fred GloverUniversity of ColoradoGraduate School of Business & Administration(303) 492-8589



CONTRACT NO: N0001492MP24018



Technical Objective:The objective is to pull together various approaches tocombinatorial optimization into a unified mathematical frameworkbased on probability theory and statistical inference. Attemptswill be made to emcompass present search procedures such assimulated annealing, genetic algorithms, enumerative tree searchand tabu search. Based on this unified framework, new heuristicprocedures will be developed, including parallel procedures.

Approach:The approach involves the development of theoretical principles ofsearch which, like axioms in a mathematical system, may be combinedin various ways to produce alternative methods. The convergenceanalysis of existing search techniques will also be combined usingtechniques from probability theory.

Progress:Several applications of a greedy randomized adaptive searchprocedure have been developed, including algorithms for graphcoloring and single machine scheduling. A method that generalizessimulated annealing and incorporates ideas from tabu search hasbeen developed. Several different forms of flexible memorystructures for tabu search have been developed and tested. Oncertain routing and partitioning problems, these have yielded thebest-known results. A principle of proximate optimality has beenformalized, which provides a link between tabu search and dynamicprogramming. Techniques for applying tabu search in nonlinear andparametric optimization have been developed.

112

TITLE: Studies on the Complexity of Linear Programming andRelated Problems

PI: Nimrod MeggidoInternational Business Machines CorporationAlmaden Research Center(408) 927-1271






Technical Objective:The objective is to develop and analyze new algorithms for linearprogramming and extensions. Particular emphasis will be onparallel complexity, strongly polynomial algorithms andprobabilistic analysis.

Approach:The approach employs recent results in complexity theory to analyzelinear programming algorithms. Professor Megiddo will generalizehis previous work in order to develop low-order polynomial-timeparallel algorithms for linear programming in fixed dimensions.This will include the use of multidimensional search techniques andparametric analysis to develop classes of linear-programmingproblems that are strongly polynomial. A probabilistic analysis ofinterior-point algorithms for linear programming will be developedusing the global properties of central paths associated with suchalgorithms.

Progress:A new algorithm for the generalized circulation problem has beenfound. It iteratively solves a sequence of uncapacitatedtransshipment problems. It can be used to either find an optimalflow or an approximation. When used to find a constant factorapproximation, it runs in strongly polynomial time. This is thefirst strongly polynomial approximation algorithm for thecirculation problem. It is not known whether the generaloptimization version can be solved in strongly polynomial time.

113

TITLE: Cutting Planes for Mixed-Integer Programs

PI: Andrew E. BoydRice UniversityDepartment of Mathematical Sciences(713) 527-4805



CONTRACT NO: N0001491F00043



Technical Objective:The objective is to develop a theory of cutting planes for mixed-integer programming problems based on Fenchel and Lagrangianduality. An important part of the research is to developprocedures for solving the separation problem for Fenchel andLagrangian cuts. Based on this theory, algorithms formixed-integer programming will follow. Computer implementation ofthe algorithms, together with a collection of test problems will bemade available.

Approach:The approach to the separation problem for Fenchel and Lagrangiancuts involves ascent procedures that take advantage of thecombinatorial structure found in typical mixed-integer programmingproblems. This requires the development of efficient proceduresfor choosing an ascent direction, handling degeneracy, andverifying optimality in the ascent procedure. Computerimplementation will be done in a modular structure that will allowother researchers to use these cutting-plane procedures in thedevelopment of new mixed-integer programming algorithms.

Progress:An integer-programming code is in its final stages and soon will bereleased through IMSL. Theoretical work includes a result that anatural extension of the widely used steepest ascent simplex pivotrule is provably finite. A second finite pivot rule having severalcomputationally attractive features was also developed. Bothprocedures have the unique advantage that they choose the pivotelement without explicit knowledge of the set of active constraintsat a point of degeneracy, thus making them attractive incombinatorial settings where the linear programming problem is notexplicitly written out.

114

TITLE: Operations Research Methods for Logical Inference

PI: John N. HookerCarnegie-Mellon UniversityGraduate School of Industrial Engineering(412) 268-3584






Technical Objective:The grantee will develop new techniques for solving logicalinference problems using methods from optimization. This work willinclude improved integer programming algorithms for propositionallogic and nonlinear programming algorithms for reasoning withuncertainty. Extensions of constraint logic programming languagesto deal explicitly with mixed integer linear constraints will alsobe derived.

Approach:The grantee will use techniques from polyhedral combinatorics todevelop improved cutting plane algorithms for propositional logic.Nonlinear programming and column generation techniques will be usedto develop methods for probabilistic reasoning. A normal form forthe encoding of logical constraints in a mixed integer programmingmodel will be developed. This will yield tighter linear programmingrelaxations for the problem.

115

TITLE: Operations Research Methods for Logical Inference

PI: Vijaya ChandruPurdue UniversitySchool of Industrial Engineering(319) 494-5419






Technical Objective:The grantee will develop new techniques for solving logicalinference problems using methods from optimization. This work willinclude improved integer programming algorithms for propositionallogic and nonlinear programming algorithms for reasoning withuncertainty. Extensions of constraint logic programming languagesto deal explicitly with mixed integer linear constraints will alsobe derived.

Approach:The grantee will use techniques from polyhedral combinatorics todevelop improved cutting plane algorithms for propositional logic.Nonlinear programming and column generation techniques will be usedto develop methods for probabilistic reasoning. A normal form forthe encoding of logical constraints in a mixed integer programmingmodel will be developed. This will yield a tighter linearprogramming relaxation for the problem.

116

TITLE: Large-Scale Optimization

PI: Gordon H. BradleyNaval Postgraduate SchoolDepartment of Operations Research(408) 646-2359






Technical Objective:This research will develop theory and algorithms for the solutionof large-scale optimization problems. One focus is on persistenceof integer and nonlinear optimization solutions, automaticexploitation of special structure and communication of modelstructure from modeler to solver. A second focus is thedevelopment of optimization and game theoretic models andalgorithms to solve network interdiction problems.

Approach:The investigators will develop common themes in modeling, solutionalgorithm design and interpretation of solutions that willsubstantially strengthen the persistence of solutions. They willalso synthesize useful results of known enumeration rules with newrules to obtain a form useful for general large-scale mixed integerenumeration. This will include developing and testing of anenumeration algorithm for problems exhibiting any or all of thefollowing features: binary variables, general integer variables,semi-continuous variables and sets of variables precluding naturalinteger solution due to their involvement in odd cycles in setpartition rows.

117

TITLE: Satisflability

PI: Peter L. HammerRutgers UniversityCenter for Operations Research(908) 932-3041






Technical Objective:The grantee will develop new techniques to analyze and solvesatisflability problems. This work will include the use ofpartially ordered sets, graph theory, polyhedral theory andpseudo-boolean optimization. Classes of satisflability problemsthat can be solved in polynomial time will be identified.Approximation procedures and decomposition techniques will bedeveloped for analyzing large-scale satisflability problems.

Approach:The grantee will use techniques from implicit enumeration,resolution and cutting planes to develop upper and lower bounds onthe worst case complexity of the satisflability problem. Inaddition, the stable set formulation of the satisfiability problemwill be used to develop new analysis techniques and solutionalgorithms. Methods based on the majorization and minorization ofBoolean functions will be developed and applied to thesatisflability problem.

118

TITLE: High Speed Heuristics for Real-Time Personnel AssignmentModels

PT: Jeffery L. KenningtonSouthern Methodist UniversityComputer Science and-Engineering Department(214) 692-3278






Technical Objective:The objective of this effort is to develop and apply newoptimization methods for solving manpower assignment problems inreal time. The results will be implemented in software andevaluated across many computational platforms, including parallelmachines.

Approach:New heuristics will be developed for solving specialized integerprogramming models underlying manpower assignment problems.Several algorithmic approaches will be tested, including greedyprocedures, local search and truncated branch-and-bound. Otherpossible approaches involve man-machine interaction. Severalhardware configurations will be tested to find one appropriate forthis application.

Progress:A highly successful algorithm for the singly constrained assignmentproblem has been devised. It is based on Lagrangean duality theoryand an elaborate scheme to update the Lagrange multipliers. Acomputer code of the two-tree version of Dijkstra's algorithm forthe one-to-one shortest path problem has been developed, and isbelieved to be the world's best code code for this pioblem.

119

TITLE: The Retention Behaviour of Six-Year Obligors

PI: Patrick C. MackinSAG Corporation (System-Analytics Group)(703) 538-4500


R&T PROJECT CODE:' 4428037---Ol

CONTRACT NO: N0001492-C-0132



Technical Objective:The grantee will develop an econometric model portraying theretention behaviour of sailors in the Navy's Six-Year Obligorprogram. An early warning system will be designed to alert Navymanpower managers of potential retention problems so that remedialactions can be taken in time.

Approach:Data will be collected from the Navy's Enlisted Master Record andother sources. Several econometric models will be built; fivepotential approaches have been identified. Each model will betested and analyzed. Based on these results, an early warningsystem will be designed.

120

TITLE: Development of Domain Specific Rule Bases for Use inRule-Based Forecasting.

PI: Fred CollopyCase Western Reserve UniversityDepartment of Management Information & Decision Systems(216) 368-2048






Technical Objective:This research will develop techniques for capturing domain-specificknowledge for use in a rule-based forecasting system. The researchis aimed at improving the accuracy of forecasts of Navy personnellength of service.

Approach:Analysts from NPRDC will be observed using concurrent verbalprotocols as they make forecasts for the length-of-service timeseries. Protocol analysis of these sessions will be used toidenify relevant contextual features and to develop rules thatspecify actions to be taken in response to particular features.After discussions with the analysts, the rules will be revised andrefined.

121

Probability and Statistics

123

TITLE: Statistics and Oceanography

PI: John R. TuckerNational Academy of SciencesBoard on Mathematical Sciences(202) 334-2421





SCIENTIFIC OFFICER: Julia Abrahams

Technical Objective:With this report, research opportunities at the interface ofstatistics and oceanography will be identified.

Approach:A group of statistics researchers and oceanography researchers willmeet to identify and write a report on basic statistics researchissues motivated by oceanographic applications.

Progress:Preliminary work on this report has begun under current funding andwill be completed under the expansion.

125

TITLE: Applied Statistics

PI: Herbert SolomonLeland Stanford Junior UniversityDepartment of Statistics(415) 497-1185






Technical Objective:The objective of this research is the study of real-world problemsinvolving estimation theory, probabilistic modeling, andmultivariate data analysis in order to develop enhanced techniquesacross a broad range of statistical applications.

Approach:A series of problems is formulated based on discussions withlaboratory personnel and presented to a panel of distinguishedstatistical researchers. Methods from estimation theory,probabilistic modeling, and multivariate data analysis aredeveloped and applied to these problems.

Progress:Problems taken from discussions with Navy Lab and NSA personnel arepresented to a panel of distinguished statistical researchers.Joint work using methods from estimation theory, probabilisticmodeling, and multivariate data analysis are applied to theseproblems and to advance statistical science on topics closelyrelated to the problem areas. For example, recent problemsaddressed single outlier detection in an acoustics application,measures of homogeneity of covariances in pattern recognition, andsome questions in queuing theory related to computer systemperformance.

126

TITLE: Contingency Tables and Sampling Schemes forAuto-Regressive Processes

PI: Yashaswini W. MittalUniversity of ArizonaDepartment of Statistics(602) 621-6209






Technical Objective:Measures of association between random processes will be developedbased on cross classification or contingency tables alsoincorporating auxillary knowledge about the variables under study.Sampling schemes for process monitoring will be developed forautoregressive processes rather than the standard independent andidentically distributed sequence situation.

Approach:The effects of unequal marginals on contingency table measures ofassociation will be corrected depending on the maximal and minimalvalues possible of the probability measure for the given set ofmarginals. Homogeneity of subpopulations will be used asauxilliary information in data compactification in contingencytable analyses. Variable rate sampling schemes related to recentdevelopments in quality control theory will be applied tomonitoring of autoregressive models.

Progress:Progress has been made on the analysis of paradoxes which arise inthe pooling of data from subpopulations. Necessary and sufficientconditions are developed for various paradoxes such as "Simpson'sparadox" in which spurious association between attributes can arisein a population as a result of amalgamation, even though theattributes are independent in the subpopulations.

127

TITLE: Stochastic Methods in Neural Computation Problems

PI: Charles E. SmithNorth Carolina State UniversityDepartment of Statistics(919) 737-2271






Technical Objective:The technical objective is to develop realistic models forvariability in the response characteristics of individual neurons.Since the individual neurons are nodes in a network, this willyield a better understanding of the overall activity of biologicalneural networks and their response to random inputs.

Approach:A point process with non-uniform intensity will be used to modelneural spike trains that exhibit dependence between events (afterhyperpolarization). An important issue is whether such pointprocesses can result from the level crossings of a continuous timestochastic process model for the cell membrane potential. This isa statistical issue referred to in the literature as a firstpassage time problem.

Progress:Maximum likelihood parameter estimation was applied to pointprocess models which describe neural spike trains. These modelsare point processes whose intensity functions are specific finitelyparameterized functions of the last occurrence time of the processup to time t. The performance of the estimators was studied onsimulated neural data obtained from a numerical neural model.

128

TITLE: Nonparametric Methods for Data Analysis in Threeto Five Dimensions

PI: David W. ScottRice UniversityDepartment of Statistics(713) 527-6037






Technical Objective:The objective of the proposed research is to develop theory andmethods for the analysis of large high dimensional data sets.Areas of interest include advanced visualization and parallelarchitectures, robust multivariate regression, non-parametricregression and kriging, and cross-validation.

Approach:New visualization techniques will be developed based on animationof average shifted histogram estimates. Transparency algorithmsthat allow several contour levels to be visualized simultaneouslyand a stereo option will also be developed. Theoretical work ondensity estimation in high dimensions will address the conjecturethat the curse of dimensionality is not so much the result ofsparse data, but rather the tendency of high-dimensional data tofall in subspaces.

Progess:The possibility of improving kernel density estimates by varyingthe window over the domain of estimation has been explored. Twogeneral approaches have been considered: to vary the window widthby the point of observation and by point of the sample observation.The first possibility is not promising, although the second is, andits properties have been investigated particularly for normal data.

129

TITLE: Regression and Stochastic Control

PI: Herman ChernoffHarvard CollegeDepartment of Statistics(617) 495-5462






Technical Objective:Objectives include the development of techniques for regressionanalysis when there are missing variables; analysis of likelihoodratios for singular testing problems such as mixtures ofdistributions and change point detection; and stochastic controlusing techniques in sequential analysis.

Approach:The approach to regression analysis with missing variables involvessolving the reduced regression and checking the residuals forproper behavior. If the missing terms model positive (negative)effects, the residuals should be positive (negative). If theresiduals exhibit poor behavior, an improved modified solution iscalculated. The approach to stochastic control is based on theconjecture that nonlinear optimal control will be more stable andless sensitive to deviations between actual and estimatedparameters than the Kalman filter.

Progess:In the context of testing whether a mixture of two binomials is asingle binomial, the distribution of the likelihood ratio isstudied. It is shown that the asymptotic distribution of twice thelog likelihood ratio corresponds to the square of the supremus ofa particular Gaussian process.

130

TITLE: Statistical Inference in Stochastic Processes

PI: Lynne BillardUniversity of GeorgiaDepartment of Statistics(404) 542-5232






Technical Objective:The objective is to develop inference techniques for non-lineartime series and space-time processes including nonlinearautoregressive processes with special emphasis on bilinear,threshold, and a wide class of exponential models. Inference willbe approached through a common methodology; least squares,conditional least squares, minimum L2 distance, and maximumlikelihood methods will be unified via appropriate approximatingquadratic forms.

Approach:For the nonlinear time series models studied a quadratic functionalof the autoregression is selected and minimized to estimate themodel parameters. This approach serves both to fit models to dataand to develop predictors. In addition hypothesis tests based onthese quadratic functionals will be developed and their performanceanalyzed.

Progess:Progress has been made on several problems in inference fornonlinear time series including:--- derived a test of homogeneity for testing the equality of theparameters in several independent nonlinear autoregressiveprocesses;--- introduced new models for spatial time series with long memorydependence and explored a number of estimation techniques for themodel parameters;--- developed sequential procedures for hypothesis tests for slopeand intercept of regression models.

131

TITLE: Asynchronous Computation of Fixed Points

PI: William F. EddyCarnegie-Mellon UniversityDepartment of Statistics(412) 268-2725

FUNDING AGENCY: - Office of Naval Research





Technical Objective:The research will focus on the development of software to supportdistributed computation. Statistical models of the performance ofdistributed systems will be developed and asynchronous iterativemethods will be studied and implemented on distributed systems.In addition, context specific approaches to the development ofdynamic graphics for the visualization of very large, highdimensional data sets will be a focus of the research.

Approach:The approach will be to model asynchronous calculationsprobabilistically in order to determine the range of possibleimprovements that one might expect through their use as well as tobetter understand their convergence properties. Specific types ofgraphics for special types of data will be developed. For example,in the context of asynchronous iterations the data is representedas high dimensional vectors evolving in time on videotape displays.

Progess:The issue of incorporating uncertainty into a rule-based expertsystem is investigated through the theory of belief functions. Theresulting computational complexity is reduced through the notion ofouter and inner approximations to the interval between "belief" and"plausibility". The properties of these approximations arestudied, and a computer implementation in the expert system contextis evaluated.

132

TITLE: Research in the Theory and Applications of BayesianStatistics

PI: Nozer D. SingpurwallaGeorge Washington UniversityOperations Research Department(202) 994-7515



CONTRACT NO: N0001485K0202



Technical Objective:The objective is to develop: (1) optimum Bayesian multi-stage andsequential procedures for the release of computer software, (2)optimal control in linear filtering in one and two dimensions, (3)statistical inference for Poisson and other point processes, and(4) a game theoretic approach to the problem of pooling expertopinion.

Approach:The general approach is Bayesian. This means that all forecasts,control strategies, point and interval estimators are based on theconditional distribution of the parameters given the data and someprior distribution.

Progess:Progress has been made on the design of optimal life-testingexperiments, optimal according to maximum information in the senseof Shannon. A paradox arises in assessing the information providedby a single item on test for failure or survival in that a survivalmay provide more information than a failure. The paradox isresolved using a subjective probability approach. In addition,prior distributions in the test context are chosen in order tomaximize information in a given test time.

133

TITLE: Time Series and Image Analysis

PI: R. D. MartinUniversity of WashingtonDepartment of Statistics(206) 685-2911






Technical Objective:The technical objective is the development of robust methods foranalyzing time series and spatial processes. Robustness is ofcritical importance when fitting models to real-world datacorrupted by impulsive noise.Also, the objective is to develop newmethods for feature detection in images using clustering andmathematical morphology.

Approach:The approach to robust time series analysis involves influencefunctionals and parsimonious nonparametric models. Influenc efunctionals measure the sensitivity of a parameter estimate to datacontamination. Low sensitivity implies high robustness.Parsimonious nonparametric models yield autoregressions of anysmooth function of the lagged time series. The approach to imageanalysis entails a radical expansion of the scope of clusteranalysis to clustering around typical curves and not just points.Methods of mathematical morphology will be incorporated.

Progess:Limitations of existing classification maximum likelihoodprocedures are overcome by developing a mechanism which allows thespecification of certain features (orientation, size, and shape) tobe common to all clusters while others may differ between clusters.For multivariate Gaussian data the key is a reparameterization ofthe covariance matrix in terms of its eigenvalue decomposition inorder to isolate parameters specifying orientation of principalcomponents from those which specify the size and shape of thedensity contours.

134

TITLE: Mathematical Methods for ObjectRecognition

PI: Stuart GemanBrown Un' 'rsityDivision uv Applied Mathematics(401) 863-3099






Technical Objective:This research addresses problems in computer vision with emphasison methods for object recognition and techniques for recoveringstructural information about a scene. A central issue isrepresentation, the association of an object with a mathematicalmodel which incorporates essential invariant features of the objectand which is flexible enough to represent different instances orpresentations of the object.

Approach:The approach involves the use of relational templates for invariantobject recognition and deformable templates for nonrigidone-dimensional objects, together with an analysis of genericaspects of deformable template models including their use in higherdimensions. In addition, Markov random field models will be usedfor recovery of intermediate level structure in a scene, motivitedby the "shape from shading" problem for visible light images aswell as by the problem of constructing surface maps from syntheticaperture radar images.

Progess:A new method for the automatic detection of road, railway, andriver networks in high resolution satellite imagery has beenproposed and analyzed. Detection is formulated as a discreteoptimization problem in which the cost functional is constructedfrom the output of local filters constructed to identify likelytrack pixels. In addition curvature and connectivity constraintsare enforced. The method appears to have significant potential forautomated cartographic systems.

135

TITLE: Acausal Stochastic Process Models:Structure and Inference

PI: Howard L. WeinertThe Johns Hopkins UniversityDepartment of Electrical and Computer Engineering(301) 338-7017






Technical Objective:The objective is to derive the probabilistic properties ofstochastic partial differential equations with specified boundaryconditions. It has been observed that for certain of theseequations a tranformation approach converts the estimation probleminto an estimation problem for a collection of stochastic ordinarydifferential equations. The question of characterizing those PDE'sfor which the transformation approach works will be addressed.

Approach:When the transformation of estimation problems for stochastic PDE'sinto estimation problems for a collection of stochastic ODE'sapplies, explicit smoothing algorithms will be developed based onGreen's functions, Schur transformations, and other eigenvaluedecompositions. If the transformation approach is not feasible, asin the case of nonuniformly spaced data, it is planned to addressthe discretized PDE as a multidimensional nearest neighbor model inthe expectation that this will be a useful perspective forestimation problems.

Progess:An efficient linear least squares algorithm for smoothing a twodimensional stochastic signal governed by the Poisson equation hasbeen derived for the case of a finite number of regularly spacednoisy observations. The method of complementary models is used toderive the optimal estimator, and then the sine transform is usedto convert the two dimensional problem into a family of coupled onedimensional problems which can be decoupled under various specialcase assumptions.

136

TITLE: Workshops--Pathways to the Future

PI: Lynne BillardUniversity of GeorgiaDepartment of Statistics(404) 542-5232






Technical Objective:The objective is to provide guidance in career advancement andresearch directions to recent women Ph.D.'s in statistics andprobability in order to increase research participation by women.

Approach:The approach is to hold a series of three workshops concurrent withthe major annual statistics meetings. Presentations will be madeby senior women researchers in statistics and probabilityemphasizing research and career paths. Individual researchprojects currently underway by recent doctoral recipients will bepresented and discussed. Finally, research funding opportunitieswill be described.

Progess:A very successful 1991 workshop was held. Several junior womenstatisticians each presented a short talk on her research and heardtalks from senior women statisticians and funding agencyrepresentatives. A number of "alumnae" from previous workshopsalso attended and participated. Plans for the 1992 meeting inconjunction with the Annual Joint Statistics Meetings are underway.

137

TITLE: Parameter Estimation and Model Selectionfor Gibbs Random Field in Image Analysisand Related Fields

PI: Chuanshu JiUniversity of North Carolina at Chapel HillDepartment of Statistics(919) 962-3917






Technical Objective:The objective is to develop estimators for the potential functionin Gibbs state models of one and two dimensional Markov processes.Both finitely parameterized models and infinitely parameterizedmodels under various dependence assumptions will be considered.

Approach:The approach will be to determine conditions on the functionalparameters appearing in the potential function that insureconvergence of empirical frequencies to the true underlyingvalues.Also, linear models will be assumed, and maximum likelihoodestimators for the coefficients and model order parameters will bedeveloped and their convergence properties studied.

Progess:Grenander's method of sieves has been applied to the problem ofconstructing strongly consistent estimators for certain infinitedimensional unknown parameters, such as interaction potentials,local characteristics, etc., in Gibbs random fields. Exponentialrates of consistency have been established by using the conditionalmixing property of the Gibbs random fields. This approach isapplicable to texture image models.

138

TITLE: Statistics and Oceanography

PI: Joseph B. KadaneCarnegie-Mellon UniversityDepartment of Statistics(412) 268-2717






Technical Objective:Stable random variables which describe heavy tailed phenomena willbe incorporated into time series models to provide models for dataarising in turbulence modeling. Statistical inference techniquesfor these models will be developed.

Approach:The statistical analysis of time series models based on stableinnovations will be pursued via least squares, M-estimation, andleast absolute deviations techniques with comparison among theseapproaches. Seasonal adjustment in the context of these modelswill be examined following preliminary work on the AR(1) case.Bayesian forecasting techniques for traditional ARMA models will beextended to the stable innovations case. The expansion provides fora special conference session at the annual joint statisticsmeetings on Statistics and Oceanography.

Progess:Progress has been made on the problem of specifying priordistributions in Bayesian statistics via the use of a class ofpriors wide enough to represent the diversity of reasonableopinion. New results on computing bounds on expectations permitthe use of one-dimensional Monte Carlo integrations to bound theexpectation of a random variable over a class of priors, even ifthe parameter space is multidimensional.

139

TITLE: Gaussian and Related Random Fields

PI: Raisa E. FeldmanUniversity of California, Santa BarbaraStatistics & Applied Probability Department(805) 961-2826






Technical Objective:The objective is to conduct research on Gaussian and related randomfields constructed as limits of sums of functionals of simplerblocks, e.g., intersection local times. This representationpermits the analysis of prediction problems for these fields. Inaddition, the computer simulation of field sample paths will beundertaken.

Approach:Previous work by the proposer showed how to construct random fieldsfrom the intersection local times of Markovian particle paths. Thegoal here is to extend this work to address intersection localtimes for martingales. This will be done using a combination ofthe stochastic calculus methods traditionally used to study localtimes for continuous semi-martingales with the spectral domaintechniques developed for local times of Gaussian processes.Simulations of random field sample paths will be used to provideinsight into sample path behavior.

Progess:The distribution of the first exit value has been derived for aclass of Markov processes. In particular the method provides thecharacteristic function of the first passage distribution for aclass of symmetric Levy processes which includes those withsymmetric stable distributions. The method combines predictiontheory for Gaussian processes with the theory of Dynkin whichrelates Markov processes to Gaussian processes associated withthem.

140

TITLE: Virtual Reality for Exploratory DataAnalysis

PI: Edward J. WegmanGeorge Mason UniversityCenter for Computational Statistics(703) 993-1691






Technical Objective:The objective is to develop parallel coordinate density plots as atool for graphical analysis and statistical inference on large andhigh dimensional data sets. Exploratory data analysis will bepursued via data set mapping or the selection of analysisprocedures from within a specified set automatically determined bythe nature of the data. Of particular interest is therepresentation of higher dimensional data in 3 rather than 2dimensions through the use of virtual reality technology at thehuman-computer interface.

Approach:The approach is to use computer implementations of statisticalprocedures as well as computer graphics displays of large and highdimensional data sets in order to gain mathematical insight.Formal proofs of the statistical properties of the analysistechniques will be derived. While previous work has concerned 2-Drepresentations of high dimensional data, the current effort willfocus on using virtual reality technology to understand highdimensional data sets through 3-D representations.

Progess:The grand tour, a method for searching for structure inD-dimensional data by projecting the data sequentially into allpossible 2-planes, has been generalized to the notion of ak-dimensional grand tour for k less than or equal to D. Thegeneralization is a technique to search for k-dimensionalsubstructures in the data. The basic algorithms are given and thek-dimensional parallel coordinate plot is used to representvisually the projections of the data into k-flats.

141

TITLE: Modeling and Estimation of Reciprocal Diffusionsand Gauss-Markov Random Fields

PI: Arthur J. KrenerUniversity of California, DavisDepartment of Mathematics(916) 752-0827






Technical Objective:The goal is to develop random field models and estimation methodsbased on reciprocal diffusions, a class of acausal random processesobtained as the solution of stochastic ordinary differentialequations with boundary value conditions. These will be extendedto the multiparameter case. A necessary intermediate step is toformulate the notion of transition functions for Markov fieldmodels based on Gibbs distributions.

Approach:Reciprocal processes have optimal smoothers (estimators) which arethemselves acausal in that they are solutions of dynamical systemswith boundary conditions whose input is the process to be smoothed.The approach is to develop such acausal smoothers for themultiparameter case. It is expected that transition functions ofMarkov fields can be used to construct multiparameter analogs toreciprocal processes having optimal estimators that resemble theunderlying field model.

Progess:It is shown that if a stationary multivariate Gaussian process isreciprocal over the entire real line, then it is necessarilyMarkov. The demonstration is for a discrete index set, and theregularity condition is imposed that the process is regular; i.e.is not completely predictable from its infinitely remote past andfuture. This resolves a conjecture in the literature.-

142

TITLE: Analysis of Temporal Symmetry in Non-GaussianRandom Fields

PI: Don H. JohnsonRice UniversityDepartment of Electrical & Computer Engineering(713) 527-6077






Technical Objective:The objective is to extend previous work on identifying temporalsymmetries and exploiting temporal asymmetries in estimationproblems for time series models to multiparameter random fields.In particular, the stochastic wave equation will be examined bothf or Gaussian and non-Gaussian excitation as it is expected thatnon-Gaussianity will correspond to asymmetries in space as well astime.

Approach:For time series, the P.I. has shown that conditional expectationsof the present value given past data and of the present value givenfuture data are the appropriate tool for identifying models withtemporal asymmetries and for enhanced estimation when temporal datais not confined to past information. The approach is to studyconditional expectations of random fields at a point given data atarbitrary subsets of the argument space.

Progess:Non Gaussian Narkov time series are analyzed by expressing theconditional density functions in terms of bivariate densityexpansions involving orthogonal polynomials. In contrast withprevious applications of these expansions, the orthogonal functionsdiffer for each of the two arguments, and this is the mechanism bywhich temporal asymmetry is incorporated into the model.

143

TITLE: Fitting Stochastic Partial Differential Equationsto Spatial Data

PI: Richard H. JonesUniversity of Colorado Health Sciences CenterPreventive Medicine and Biometrics(303) 270-6860






Technical Objective:The objective is to develop methods for fitting stochastic partialdifferential equation models to unequally spaced spatial data.Based on these models, techniques for predicting random fields atunobserved locations with well defined confidence intervals will bederived.

Approach:The approach is based on the observation that fitting atwo-dimensional smoothing spline corresponds to fitting the Laplaceequation driven by white noise. This correspondence will beapplied to other stochastic partial differential equations. Thefirst extension will be to incorporate trends in random fields byconcentrating the linear parameters out of the likelihood. Thenthe methodology will be extended to more complicated errorstructures.

Progess:Progress has been made on methods for fitting continuous ARMAmodels to unequally spaced observations in 2 dimensions. These aremodels with rational 2 dimensional spectra. For Gaussian inputnoise and observational errors, maximum likelihood methods are usedto estimate the ARMA parameters and the regression coefficients ofthe deterministic trend. When the number of observations is toolarge for exact maximum likelihood estimation, approximate maximumlikelihood is used based on nearest neighbors.

144

TITLE: Morphological Modeling and Analysis ofDiscrete Random Shapes

PI: John I. GoutsiasThe Johns Hopkins UniversityDepartment of Electrical and Computer Engineering(301) 338-7871






Technical Objective:The objective is to develop a unified theory for stochasticmathematical morphology particularly for discrete random shapes.Towards this end, research will be conducted on a theory of randomsets defined on discrete spaces, random set models which arecapable of describing important image characteristics, statisticalinference techniques which will allow an optimal fit of the modelsdeveloped to real image data, and image analysis techniques at theinterface of mathematical morphology and random set theory.

Approach:The approach is to define and analyze the discrete generatingfunctional for a discrete random set in terms of probabilities ofhitting each element of a family of test sets. These probabilitiescan be estimated directly from the given image data, and this isnot the case for the corresponding quantities in the older,continuous random set theory. Morphological skeletons will formthe basis for a-new approach to random shape modeling.

Progess:A general theory for the representation of discrete binary imageshas been developed. The theory is based on the generation of a setof nonoverlapping segments of an image via repeated erosions andset transformations, which in turn produces a decomposition thatguarantees exact reconstruction. The morphological imagerepresentation transform is introduced and related to various formsof digital morphological skeletons used in a variety of approachesto image representation.

145

TITLE: Probability, Statistics, and Convex Geometry

PI: Richard A. VitaleUniversity of ConnecticutDepartment of Statistics(203) 521-7727






Technical Objective:Ideas from convex geometry will be applied to stochastic problemsin order to obtain new theoretical results about statisticalprocedures and to improve the modeling of random shapes. Inparticular, notions of sphericity will be used for estimation andhypothesis testing problems on directional data. Also, new modelsof random closed curves will be developed for use in imageprocessing and shape modeling.

Approach:One approach to the development of statistical methods fordirectional data is the construction of convex structures about thedata combined with tests for sphericity of the structure. Sums ofrandom vectors selected according to various schemes sweep outflexible classes of closed curves and thereby generate randomshapes for analysis.

Progess:A novel approach from stochastic geometry is used to study theproperties of random matrices, for example expected absolutedeterminants. In particular a matrix whose columns are independentand identically distributed copies of a random vector has anexpected absolute determinant given in terms of the volume of theexpected value of a particular random set constructed from therandom vector. The result may be interpreted as a kind ofinterchange between volume evaluation and the taking ofexpectations.

146

TITLE: Applications in Random Fields and Image Analysis

PI: Loren D. PittUniversity of VirginiaMathematics Department(804) 924-4926






Technical Objective:The project will develop techniques for the prediction of randomfields. The optimal estimator is given by the solution to apartial differential equation with stochastic boundary conditionswhen the field itself is stationary Gauss-Markov. Approximatesolutions to the predictor equations will be developed.Also, anotion of shape in three dimensions based on Markov processes ongraphs will be developed and applied to create prior distributionsfor image analysis.

Approach:Approximate solutions to the stochastic prediction equations willbe analyzed from the viewpoint of stochastic approximation theoryin order to determine accuracy compared with the true optimalpredictor. Considerations of tradeoff between accuracy and costwill be investigated as a function of number of boundary datavalues incorporated into the predictor. Also, shape informationwill be analyzed via limit theorems for compact manifolds andstochastic differential equation based methods.

Progess:Progress has been made on the prediction of Gauss Markov randomfields from observations on the boundary of a general domain. Theestimator is closely related to the solution of a generalizedelliptic boundary value problem. To overcome the difficultiescaused by the irregular boundary, an equivalent boundary valueproblem has been introduced which can be solved via anapproximation scheme. In other work, families of deformations ofsmooth manifold templates are constructed using the machinery ofdifferential geometry.

147

TITLE: Theory and Applications of Random Fields

PI: Gary L. WiseThe University of Texas at AustinElectrical and Computer Engineering Department(512) 471-3356






Technical Objective:The objective is to develop mathematical techniques for theanalysis and representation of random fields. In particularprediction theory in the Wiener and Kalman formulations will beaddressed for higher dimensions, both in the standard and robustsettings.

Approach:The Wiener filter appears most amenable to generalization to higherdimensional problems via Kolmogorov's analysis. Necessary is theWold decomposition of the field and a particular factorization ofthe spectral density. In addition, dependency properties of fieldswill be studied, with special attention to the effects ofdiscretization of the parameter space on the field of interest.

Progess:The problem of minimum mean squared error estimation of a randomvariable given random field (partially ordered) observations isconsidered. Several "counterexamples" are constructed whichhighlight the role of regularity conditions in these types ofproblems. One example exhibits a random variable which can beprecisely written as a nonzero function of the random field at anypoint in space yet the corresponding estimator is zero almostsurely.

148

TITLE: Optimal Rank-Based Procedures in Time Series Analysisand Random Fields

PI: Madan L. PuriIndiana UniversityDepartment of Mathematics(812) 855-9537






Technical Objective:The technical objective is the development of methods for timeseries and spatial process analysis that are less dependent onassumptions regarding the underlying distribution of the data thanconventional model-based parametric methods. These will be morerobust against departures from normality.

Approach:The approach employs nonparametric methods, including rank-orderstatistics, central-order statistics, serial-rank statistics, andsigned-rank statistics. These will be applied to problems in timeseries and spatial process analysis, including model identificationand tests for pure randomness. Robustness of the techniques, largedeviations and Bahadur efficiencies, and small sample performancewill all be derived.

Progess:Asymptotic normality has been established for L-statistics for alarge class of time series. A new approach to the specification ofthe dependence structure known as m(n)-decomposibility avoids thedifficulties of the usual mixing conditions and is particularlyappropriate for linear, bilinear, and processes with a finite orderVolterra series expansion.

149

TITLE: Statistical Regularization of Inverse Problemsin Physical Oceanography

PI: Jose MouraCarnegie-Mellon UniversityElectrical and Computer Engineering Department(412) 268-6341






Technical Objective:The goal is to develop a stochastic relaxation approach to thereconstruction of random fields from sparse, noisy measurements.Sources of randomness include random forcing terms, random initialconditions, and random measurement noise. The field models areMarkov random fields obtained by the differencing of stochasticpartial differential equations with particular emphasis onNavier-Stokes type equations.

Approach:For the Markov random field models obtained as discretized versionsof stochastic partial differential equations, a stochastic matrixequation of particular form is to be solved for the unobservedrandom field. Techniques for fast recursive solutions which exploitthe special structure of the matrices will be developed using ideasfrom stochastic relaxation and simulated annealing.

Progess:Necessary and sufficient conditions have been obtained for theparameter space of a noncausal Gaussian Markov random field byanalysis of the eigenstructure of the inverse of the covariancematrix, known as the potential matrix. The potential matrix isconveniently decomposed as the sum of two matrices, one of whichexhibits the Markov structure in a sparse block banded form whilethe other exhibits the type of boundary conditions assumed and isalso sparse and highly structured.

150

TITLE: Probabilistic Modeling and Statistical Inference forRandom Fields and Space-Time Processes

PI: Alan S. WillskyMassachusetts Institute of TechnologyElectrical Engineering and Computer Science Department(617) 253-2356






Technical Objective:The goal is to fit multiple scale random field models from datataken at the boundary of a region based on wavelet transform ideasand hierarchical data structures. This will involve thedevelopment of techniques for multiscale estimation and inference,particularly using coarse resolution data to interpolate fineresolution fluctuations. Markov random fields on lattices will beused as hierarchical data models.

Approach:Boundary value stochastic models lead to a class of Markov randomfields (MRFs) with special structure whose parameters depend on theunderlying stochastic partial differential equation.Multiresolution features of the MRFs will be used to develop fastestimation algorithms moving inward from boundary data usingwavelet type techniques. Also, closed curves in higher dimensionswill be developed to "snake" in from the boundary data and describethe evolution of stochastic geometric features within the region ofinterest.

Progess:Recursive estimation techniques have been developed for a generalclass of linear differential equations driven by white Gaussiannoise. The special feature of this class of models is that thedimensionality of the system is variable with time. Thus thisclass of models is a precurser to the development of correspondingresults for 2-dimensional problems in which the natural directionof propagation is inward or outward from the boundary, andtherefore the boundary and the dimensionality of the problem changein size.

151

TITLE: Probabilistic Analysis of Random HyperbolicEquations

PI: Rene CarmonaUniversity of California, IrvineMathematics Department(714) 856-5506






Technical Objective:The goal is to derive Markov properties of solutions to stochasticpartial differential equations and exploit such properties in thecontext of filtering problems. The nature of the boundary withrespect to which Markov properties hold, particularly when theboundaries are themselves random, is an important issue.

Approach:A variety of approaches will be employed including randomShrodinger operators, function valued processes, eigenfunctionexpansions, and limits of discretized field approximations. Thesewill lead to an understanding of Markov properties of thestochastic pde solutions. The Markov property will be exploited insolving the filtering problem.

Progress:A particular stochastic partial differential equation of parabolictype is considered as a model for non-interacting particle systems.Existence and uniqueness of the solution are established, and aclosed form expression for the moments is obtained.Asymptotically in time, the behavior of the solution and itsmoments is intermittent, that is, there are places where theconcentration of particles is very high, the distances betweenthese places are very large, and most of the mass of the ensembleof particles is concentrated in these places.

152

TITLE: Random Fields and Stochastic PDE's inOceanography and Geophysical FluidDynamics

PI: Wojbor A. WoyczynskiCase Western Reserve UniversityDepartment of Mathematics and Statistics(216) 368-2880






Technical Objective:The objective is to conduct research on several problems instochastic modeling for oceanographic applications. In particularthese topics are: nonlinear amplitude equations for geophysicalwaves and instability of zonal flows in presence of external noise;modeling complex topography--random, fractal, and "hightemperature" boundaries for ocean basins; random vortex models ofturbulent oceanic flows and the "internal noise" randomization ofthe Navier-Stokes equation; and sparse sampling and predictionproblems for non-Gaussian random fields.

Approach:Mean field theory will be extended to 2-dimensional problems andwill be analyzed in the presence of non-Gaussian models for oceanbasins and boundaries. In addition to ocean wave propagationproblems in domains with highly irregular boundaries, the inverseproblem will be examined in this context. Random Schrodingeroperators decribe the evolution of the spectral content of themodel and will be employed in both the forward and inverse problemanalyses.

Progess:Progress has been made on an analysis of the effect of randombottom topographies on the propagation of long period, linearsurface waves, that is, topographic Rossby waves. Both ocean depthand its derivatives enter as coefficients in the governingequations so that perturbation methods which can handle smallrandom coefficients are not appropriate for the incorporation ofthe effects of the derivatives. Instead a new method is proposedto substitute an "average slope" for the random derivative terms.

153

TITLE: Random Fields Governed by StochasticPartial Differential Equations and theirApplications to Oceanography

PI: Boris L. RozovskiiUniversity of Southern CaliforniaDepartment of Mathematics(213) 740-6117






Technical Objective:The objective is to solve quasi-geostrophic type and other relatednonlinear partial differential equations when forced by a randomfield, particularly multiparameter white Gaussian noise. Questionsof steady state solutions and sample path behavior will beaddressed. Statistical estimation in the context of linearizedapproximations to these equations will be developed.

Approach:Numerical experimentation has suggested conditions on the terms inthe equations which guarantee the existence of a steady-statesolution. This part of the problem will be addressed via theconstruction of the invariant measure. Large deviation techniqueswill be used to gain insight into the sample path behavior of thesolutions. Maximum likelihood and other techniques for parameterestimation will be pursued for models with random parameters.

Progess:A new approach to the uniqueness of solutions to a class ofstochastic partial differential equations has been developed byworking in the space of signed measures rather than the space ofprobability measures. The equations studied are the Kushner andZakai equations of nonlinear filtering.

154

TITLE: Stochastic Models for Ion ChannelKinetics

PI: John A. RiceUniversity of California, BerkeleyDepartment of Statistics(415) 642-3357






Technical Objective:The objective is to derive maximum likelihood estimators forparameters such as conductance and subconductance levels from thekind of digitized data which arises in patch clamp recordings,without going through an intermediate signal reconstruction stage.In addition problems of identifiability for Markov models ofchannel kinetics will be addressed in order to study questionsrelated to the nonuniqueness of models consistent with observablerecords of patch clamp recordings.

Approach:Patch clamp recordings are modeled via a smoothed two state Markovchain with data dependent smoothing. The extension to anunderlying multistate model and the maximum likelihood estimationof the model parameters which correspond to neural features will beaddressed. For a family of such models, goodness-of-fit techniqueswill be developed and analyzed. Issues of identifiability will bepursued through analysis of the eigenstructure of the rate matrixof the underlying Markov chain.

155

TITLE: Statistics Methods for the Estimationof the Motion of Rigid Bodies

PI: Theodore ChangUniversity of VirginiaDepartment of Mathematics(804) 924-3729






Technical Objective:The objective is to develop a variety of different statisticalmethodologies for the analysis of the rotation and Euclidean motionof rigid bodies in the plane and on the surface of the sphere. Inparticular confidence bands for paths of rotations, computerintensive methods for constructing confidence regions of rotationsor Euclidean motions, robust estimation techniques, and estimationtechniques for misspecified models will be examined.

Approach:Reasonable error models for data do not lead to independentestimates of the translational and rotational part of the motion,and thus previous work on confidence bands for fitted rotationswill be extended to the case of the group of Euclidean motions ofp-dimensional space. The quality of the asymptotics used todevelop confidence regions is not well understood due to the roleof the geometry of the data points, and computer intensive methodssuch as bootstrap confidence regions should be applicable to thisclass of problems.

156

TITLE: High Level Excursions of a Random Fieldwith Application to Sea SurfaceWhitecap Modeling

PI: Tailen HsingTexas A&M UniversityDepartment of Statistics(409) 845-3160






Technical Objective:The objective is to derive expressions for high level excursionsfor various models of random fields satisfying types of dependencestructure. Analysis of these expressions will lead to techniquesfor fitting models to data, particularly satellite data of whitcapoccurrences.

ApproachcA random measure, called the excursion random measure, whichsimultaneously describes the local and global behavior of highlevel excursions of a random field will be characterized forfamilies of random fields exhibiting a dependence structuredescribed by types of mixing conditions. The fitting of models todata will be developed both nonparametrically and parametricallythrough Gaussian and stable field models.

157

TITLE: Statistical Problems in Single NeuronActivity

PI: Satish IyongarUniversity of PittsburghDepartment of Mathematics and Statistics(412) 624-8341






Technical Objective:The objective is to develop techniques f or the estimation ofphysically significant parameters in neural models from highdimensional data. Dependence between neurons in a multidimensionalpoint process setting will be incorporated.

Approach:Graphical techniques, projection pursuit methods, and specificparametric approaches will be used to estimate the dependence andrelated parameters of multiparameter point process neural models.One model of particular interest is based on level crossings ofn-dimensional Brownian motion as a potentially tractablen-dimensional diffusion process.

Progess:For a multivariate point process model of the simultaneous activityof a collection of neurons, a gravitational clustering algorithm isused to detect and describe the nature of the neural interactions.Using simulation, it has been shown that the presence of noise andof time varying interactions can affect the performance of thealgorithm, while the effect of charge dynamics appears to benegligible.

158

TITLE: Stochastic Methods for Analyzing IonChannels and Spike Train Data

PI: Grace L. YangUniversity of MarylandDepartment of Mathematics(301) 405-5173





SCIENTIFIC OFFICER: Julia Abrahais

Technical Objective:The objective is to develop statistical procedures for analyzingvoltage activated channel experiments with special emphasis onmulti channel data. A class of models given by the superpositionof several independent Markov chains will be investigated.Measurements from spike trains will be modeled through the use ofnonstationary, non Poisson point processes with refractory periods.The effects of the refractory periods in the model on the neuronfiring rate will be investigated.

Approach:A kinetic model for channel experiments based on a superposition ofMarkov chains will be fit to data by means of Le Caml's large sampleoptimal estimation procedure. For this class of models themarginal and joint distributions of neurally significant parameterssuch as open dwell time, first passage time, etc., will be derived.Identifiability of the models will be addressed, and it is expectedthat these mixture models will not necessarily be identifiable,leading to difficulties with standard parameter estimationtechniques. Appropriate modifications will be investigated.

159

TITLE: Computational Properties of AuditoryNeurons

PI: Malvin C. TeichColumbia University in the City of New YorkDepartment of Electrical Engineering(212) 854-3117






Technical Objective:The doubly stochastic Poisson process driven by fractal shot noisewhich models spontaneous neural firings will be extended toaccomodate time varying and information carrying signals as modelsfor the refractory behavior encountered in biological neurons.Techniques for evaluating the goodness-of-fit of models in theseclasses to data will be examined. Estimation techniques which areexplicitly designed to distinguish between stochastic variabilityand deterministic components to the temporal variability of thedata will be developed.

Approach:The extended model will be a triply stochastic Poisson process withthree forms of stochasticity arising from rate variationsassociated with adaptation and/or stimulus variability, abiophysical mechanism involving long term correlations, and anaction potential generation mechanism involving intrinsic neuronfluctuations and refractoriness. The moment generating functionalfor cascades of filtered Poisson point processes will form thebasis for the analysis.

160

TITLE: Particle Systems Over Stochastic Flows

PI: Erhan CinlarPrinceton UniversityDepartment of Civil Engineering & Operations Research(609) 258-5995






Technical Objective:Questions related to the theory of stochastic flows will beaddressed with emphasis on vortex approximations to stochasticflows, the effect of a flow on a system of particles drifting withthe flow, and the description of the flow from the point of view ofobservers riding on those particles. The motivation comes fromproblems in oceanography: stochastic modeling of ocean currentsand turbulent mixing, models for the motion of drifters employed indata collection, etc.

Approach:A stochastic flow is a family of random transformations F(s,t) ofa space into itself such that F(t,u) F(s,t)-F(s,u). The flowsencountered in fluid mechanics are specified via differentialequations expressing physical relationships among position,velocity, etc. For turbulent flows the velocity fields are randomfields over space and time. Flow models of particular relevance inoceanography will be analyzed by relating the model specificationto its stochastic behavior.

161

TITLE: Random Fields in Oceanography Workshop

PI: Arthur J. MarianoUniversity of MiamiMeteorology and Physical Oceanography Department(305) 361-4193






Technical Objective:The objective of this workshop is to advance the state of knowledgein the area of random fields for oceanographic modeling withemphasis on stochastic partial differential equation models andprediction and interpolation for them.

Approach:A research workshop will be held on Random Fields for OceanographicModeling. Tutorial and research presentations will be made.

162

TITLE: Computing Science and Statistics:Symposium on the Interface

PI: H. J. NewtonInterface Foundation Of North America(409) 845-3141



CONTRACT NO: 0001492J1459



Technical Objective:The objective is to advance the state of knowledge in statistics atthe interface with computing sciences with special emphasis oncross-disciplinary applications.

Approach:A reeearch conference will be held, the 1992 Symposium on theInterface of Computing Science and Statistics. Invited andcontributed papers will address visualizing stochastic algorithms,visualizing multivariate data, geographic information systems,visualization in climate research, among other topics.

163

jr&

Signal Analysis

165

TITLE: Spectral Analysis, Estimation, and Prediction ofMultiple Harmonizable Random Fields and Time Series

PI: Malempati M. RaoUniversity of California, RiversideMathematics and Computer Science Department(714) 787-5944


R&T PROJECT CODE: 411m001---08




Technical Objective:The technical objective is the development of new techniques forspectral analysis and characterization of harmonizable time seriesand random fields. Problems of detection, classification, andprediction are of particular interest.

Approach:The structure theory of harmonizable stochastic processes will beextended to random fields with emphasis on frequency domainproperties. Methods for estimation of the associated spectraldensity functions will be developed and their asymptotic propertiesderived. Rational and factorizable densities are of particularinterest. This will permit the development of techniques forprediction of isotropic random fields in n-dimensional Euclideanspace and on n-spheres.

Progress:In recent work it was shown that the concept of L(2,2)-boundednessintroduced by Bochner leads to a class of generalized harmonizablerandom processes. These generalized harmonizable processes arenoteworthy because of the straightforward nature of the linearfiltering problem for the class. A natural extension ofL(2,2)-boundedness brings in the Cramer-Karhunen processes, but forthese latter processes the linear filtering problem presentsdifficulties. Thus the work shows the key role played byL(2,2)-boundedness in linearfiltering.

167

TITLE: Model-Based Analysis and Applications of RandomSignals and Fields

PI: Rui deFigueiredoUniversity of California, IrvineElectrical & Computer -Engineering Department(714) 856-7043



CONTRACT NO: N0001491a1072



Technical Objective:The first objective is the development of improved methods fordetecting and classifying transient and nonstationary signals innoise. The second objective is the development of new stochasticmodels and estimators for the velocity field of a fluid in motion.

Approach:The approach to signal analysis uses scale space representations,the Haar transform being particularly attractive due to itscomputational simplicity. Scale space representations provide abasis for automating detection and classification, for example, bya neural network. The approach to estimating the stochasticvelocity field of a moving fluid is based on the noisyNavier-Stokes equation. For the low Reynolds number case, thenonlinear terms will be lumped with the noise and the linear partestimated using generalized splines. For the high Reynolds numbercase, a nonlinear filtering formulation will be used.

Progress:A two-stage design methodology for artificial neural networks wasintroduced. The first stage is a recursive least squares algorithmwhich determines the optimal configuration of the network. Thesecond stage involves the fine tuning of the synaptic weightsacquired during the first stage. In other work, the convergence ofan expansion method for calculating the innovation process requiredfor the solution of nonlinear filtering problems was established.

168

TITLE: Modeling, Detection and Estimation of Transientand Non-Stationary Signals

PI: Stuart C. SchwartzPrinceton UniversityDepartment of Electrical Engineering(609) 258-4618






Technical Objective:The technical objective is the development of new methods forcharacterizing nonstationary signals. These will yield newtechniques for detecting and classifying transient signals innoise.

Approach:The Gabor representation gives a signal as a series expansion in anon-orthogonal but time-localized basis. The biorthogonal functionassociated with the Gabor window greatly simplifies computation ofthe Gabor coefficients. However, an alternative maximum likelihoodapproach appears to offer estimates with lower mean-square errorwhen the signal is received in noise. For quickest detection, theCUSUM test first considered by Page will be generalized andanalyzed as a method for detecting transient signals in noise.

Progress:The CUSUM test for quickest detection of a change in thedistribution of a random sequence was generalized to arbitrarydiscriminant functions. Perfomance was derived in terms of theworst-case expected delay in detection of a change and the meantime between false alarms when no change occurs. A simple analyticformula was obtained which can be useful in comparing theperformance of different discriminant functions for variousdistributions.

169

TITLE: Characterization of Wideband Transient AcousticSignals

PI: Thomas W. ParksCornell UniversitySchool of Electrical Engineering(607) 255-7122

FUNDING AGENCY: Office of Nal Research


CONTRACTNO: N0001491J1161



Technical Objective:The technical objective is the development of more effectivemethods for detecting and classifying transient signals.

Approach:Conventional scale-space analysis and more modern wavelet transformtechniques yield filters whose bandwidth is proportional to theircenter frequency ("constant-Q filters"). Cochlear models alsoemploy constant-Q filters, but with several interestingdifferences: cochlear filters are more nearly minimum phase, groupdelay is much less for high frequencies than for low frequencies,and the magnitude of the frequency response has much sharper cutoffat the high end of the frequency band than at the low end. It isnot clear how such properties can yield improved detection andclassification in general applications.

Progress:It was shown that convolving the Wigner distribution with a generalsmoothing function is equivalent to computing a weighted sum ofspectrograms. For approximation in the least-squares sense, thebest set of windows is the set of eigenfunctions of a linearoperator determined by the smoothing function, and the weights arethe eigenvalues of the operator. When the smoothing function isthe indicator function of a rectangle in time and frequency, thelinear operator is related to the time-limiting and band-limitingoperators studied by Landau, Pollak, and Slepian.

170

TITLE: Statistical Learning Theory and Algorithms

PI: John E. MoodyYale UniversityDepartment of Computer Science(203) 432-1200


R&T PROJECT CODE: 411p005---04




Technical Objective:The two characterizing features of a neural network are itstopology and its learning algorithm. The objective here is todevelop new methods for self-organized learning.

Approach:The approach involves the k-means clustering algorithm. For agiven data set of n points, the k-means clustering algorithm findsthe set of k cluster centers (means) and associated widths(standard deviations) that minimizes the sum of the squareddistances from each data point to its cluster center. Thetechnique will be tested on the problem of predicting the chaotictime-series generated by the Mackey-Glass differential delayequation.

Progress:Many of the most popular schemes for recursive real-time learning,including the least-mean-squares (LMS), k-means clustering, andback-propagation algorithms, involve optimizating an objectivefunction via stochastic gradient descent. The learning rateschedule d(t) controls the size of the update made at time t.Fixed learning rate schedules d(t)=d yield vector estimates that donot converge but instead exhibit random residual misadjustment.The running average schedule d(t)-d/(l+t) taken from the theory ofstochastic approximation (Robbins-Monro) overcomes this shortcomingbut may yield very slow convergence. Recent work under this granthas lead to the formulation of a new adaptive learning rateschedule. The new adaptive schedule is based on the fact thatcorrelation in the update vectors indicates the learning rate istoo small. Preliminary experiments are encouraging as to thepotential for achieving optimal rates of convergence.

171

.. .. .....

TITLE: Neural Computations: Temporal Aspects

PI: Pierre BaldiNASA/Jet Propulsion LaboratoryCalifornia Institute of Technology(818) 354-9038



CONTRACT NO: N0001492MP24004



Technical Objective:The objective is to develop statistical models for the activity ofneural networks. Specific areas of interest include optimizingfeed-forward networks with a hidden layer, neural systems employingbroadly tuned sensory maps such as those found in echolocatingbats, and optimal input-output pairs for training robotic neuralnetwork controllers to handle inverse kinematics problems that areill-defined.

Approach:Feed-forward networks with a single hidden layer are modeled bysuccessive matrix multiplications where the matrix entries give theinterconnection weights. The matrices that minimize the globalerror can be expressed in terms of the principal eigenvectors ofproducts of the auto- and cross-covariances of the input anddesired output. Sensory maps with enhanced resolution are derivedin terms of discrete receptor cells with parameterized tuning. Theinverse kinematics problem is modeled in terms of rectangularmatrix inversion. The Noore-Penrose pseudo-inverse provides astarting point for the analysis.

Progress:The concept of contrastive learning was developed as a family oflearning algorithms for neural networks. During learning thenetwork oscillates between two phases. One phase has a teachersignal and one phase has no teacher signal. The weights areupdated according to a learning rule that corresponds to gradientdescent on a contrast function that measures discrepancy betweenthe free network and the network with the teacher signal. Apossible analog implementation, simulation results, and extensionswere obtained.

172

TITLE: Statistical Learning Networks

PI: Andrew R. BarronUniversity of IllinoisDepartment of Statistics(217) 333-6216






Technical Objective:The statistical properties of artificial neural networks, adaptivelearning networks, and related nonparametric estimation methodswill be derived. This will allow the performance of learningnetworks to be studied analytically. In addition, algorithms forefficient network estimation will be developed.

Approach:Sequences of network classes will be derived for which theapproximation error converges to zero. The rate of convergence ofthe error will be related to the smoothness properties of theassociated network functions.

Progress:For a class of artificial neural networks and target functionssatisfying a given smoothness condition, the mean integratedsquared error was shown to be bounded by O(1/n)+O(nd/N)logN, wheren is the number of nodes, d is the input dimension of the network,and N is the number of training observations. The twocontributions to this total are approximation and estimation error,respectively. Approximation error refers to the distance betweenthe target function and the nearest function realizable by a givenneural network architecture, while estimation error refers to thedistance between this nearest realizable function and the estimateactually generated by the network. The order of the bound isminimized with n-SQRT{N/(dlogN)}.

173

TITLE: Mathematical Models of Auditory Processing andRecognition

PI: Shihab A. ShammaUniversity of MarylandElectrical Engineering Department(301) 405-6842


R&T PROJECT CODE: 411s001---04




Technical Objective:The objective is to develop mathematical models for the processingand recognition of acoustic signals by the auditory nervous system.Both early auditory processing (front-end processing by thecochlea) and central auditory processing (extraction ofperceptually significant features by the auditory cortex) will beaddressed.

Approach:The approach is both mathematically rigorous and respectful ofauditory neurophysiology. For monaural processing, wavelettransforms will be used to model the cochlea, particularly itsability to suppress noise and enhance spectral features. Forbinaural processing, spatial correlation models employing lateralinhibition will be used, in contrast to the more traditionalidealized tap-delay line models.

Progress:Mappings of the spatial distribution of responses in the primaryauditory cortex revealed that both the gradient of the acousticspectrum and sensitivity to FH sweep direction are encoded in anorderly manner along isofrequency planes. Psychoacoustical testsalso demonstrate a potential perceptual correlate of the gradientmaps, namely the threshold stability and heightened sensitivity ofhuman subjects to the detection of changes in the symmetry ofspectral peaks.

174

TITLE: mechanisms for Bioacoustic Signal Classification

PI: Richard A. AltosChirp Corporation(619) 453-4406






Technical Objective:The objective is to develop new models for bioacoustic signalprocessing. Specific goals include creation of a statisticalframework for cochlear preprocessing, determining the relationshipbetween cochlear preprocessing and animal echolocation, waveletanalysis as a model for bioacoustic processing and as a signalprocessing tool (e.g., for the design of wideband active sonarsignals), and the role of cortical maps in signal classification.

Approach:A cochlear model is proposed in which a feedback loop is used tosynthesize a Wiener filter. Under the new model, data can beprocessed continuously (rather than in blocks) and the filterbandwidth can be readily adapted to the signal/noise environment.Wavelets will be used to design active sonar waveforms withspecified properties. An example of the utility of the approach isthe derivation of a waveform for testing a specific Doppler (scale)hypothesis. Finally, lateral inhibition and Hebbian learning willbe used to model the development of cortical maps for signaldiscrimination.

Progress:Interactive activation (IA) networks are cognitive models for datainterpretation. Competition and association of different featuresand hypotheses are represented by mutual inhibition and excitationof nodes. Interpretation of nodal activations as likelihoods orprobabilities relates IA networks to statistical hypothesistesting. A recently developed formulation of IA networks in termsof Bayesian sequential hypothesis tests shows that, whilesuboptimal, IA networks may be more robust with respect to errorsin estimated probability distributions.

175

TITLE: Bioacoustic Signal Detection and EstimationUsing Wavelet Theory

PI: Benjamin FriedlanderUniversity of California, DavisDepartment of Electrical Engineering(916) 752-7394






Technical Objective:The ability of animals such as bats, whales, and dolphins to useacoustic signals to locate and recognize objects motivates work inbioacoustic signal processing. Since the process by which animalsmanipulate and interpret acoustic signals cannot be directlyobserved, inferences may be made using modern signal processingtheory. The goal of this project is to develop detection,estimation, and classification algorithms based on models of theauditory processing mechanism.

Approach:The relationship between multiscale signal analysis and currentmodels for the peripheral processing in auditory systems motivatesan approach based on the wavelet transform. The wavelet transformdescribes a signal as a superposition of shifted andcontracted/dilated versions of an analyzing waveform ("wavelet")often chosen to have bounded support. Signal enhancement andacoustic imaging will be addressed in the wavelet domain.Comparisons to conventional spectral domain methods will beperformed using real bioacoustic data.

Progress:The problem of detecting short duration transients characterized bya model with unknown parameters was addressed. Both linear andnonlinear signal models were considered. The transients undergo alinear transform prior to the application of a detection algorithm.Examples of such transforms include the short-time Fouriertransform, the Gabor transform, and the wavelet transform. Closedform expressions were derived for the worst-case detectionperformance across all transients in a given class.

176

Title: Structure, Estimation and Prediction for RandomSequences and Fields

PI: Murray RosenblattUniversity of California, San DiegoDepartment of Mathematics(619) 534-2634






Technical Objective:The objective of the research is the development of random fieldsolutions for classes of ordinary and partial differentialequations subject to random initial and boundary conditions andforcing functions. The Burgers and Navier-Stokes equations are ofparticular interest. Techniques for prediction and estimation willbe developed.

Approach:The approach involves both the time and spectral domains. In thetime domain, the differential equations of interest will bediscretized yielding systems of difference equations. Parametricsolutions for the discrete systems will be derived, and therelation between stochastic solutions of the discrete systems andthe corresponding differential equations will be determined. Inthe spectral domain, high order cumulant spectral estimates will beused to measure nonlinear transfer of energy between differentwavenumbers in random fields.

Progress:Derived the asymptotic behavior of a nonparametric measure ofindependence for the component random variables of independent andidentically distributed bivariate random vectors. This involvedapplication of a central limit theorem for degenerate U-statistics.Asymptotic normality is proved for the case of independentcomponents.

177

Title: Matched Subspace Filtering for Detection,Estimation, and Time Series Analysis

PI: Louis L. ScharfUniversity of ColoradoElectrical & Computer Engineering Department(303) 492-8283




CURRENT END r)ATE: 30 SEP 1994


Technical Objective:The objective is to develop improved methods for signal detectionand source localization. Multirank estimators for multidimensionalspectra (e.g., frequency-wavenumber and range-depth spectra) andmatched subspace filters for matched field processing will bedeveloped.

Approach:The linear statistical model is used to derive low rankapproximants for random and deterministic signals. By matching tothe structure of this low rank subspace, a matched subspace filteris derived. The dimension of the subspace provides a mechanism forreducing noise at the expense of introducing bias, but with a netgain in signal-to-noise ratio. These concepts are particularlypowerful for known modal signals, for which the low rank subspaceis determined by the dominant modes. The problem is much moredifficult when the modes are not known, e.g., when the medium isdeterministic but modeled with error or subject to stochasticspatial/temporal variation.

Progress:Showed the relationship between low rank modeling and multiplewindow spectrum estimation via maximum likelihood estimates ofstructured covariance matrices. The power in a narrow spectralband is estimated using a low rank signal plus noise covariancemodel; This model is swept through the entire frequency band toobtain an estimate of power as a function of frequency. Theresulting spectrum estimates are given by weighted combinations ofeigenspectra. Each eigenspectrum results from projecting the dataonto an orthogonal component and squaring. The multiple windowspectrum estimates of Thomson correspond to a particular choice forthe low rank signal model.

178

Title: High Order Spectral Analysis of Random Signals

PI: Keh-Shin LiiUniversity of California, RiversideDepartment of Statistics(714) 787-3836




CURRENT END D.%TE: 30 SEP 1994


Technical Objective:The technical objective is the development of more effective highorder spectral methods for detecting, estimating, and classifyingnon-Gaussian signals in non-Gaussian noise. Of primary importanceare better methods for high order spectral density estimation.

Approach:The approach is based on high order cumulant polyspectra. K-thorder (k greater than 2) spectral density estimation will beaddressed using both non-parametric and model-based approaches. Inparticular, new non-parametric estimation procedures have beenproposed with faster rates of convergence than currently knownprocedures.

Progress:High order spectral density estimation from irregularly spaced datawas addressed. Consistency results and alias-free sampling schemeswere derived. In particular, it was shown how the information ofthe sampling process comes into play with respect to consistentestimation of the bispectrum of a continuous time process.Estimates under a Poisson sampling scheme were derived in detail.

179

Title: Numeric and Symbolic Signal Representationand Processing

PI: Alan V. OppenheimMassachusetts Institute of TechnologyResearch Laboratory of Electronics(617) 253-4177

FUNDING AGENCY: Defense Advanced Reasearch Projects Agency





Technical Objective:The technical objective is to develop and analyze numerical andsymbolic algorithms for processing and interpretation of signalsand images. This will yield an interactive computing environmentfor signal processing system design. The ultimate goal is thedevelopment of a fully automated system for signal processingsystem design.

Approach:Techniques for symbolically manipulating and evaluating signalprocessing expressions will be developed. The problem of findingfunctionally equivalent expressions that can be implemented atreduced computational cost is of particular interest. Another areaof emphasis involves automatic compilation of signal processingalgorithms, including those that use extensive linear algebra, toarchitectures such as systolic or cellular arrays. Applicationsinclude noise cancellation for speech and sonar signal enhancement.

Progress:Using maximum likelihood estimation, a wavelet-based Weiner filterwas introduced for reconstructing fractal signals received innoise. It gives a least-squares reconstruction for fractionalBrownian processes in terms of the associated sequences of dataexpansion coefficients and expected signal and noise energy atdifferent scales. The technique is applicable in a broad range ofsignal and image nrocessing settings and provides a basis for newapproaches to con&unication signal design and reception.

180

Title: Nonstationary Time Series

PI: Benjamin KedemInterdisciplinary Science Applications Incorporated(301) 468-8912






Technical Objective:The objective is to develop techniques for analyzing nonstationaryand non-Gaussian time series. Such techniques have the potentialto yield performance superior to methods based on the powerspectrum, which is only optimal for stationary Gaussian signals.

Approach:The notion of high order crossing will be extended to filteredsignals, with exponential smoothing filters of particular interest.The spectral and high order crossing crossing p::pt-..ies ofrecursively filtered random processes will be determ. -e.

Progress:The contraction mapping method for estimating narrowbandfrequencies in noise was developed, analyzed, and applied. Themethod employs parametric filters that are adaptively tuned bysuccessive zero-crossing counts or first order sampleautocorrelations yielding convergent frequency estimates. In anapplication to a well known data set, the method successfullydetected a diurnal cycle that other methods often fail to detect.

181

Title: Efficient Signal Processing Algorithms - Analysisand Implementation

PI: Bede LiuPrinceton UniversityDepartment of Electrical Engineering(609) 258-4628

FUNDING AGENCY: Naval Research Laboratory





Technical Objective:The technical objective of this effort is to develop and analyzenew algorithms and architectures for very fast on-chip signal andimage processors. These will yield improved adaptive andmultidimensional filters with payoff for surveillance andcommunications. The development of efficient techniques forperforming the a posteriori optimization required in Markov randomfield image segmentation based on the Gibbs distribution is ofparticular interest.

Approach:For least-mean-squares (LMS) adaptive filters, a design in whichthe multiplier is constrained to be a power-of-two or a sum ofpower-of-two terms will be analyzed. A second simplification to beconsidered places a power-of-two quantizer at the input to themultiplier. This reduces the multiplication to a simple shift.Similar modifications of the variable step size LKS algorithm willbe studied. For image analysis based on Gibbs-Markov randomfields, a multiple resolution segmentation approach will be used.Implementations of these algorithms in very fast low power chipswill be developed.

Progress:Adaptive smoothing filters are used for the removal of narrowbandinterference and in spectral estimation. Two popularleast-mean-square (LKS) adaptive filters, one that is constrainedto have symmetry and one in which each tap adapts independently,were evaluated. The constrained filter requires only one-half asmany multiplications as the unconstrained filter for adaptation.Moreover, the analysis shows that the constrained filter has betterconvergence characteristics than the unconstrained version.

182

Title: Statistical Inference from Sampled Data

PI: Elias MasryUniversity of California, San DiegoElectrical & Computer-Engineering Department(619) 534-3815






Technical Objective:The objective of this effort is the development of techniques forsampled data systems, including those involving probability densityestimation, filtering and prediction of stochastic signals innoise, spectrum estimation, and design and evaluation of verticalarrays for range and depth localization of a radiating source in anoisy ocean.

Approach:The statistical properties of probability density estimators basedon B-splines, the method of sieves, and truncated singular valuedecompositions will be derived in the case of noisy data. In thearea of adaptive filtering, attention will turn to the adaptivesign algorithm, a variant of Widrow's LMS algorithm that uses onlythe polarity of the output error to update the coefficientestimate. Finally, will evaluate several schemes for estimatingthe range and depth of a radiating source using a vertical array.Optimal non-uniform sampling designs that make use of the spatialcovariance structure of the noise, in contrast to theequally-spaced design, will be derived.

Progress:The Li and L2 strong consistency of recursive kernel densityestimators was established for mixing and ergodic stationaryprocesses. Rates of convergence were obtained in the L2 case formixing processes. Finally, the notion and properties of Hilbertspace valued mixingales were developed, and strong laws of largenumbers were derived.

183

Title: Signal Detection Using Polyspectra

PI: Melvin J. HinichUniversity of Texas at AustinApplied Research Laboratories(512) 835-3278






Technical Objective:The technical objective is the development of more effectivemethods for detecting, classifying, and localizing nonlinear,nonstationary, and transient signals received in additive noise.In addition, techniques for defining white noise filters yieldingoutputs with specified polyspectra will be derived.

Approach:The approach to localization involves cross-polyspectral processingof the signals received by adjacent sensors in a horizontal lineararray. The finite Fourier transform is applied to the output ofeach sensor, the cross-bispectrum and cross-trispectrum arecomputed for adjacent sensors, and the phases of thecross-polyspectra are fitted to yield estimates of the propagationdirection as a function of frequency.

Progress:A bispectral approach to deconvolution of voiced speech when thesignal is not minimum phase and the sampling is not synchronouswith the pitch frequency was introduced. Voiced speech is modeledas the output of a linear filter driven by a sequence of impulses.Results of tests on both synthetic and real speech show that thealgorithm is capable of estimating the phase as well as theamplitude of the linear filter that generates the speech waveform.

184

Title: Information and Stochastic Systems

PI: Charles R. BakerCarolinian Systems Research Corporation(919) 929-3543






Technical Objective:The technical objective is the development of new techniques fordetecting and classifying random signals and improved models forthe non-Gaussian communication channel.

Approach:Likelihood ratio tests for the detection of random signals innon-Gaussian noise will be derived using a model based on a sum offiltered independent increment processes. This model will also beconsidered in a study of multi-user communication channels.

Progress:Two new signal detection algorithms, one fully adaptive to thesignal-plus-noise (S+N) process, were evaluated using bothsimulated and passive sonar data. The algorithms make noassumptions regarding the statistical properties of the S+Nprocess, but instead require estimating the drift function of anunderlying diffusion model as described by Cramer-Hida multiplicitytheory. Comparisons with several reference algorithms show thepotential of the approach.

185

Title: Modulation and Coding Techniques for Atmosphericand Underwater Optical Communications

PI: Evaggelos GeraniotisUniversity of MarylandElectrical Engineering Department(301) 405-3646






Technical Objective:The technical objective is to develop effective methods foratmospheric and underwater optical communications. This includesthe development of effective modulation and coding designmethodologies.

Approach:The work begins with identifying the most appropriate models forthe atmospheric and underwater optical channels. Analytic andcomputational techniques for evaluating the performance of variousmodulation schemes capable of dealing with the intersymbolinterference and statistical amplitude fluctuation characteristicof each of these channels will then be derived. Finally,wavelength division schemes for random multiple-access in thefree-space setting will be developed.

Progress:Analyzed the performance of coherent dense wavelength divisionmultiple access (WDMA) schemes for optical networks. The effectsof interference from other signals due to instability in the lasercarrier frequency, phase noise, and thermal noise are all takeninto account. Dense WDMA is then coupled with direct-sequencespread-spectrum (DSSS) modulation in order to mitigate the effectsof interference from other signals. The performance resultsquantify for the first time the multiple access capability of denseWDMA and the performance advantages offered by hybrids of DSSS andWDMA.

186

Title: P-th Order Weakly Stationary Processesand their Harmonic Representation

PI: Abol G. MiameeHampton UniversityDepartment of Mathematics(804) 727-5549






Technical Objective:While there is much work relating to second order stochasticprocesses, there is little relating to general p-th orderprocesses, p less than two. Such processes provide models forimpulsive signals such as acoustic signals encountered undercracking ice in the arctic and static discharges from thunderstormsat radio frequencies. This work aims to develop the harmonicrepresentations that can serve as the basis for detection andprediction methodologies for p-th order processes.

Approach:For a second order stationary stochastic process X(t), the shiftoperator is unitary on the Hilbert space H-span{X(s)}. Thespectral theory of unitary operators then gives a harmonicrepresentation for X. For p-the order stationary processes, p lessthan two, H is a Banach space but not a Hilbert space and the shiftoperator is isometric but not unitary. Spectral representationsfor operators on Banach spaces will be used to develop harmonicrepresentations for p-th order stationary processes, p less thantwo.

Progress:The correlation-autoregressive (CAR) class of stochastic processeswas introduced. In short, a process X(n) with autocorrelationfunction R(m,n)-E{X(m)X(n)) is CAR when there exist scalars a(i),i-1,...,r, such that

rR(m,n) - SUM a(i)R(m+in+i).

i-1

Initial results include the development of a useful representationfor the correlation function of a CAR process.

187

Title: Detection and Classification of Signals andNoise with Long Memory

PI: Murad S. TaqquBoston UniversityDepartment of Mathematics(617) 353-3022






Technical Objective:Long-range dependence, evidenced by self-similar sample paths and1/f power spectral density, has been observed in many physicalphenomena, including underwater acoustic ambient noise. Thetechnical objective of this project is to develop new statisticaltests for long-range dependence and new methods for estimating theasymptotic behavior of the power spectrum near zero for stochasticprocesses with long-range dependence.

Approach:For time series with spectral density given by f(w)=C*exp{-d*ln(w)}in the neighborhood of zero, a parametric family of models, forwhich the exponent d is one of the parameters, will be derived.Robustness of the resulting maximum likelihood or method of momentsestimators will be addressed since non-Gaussian data withlong-range dependence typically obey non-central limit theoremswhose speed of convergence is a function of the degree oflong-range dependence.

Progress:While the covariance function characterizes the dependencestructure of a stationary Gaussian process, a non-Gaussiansymmetric alpha-stable (SAS) process has infinite variance and soits covariance function does not exist. It was proposed to use thefunction r(t) defined as the difference of the joint characteristicfunction of (X(0),X(t)) and the product of the characteristicfunctions of X(t) and X(0) as a measure of dependence for SASprocesses. It was shown that r(t) is particularly useful as t goesto infinity for distinguishing among the asymptotic structures.

188

~~~~~~~~~~~. . . . .. . . . . . . . . . . . . . . . . . ... .. .. ....... ... . . ..... ......... . . . . .

Title: Information Theory of Wideband Channels

PI: Sergio VerduPrinceton UniversityDepartment of Electrical Engineering(609) 258-5315






Technical Objective:The goal is to develop an information theory for multi-user randomaccess wideband channels that incorporates the cost of transmittinga single bit. This will allow the calculation of transmission ratebounds for shared optical channels and the subsequent design ofcoding strategies that yield performance approaching these bounds.

Approach:The first step is to develop a general criterion that characterizesthe most economical way to communicate reliably. Preliminary workshows that the tools of Shannon theory are inadequate. Forexample, the role of mutual information is taken over byKullback-Leibler divergence. Results on the asymptotic performanceof binary hypothesis tests can then be used in the proof of codingtheorems.

Progress:Recent work addressed multiple access channels with memory. Thecapacity region and optimal spectral densities were determined fora Gaussian linear multiple access channel where the inputs of theusers pass through respective linear systems and are thensuperimposed before being corrupted by an additive Gaussian noiseprocess. This involved the formulation of a geometric method whichcan be viewed as a generalization of the single-user water-fillingargument. In contrast to the memoryless channel, the capacityregion of the channel with memory is not in general a pentagon.

189

Title Image Processing Applications of Seamless Resampling

PI: Raoul LePageMichigan State UniversityDepartment of Statistics and Probability(517) 353-3984



CONTRACT NO: N0001491J1087.



Technical Objective:The goal is to develop new techniques for analyzing alpha-stableprocesses. Such non-Gaussian processes provide realistic modelsfor a wide range of physical phenomena, particularly those thatexhibit impulsive behavior. The development of a viable spectraltheory and associated methods for spectrum estimation are ofprimary interest. In addition, these techniques will be applied toimage processing.

Approach:The approach is based on a series representation for alpha-stableprocesses in which each term is the product of a random sign, analpha-stable point process, and a simple stationary process withclassically defined spectrum. This representation permitsevaluation of spectrum estimation by both standard kernel methodsand by a newly derived statistical resampling (bootstrap) techniquethat has been called "seamless resampling". In particular,seamless resampling facilitates estimation of the samplingdistributions of contrast deconvolution operations on noisy imagedata.

190

Title: 1992 NATO Advanced Study Institute onWavelets and their Applications

PI: James S. ByrnesPrometheus, Inc.(401) 849-5389






Technical Objective:The objective is to encourage and enhance U.S. research efforts inthe area of wavelet analysis.

Approach:Five young U.S. researchers (senior graduate students andpost-docs) will receive travel grants to the NATO Advanced StudyInstitute on Wavelets and their Applications to be held 16-29August 1992 in Ii Ciocco, Italy.

191

Title: Mathematical Algorithmic Research for FullSpectrum Sonar Signal Processing

PI: G. C. CarterNaval Underwater Systems Center Detachment, New LondonCode 104(203) 440-4476






Technical Objective:The goal is to develop signal processing techniques appropriate foruse in a stochastic acoustic wavequide. Detection and localizationare of particular interest.

Approach:The first step is to model stochastic variation in the waveguidetransfer function through the specification of random delay andattenuation coefficients in the multipath channel. In thissetting, the performance of the conventional deterministic-channellog-likelihood ratio detector (LRD) deteriorates rapidly withincreasing variance of the random multipath channel modelparameters. The appropriate random-channel LRD will be determinedand evaluated.

Progress:The likelihood ratio detector for the random multipath channel wasdeveloped and analyzed under the assumption of low signal-to-noiseratio (SNR). Using Monte Carlo techniques, the performance of thedetector was evaluated for a single source-single sensorconfiguration and channel with random multipath delay uniformlydistributed between 90 and 110 msec. The signal and noisesequences have spectra flat from 0 to 100 Hz with spectral heights1 and 10, respectively (SNR--10dB). The random multipath channellikelihood ratio detector outperforms both previously developed adhoc random channel detectors and the deterministic channellikelihood ratio detector in this setting.

192

Title: Adaptive Array Processing In UncertainInhomogeneous Media

PI: Arthur B. BaggeroerAassachusetts Institute of TechnologyDepartment of Ocean Engineering(617) 253-4336



CONTRACT AO: N0001491J1628



Technical Objective:This project addresses the most important issues encountered inbeamforming and matched field processing with large arrays in realunderwater acoustic environments. Specifically, randomenvironmental fluctuations and uncertainty limit the ability tocoherently process the acoustic field. This will be addressed bythe development of robust processors based on realistic randompropagation models.

Approach:The approach is two-fold. First, robust array processors will bedesigned using a minimax performance criteria, where the arrayweights are those which optimize performance over a range ofenvironmental parameters. Second, an inherently stochasticapproach termed stochastic beamforming will b6 developed.Specifically, random ocean acoustic propagation models will be usedto derive stochastic models for signals received at both horizontaland vertical arrays, and an appropriate det3ction theory will bedeveloped and applied to this random signal in noise problem.

Progress:An adaptive minimax matched field processor was developed whichyields the interference rejection characteristic of adaptiveprocessors while limiting the sensitivity of the processor toenvironmental rismatch. The derivation of the processor is basedon several nef theorems regarding the solutions to a class ofminimax approximation problems. The theorems lead to thedevelopment of a technique for solving for the optimal arrayweights and the associated estimate of the power emitted by thesource at the array focal point.

193

Title: Gabor Expansion/Zak Transform Classificationin Multipath Underwater Acoustics

PI: Jose' M. F. MouraCarnegie-Mellon UniversityElectrical & Computer Engineering Department(412) 268-6341

FUNDING AGENCY: Defense Advanced Reasearch Projects Agency





Technical Objective:The objective is to develop array processing techniques fordetection, localization, and classification in the underwateracoustic waveguide. Multipath propagation and aperiodic signalsare of particular interest.

Approach:The approach involves integrating matched field processing withGabor signal analysis. Matched field processing will allow theenergy in acoustic multipath arrivals to be coherently combined soas to yield maximum array gain. Subsequent processing by theGabor/Zak transform will allow analysis of complex aperiodic (e.g.,transient) signals.

Progress:A simple algorithm was developed for recovering the multipathstructure in a horizontally stratified waveguide. The soundvelocity is approximated by a multilinear profile. Though thealgorithm currently assumes ray acoustics, the extension to modalpropagation is in progress and further extensions to more complexenvironments are planned. In addition, work has begun on applyingthe Gabor and wavelet transforms to real acoustic data (theunclassified DARPA Standard Transient Data Set 1).

194

Title: Higher-Order Spectra in Signal Processing

PI: C. L. NikiasUniversity of Southern CaliforniaElectrical Engineering Department(213) 740-4445






Technical Objective:The goal is to develop end extend high order cumulant spectralanalysis methods. In particular, issues of robustness ofpolyspectral estimates, stochastic propagation effects incross-polyspectra-based array processing, and nonstationarity willbe addressed. In addition, a high order spectral theory foralpha-stable processes will be developed.

Approach:Non-parametric methods will be used to develop robust polyspectralestimators. High order polyspectral bearing estimation will beaddressed in the context of nonlinear transfer function models thathave random parameters. Wigner polyspectra will be developed andapplied to problems of nonstationary signal analysis. The notionof the "covariation" (vs covariance) function for an alpha-stable(vs Gaussian) process will be used as the basis for defining highorder spectra for alpha-stable processes.

195

Title: Matched Field Source Localization in aRandom Ocean Channel

PI: Jeffrey L. KrolikUniversity of California, San DiegoScripps Institute of Oceanography(619) 534-7925






Technical Objective:The goal is to improve the performance of sonar signal processingsystems by fully exploiting attainable knowledge of the oceanacoustic propagation channel. In particular, issues relating toacoustic propagation in the random ocean waveguide, representationof oceanographically realistic sound speed profile perturbations,and the design of rbust array processing techniques will beaddressed.

Approach:A realistic model for acoustic propagation in a random oceanchannel will provide the basis for matched field processing methodswhich are robust to perturbations in a range dependent sound speedprofile. The model combines adiabatic normal mode theory and firstorder perturbation analysis. This full wave model will be used toassess the utility of multiply-constrained maximum likelihoodbeamforming and optimal quadratic processing.

196

Title: Beamforming on Transient and Non-Stationary Signalsin Inhomogeneous Acoustic Environments and Data DrivenBoundary Value Problems

PI: Stephen L. HobbsNaval Ocean Systems CenterSignal Processing Technology Branch(619) 553-2018






Technical Objective:Array processing (beamforming) methods will be developed fornonstationary signals in realistic ocean environments. Models forsuch environments involve the use of stochastic boundary valueproblems. In this context the objective is to evaluate statisticalproperties of the random field which is obtained when the boundaryvalues are replaced by discrete, noisy observations.

Approach:Beamforming techniques will be developed for nonstationary signalsthat exhibit spectral correlation. Transient signals will beaddressed through the use of the Gabor representation andappropriate time-frequency transforms. For stochastic boundaryvalue problems, smoothing the discrete observations on the boundaryyields a continuous random boundary function. The statisticalproperties of the boundary are used to derive the statisticalproperties of the interior by the classical methods of mathematicalanalysis.

Progress:The asymptotic distribution of location estimates of a source underweak assumptions about the signal processing method and ambientnoise was derived. The location estimates are the peaks of thearray output (ambiguity surface) while the signal plus noiseprocess is assumed strictly stationary. The estimates are shown tobe consistent, asymptotically normal, and the asymptotic covariancewas obtained. Use of these statistics was illustrated by theconstruction of an approximate confidence region for the truesource location.

197

Title: Research on Nonstationary Random Processes andApplications of Spectral Coherence

PI: Harry L. HurdHarry L. Hurd, Associates, Inc.(919) 846-9227

FUNDING AGENCY: Naval Undersea Warfare Center, RI





Technical Objective:The goal is to develop new spectral analysis methodologies fornonstationary stochastic processes, particularly those that providerealistic models for transient, broadband, and chaotic signals. Inaddition, recently developed spectral coherence based analysistechniques will be refined and evaluated in tests on real worlddata.

Approach:The approach is based on the notion of spectral coherence as itarises in the context of harmonizable and periodically correlatedstochastic processes. That is, whereas the spectral representationtheorem shows that stationary stochastic processes are generated byintegrating exp(iwt) with respect to a random spectral measure B(w)that has independent increments, the spectral measure fornonstationary but harmonizable processes has correlated increments(spectral correlation). These concepts will be extended to chaotic(i.e., deterministic) signals. In particular, preliminary workshows that the output of a periodically perturbed dynamical systemmay exhibit spectral coherence as previously observed inperiodically correlated stochastic signals. The periodicperturbation may be either deterministic or random in nature. Thespectral coherence structure of such chaotic processes may serve tocharacterize them and so provide a basis for new analysismethodologies.

198

Title: Ocean Acoustic Signal Processing: AModel-Based Approach

PI: James V. CandyU.S. Department of Energy(510) 422-8675


R&T PROJECT CODE: 4119383---0l




Technical Objective:The goal is to develop methods to make it computationally feasibleto estimate ocean acoustic environmental parameters frompressure-field-only measurments on-line.

Approach:The approach is based on the normal mode model for acousticpropagation. A state-space formulation is used to provide arecursive estimator for the sound speed profile. The validity ofthe approach and approximations (e.g., piece-wise linearity of thesound speed profile) associated with the implementation of therecursive estimator (extended Kalman filter) will be determined bytests using real ocean acoustic data.

199

Title: New Time-Frequency and Time-Scale Methods forBio-Acoustical Signal Processing

PI: G. F. Boudreaux-BartelsUniversity of Rhode IslandElectrical Engineering Department(401) 792-5805






Technical Objective:The objective is to develop more effective methods for analyzingtransient and nonstationary signals received in noise. Detection,characterization, and classification are of particular interest.

Approach:The approach is based on time-frequency representations such as theWigner distribution and time-scale representations such as thewavelet transform. Connections between the two types ofrepresentations will be developed and used to create new types oftransforms with the favorable properties of each. Constant-Q(i.e., proportional bandwidth) properties, which are characteristicof auditory processing in mammals, are of particular interest.

Progress:A generalization of the Choi-Williams (time-frequency) ExponentialDistribution (ED) and a new Butterworth distribution (BUD) wereintroduced. The Generalized Exponential Distribution (GED) and BUDwere shown to satisfy all the desirable properties of the ED whilebeing superior by reducing the appearance of spurious cross-terms.In related work, the various constant-Q transforms introduced byAltes, Flandrin, and Bertrands were placed within the commonframework of hyperbolic-class distributions, characterized by scaleinvariance and hyperbolic time shift properties.

200

Division Projects

201

TITLE: Mathematical Olympiad Training Session

PI: Marcia P. SwardMathematical Association of America(202) 387-5200






Technical Objective:The objective is to prepare a team of U.S. high school students tocompete in the International Mathematical Olympiad to be held 10-22July 1992 in Moscow, Russia. Those selected will be introduced toa wide range of mathematical problems and theory at theMathematical Olympiad Training Session, 9 June to 7 July 1992 atthe U.S. Naval Academy in Annapolis, Maryland.

Approach:The selection process starts with the 400,000 high school studentswho take the annual American Mathematics Examination. Those withthe top scores (about 4,000) procede to the American InvitationalMathematics Examination, and from this group 140 participate in theUSA Mathematical Olympiad (USAMO). The 24 USAMO finalists thenparticipate in the Mathematical Olympiad Training Session, a fourweek workshop of intensive university-level training. This servesto familiarize them with more advanced mathematical problems,theory, and methodology.

Progress:In the 1991 International Mathematical Olympiad, held 12-23 July inSigtuna, Sweden, the U.S. team finished fifth with a score of 212.The U.S. team trailed the USSR (241), PRC (231), Romania (225), andGermany (222).

203

TITLE: Board on Mathematical Sciences (BMS)

PI: John E. LaveryNational Academy of SciencesBoard on Mathematical Sciences(202) 334-2421






Technical Objective:The Board on Mathematical Sciences is designed primarily to keepONR and the other federal funding agencies up-to-date on the latestdevelopments in the mathematical sciences, including pure andapplied mathematics and probability and statistics. The Boardidentifies and analyzes the most important trends and discoveries.

Approach:The Board provides information to ONR regarding new developments inmathematics and cross-disciplinary areas of Navy interest. Inaddition, it coordinates activities with the academic mathematicscommunity, federal funding agencies, and professional societies.

Progress:The Board issued the report "The Mathematical Sciences: A Resourcefor American Technology and Economic Competitiveness", NationalAcademy Press, 1991. The report traces the role of mathematicalmethods in the aircraft, computer, petroleum, automotive, andtelecommunication industries. It emphasizes the importance oftechnology, transfer from mathematicians to engineers and designers.The President's Science Advisor cited the report in his address tothe 1991 International Conference on Industrial and AppliedMathematics.

204

TITLE: Mathematical Sciences Education Board (MSEB)

PI: Ray C. ShiflettNational Academy of Sciences1.chematical Sciences Education Board(202) 334-3294

?FUNDING AGENCY: Office of Naval Research

R&T PROJECT CODE: 4110003-..05




Technical Objective:The objective of this research is to develop improved methods formathematics education. These will have payoff in all areas ofscience and so will strengthen the scientific infrastruture of theUnited States.

Approach:The MSEB will carry out research on mathematics curriculum,instruction, testing, and teacher education. The application ofcomputer technology to each of these areas, particularlyinstruction and testing, will be a major focus. A "holistic"approach, spanning all aspects of mathematics education, will beemployed.

Progress:The NRC is preparing to expand MSEB's nationwide electronic andhuman networks to include science as well as mathematics. Theprincipal networks involved are

- State Mathematics Coalitions- Mathematical Sciences Education Leadership Network- Corporate Council for Mathematics and Science Education- Alliance to Involve Minorities in Mathematics

It is the NRC's intention to develop each of these into a nationalnetwork serving both mathematics and science, rather than justmathematics as at present. The NRC effort will be headed byDr. Kenneth M. Hoffman, the former MSEB Executive Director and nowNRC Associate Executive Officer for Education.

205

TITLE: A Summer Program in Mathematics and Computer Science forAcademically Oriented Students

PI: Bernis BarnesUniversity of the District of ColumbiaDepartment of Mathematics(202) 282-3171



(CXNIACT NO: N0001492J1322

CUFR&RENT END DATE: 31 DEC 1992


Technical Objective:This effort aims to improve the United States' technology manpowerbase by creating a framework for academically talented students topursue research in mathematics, statistics, and computer science.

Approach:The PI has conducted similar programs during the ten previoussummers and the same plan will be utilized for the summer of 1992.He will first conduct a survey of eligible students that have thegreatest promise in mathematics and computer science.Approximately forty will be selected to take part in a five weeksummer research program in mathematics, statistics, and computerscience. A follow-up survey will be conducted to assess thesubsequent academic success of each student.

Progress:Thirty-seven students enrolled in the 1991 UDC Summer Program inMathematics and Computer Science and thirty-six completed itsuccessfully. The students participated in a variety of classroomactivities and made field trips to the Naval Research Laboratoryand the David Taylor Research Center. Subject areas coveredincluded geometry, algebra, computer programming (BASIC), andstatistics.

206

TITLE: Workshops to Encourage and Support Young Womenin Mathematics

PI: Jill P. MesirovAssociation for Women in Mathematics(617) 876-1111






Technical Objective:The objective is to increase the rate of retention of young womenmathematicians in research-oriented careers. This will benefit themanpower base and scientific infrastructure of the U.S.

Appriach:The approach is to hold a series of workshops concurrent with themajor annual mathematics meetings. These will bring together youngresearchers with more senior researchers. Ten young mathematicianswill make technical presentations on their current research. Themore senior participants will provide feedback and encouragement.

Progress:A workshop was held 7 JULY 1991 in conjunction with ICIAM 91 inWashington DC. The workshop featured technical presentations by 10women post-docs, a poster session with 12 women graduate students,and a panel discussion on research opportunities and support thatincluded representatives from ONR (Dr. Julia Abrahams) and NSF (Dr.Deborah Lockhart).

207

INDEX

209

INDEX

Ablowitz . . . . . . . . . . . . . . . . . . . . . . . . . 25

Babuska . . . . . . . . . . . . . .* . . . 0 0 . 0 0 . 475Baggeroer .o . . o . . o o . o . . o . . . . . o . o 193

Barnesk . o o . . . . . . . 0 . . 0 . 0 . 0 . . . . 0 75 . 20Baker . o . o o . . . . o . .o . . o . 0 . . . . . . 185Balas . o . o . o . . . o . . . . . . . . . . . 0 . 0 . . 107Baldi . o . . . . . . o . o . . . . . . o . . . . . . . 172Bank . . o . .o . . . . . . . . o . . o . . . . . . 79Barnes . .o .o . . o . . .o . o . o 206BarC on . . o . . . . . . . . . . . o o . . . . . o . o . 173Bellout . . . . o . o o o . . . . . o o o . . 0 . 0 0 . 16Billard . o . . . . . . . . . . . . . . o . o . . . . 16137Bland .i.do . . . . o o . . . o . . o . . o o . . . o . o 109Bleistein o. .. o o . o o o o o . . o . o . . .oo. 12Block . . . . . . o . . . o . o . . o . . o . o o . 32Bogart . .o o o.. . o o oo . . o . . o o o o . . . . o 43Boudreaux-Bartels o.o. o... o... o.o. 200Boyd . . . .. . o . o . oo o o . . . . . . o 114Bradley . . . . . o . . o . . . . . . . . . o o o . . 117Browning . . . . . . . . . o . . . . . . . . . o o . ... 92Byrnes . o. . . . . . . . o . . . . . . . o . o . .o. 191Candy . . . . . . . . . . o o . . o . .o o . o . . 199Canning . o . o . . o . . . . . . o o . o . . . 86Carmona . . . o. o o . . o . . o o . o . . . . . . ... 152Carter . . .. . 9 . . . . .o . . o . . o . 142Chan oo. o. oo84Chandru . . o. o. .. o.o..o..o.. .o. 116Chang . . . o. o. . . o . . . . . . . o . o o o.. 156Charnes o. o. .. oo . . o o . . o o . o o . o . ... 68Chartrand . o..o o o. .o .. . o o o .. o . . ... 42Chen . . . . . . o . .o o o . o . o . o . . . . ... 21Cheney . . . . . . . o . . . . . . . o . o.... .. .o .. 5Chernoff . . o. o . o o . . . . . o . o . . . . . o ... 130Cinlar . . o.. o o . o o . o . . o . . . . . . . . . ... 161Collopy 0. . . . . . 0 0 . . .. . . . . . . 121Cozzens . . o o.. o . . o . . . . o o . o . . . . . oo. 54Dafermos o. . . . .. .. .. o o o . . .. . o . .o 30Dearholt o. o.. o . . o o . . . . o o . . o o ... 66deFigueiredo . oo. ..... oooo.ooo.o 168Demmel o. o... *o*.. oo..o. 73Deogun o. o.... o" . 60

Devote . o o.. . o . . . . . o o o . .o. 18Duffus . . o.. o o . . . . . . o . .o o . . oo. 38Eddy . . . .- 132

Faudree . o. o . . . o .o . o o . o . . . . o . o 5Feldm~an . o. . . o.o.0.00..0 .. 0 ... 140Fiacco o. .. ooo.o.. .... 0 103

211

Foias . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Friedlander . . . . . . . . . . . . . . . . . . . . . . . . 176Garcia . . . . . . . . . . . . . . . . . . . . . . . . . . 94Geman . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Geranootis . . . . . . .. . . . . . . . . . . . . . . . . 186Glover . . . . . . . . . . . . . . . . . . . . . . . . . . 112Goldberg . . . . . . . . . . . . . . . . . . . . . . . . 99Golomb . . . . . . . . . . . . . . . . . . . . . . . . . . 49Goodman . . . . . . . . . . . . . . ........ . . . . . . . . 29Goutsias F . . . . . . . . . . . . . . . . . . . . . . 145Greengard K . . . . . . . . . . . . . . . . . . . . . 88GHozburger . . . . . . . .......... . . . . . . . . . . 23Hammer . . . . . . . . . . . . . 118Hansen . . . . . . . . . . . . . . . . . . . . . . . . . . 63Harary . . . . . . . . . . . . . . . . . . . . . . . . . 50Henshaw . . . . . . . . . . . . . . . . . . . . . . . . . 490Herron . . . . . . . . . . . . . . . . . . . . . . . . . . 36Hinich . . . . . . . . . . . . . . . . . . . . . . . . . . 184Ho . . . . . . . . . . . . . . . . . . . . . . . . . ... 102Hobbs . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Hochbaum . ........................ 108Hoffman, F . . . . . . . . . . . . . . . . . . . . . . . . 52Hoffman, K ........................ 105Hooker . . . . . . . . . . . . . . . . . . . . . . . . . 115Hsing . . . .157Hurd . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Iyengar . . . . . . . . . . . . . . . . . . . . . . . . . . 158Jacobson . ........................ 40Janowitz . . . . . . . . . . . . . . . . . . . . . . . . . 41Jawerth . . . . . . . . . . . . . . . . . . . . . . . . . . 82Jensen . . . . . . . . . . . . . . . . . . . . . . . . . . 78Ji . ... 138Johnson, C 51Johnson, D 00......... 143Jones, C o. 30Jones, R . 144Kadane . . . . . . . . . . . . . . . . . . . . . . . . . . 139Kedem . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Kennington .o...................... 119Kierstead . ........................ 36King . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Kleinman . ........................ 91Klibanov . ........................ 26Knight . . . . . . . . . . . . . . . . . . . . . . . . . . 95Knightly . ........................ 11Kreiss . . . . . . . . . . . . . . . . . . . . . . . . . . 83Krener . . . 142Kriegsuan . ........................ 14Krolik . . . . . . . . . . . . . . . . . . . . . . . . . . 196Lai . . . 0 0... 61Laskar . . .. 55

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Lavery . . . . . . . . . . . . . .• . . 0 . . 0 . 0 . 0 0 204Lawson . . . . . . . . . . . . . . . . . . . . . . . 57LePage . . . . . . . . . . .• 0 . . . . 0 . : : : 190Lia .u.li . . . . . . . . . . . . . 0 . . .0 22LiiM .c r . . . . . . . . . . . o o . * . . * . 179Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Lucierll . . . . . . . . . . . . . . . . . . . . . . . . . 20Lundgren . . . . . . . . . . . . . . . .. . . . . . 56Mackin . . . . . . . . . . . . . o . . . . . . . . . 120MaJda . . . . . . . . . . . . . . . . . . . . . . . . . . 7Malek-Madani . . . . . . . . . . . . . . . . . . . . . . . 15Mariano . . . . . . . . . . . . . . . . . . . . . . . . . . 162Martin . . .• 134paseyn i . . . . . . . . . . . . . . . . . . . . . . . . . 183McKee . .. ........................ 44

McLaughlin . . . . . . . . . . . . . . . . . . . 76McMorris . . . . . . . . . . . . . . . . . . . . . . .6Meggido, . . . 113Mesirov . . . ....................... 207Miamee . . .• 187Mitchell . . .111 ......... lMittal . . .• 127Moody . .. ........................ 171Morenoba . . . . . . . . . . . . . . . . . . . . . . . . 748Moura . . .. . 5, 194Murray . . . . . . . . . . . . . . . . . . . . . . . . 6 201Nachman . . . .17Ne~wton .• . .• 163Nikias . . .• 195Ockendon . ..................... 9Oliger . . .• 7i, 77Oppenheim . . ....................... 180Osher . .. ........................ 85Parks . .. ........................ 170Parlett . . . ....................... 74Pitt . .... 147Pluammer . . . ....................... 37P u r il . . . . . . . . . . ... .... 1 4 9Ramm . . . . . . . . . . . . . . . . . . . . . . . . ... 13Rao . . . . . . . . . . . . . . . . . . . . . . . . . ... 167Ratliff . . . ........ 0.000.0........ 104Rauch . .. ........................ 28Reid . . . . . . . . . . . . . . * . 0 . . 65Renardy . . . ....................... 31Rheinboldt . .................... 76Rice . . . . . . . . . . . . . . . . . . . . . ... 155Ringeisen . . . . . . . . . . . . . . . . . . . . . . . . . 47Rokhlin . . . . .80 .......... sRose . .. ........................ 81Rosenblatt . ....................... 177Rothblum . . ....................... 6Rozovskii . . ....................... 154

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Scharf .. . .. . . . . 178Scheinerman . . . . . . .. . . . . . . . . . . . . . . . . 39Schuetz . . . . . . . . ..... . . . . . . . . . . . 87Schultz . . . . . . . . . . . . . . . . . . . . . . . . 89Schwartz . . . .. ....... . . . . . ... . 169Schwen . . . . . . . .......... . . . . . 59

Shauma . . . . . . . . . . .. .. ..... . . . . . . . . . 174

Shiflett . . . . . . . . . .. .. .. .. ..... . . . . . 205

S ingpurwalla . oo................ o . .o . o . 1334

Solom on . . o . o o . . o . . . . . . . . o . . . .. . . . .126Solomon . . . . . o o . . . . . . 126Stuart . . . . . . 96Svobodny ooo.. 24Sward c .. . .... . . . . . . . . . . . . . . . . . 203Symes . . . . . . . . . . ... . . . . .18Szymczak .9o. .. . . .......... .3

Taqqu . . . . .......... ... . 188Tavarer . . . . . . 64Teich ....... . . . 160TrickW t . . . . .o00 . .o o .o o 72Tucker ...................... . . .. 125Verdu ..... ..... . .. 189Vitale ..... 146Walstad . . . 72Wegman . . . . . . .o . . . . ........... . 141Weinert . i . . . o . . . . . . . . . . . . . . . . . . 136Wiggins . . . ....... . . . . . . . . 27Wi1f- o .o o o . o . . 35Wi11sky . . . .. o o .o . . . . . . . . . . . 151wise o. . . . . . . o o . .. o . o o . o o o 148Wollkind o . . . . . . . . . o o . . . . . . . 10Woyczynski ... . .. ... .. .. .. ... . 153

Yang ....... . . . . . . . . . ........... 159

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