V International Conference on ApproximationMethods for Design and Control

MATHEMATICAL METHODS AND

MODELING IN ENGINEERING AND LIFE

SCIENCES

http://www.unsam.edu.ar/icamdc/

November 7-9, 2018 - Buenos Aires - Argentina

Organized by:

Promoted by:

Contents

1 Administrative informations 1

Organizing committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Web page of the conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Lunch on Wednesdy 7th and Thursday 8th . . . . . . . . . . . . . . . . . . . . 2

Social event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Location of the meeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Schedule of the conference 4

3 Abstracts of the talks 6

A. Ackleh - A Model for Structured Population Dynamics with IndefiniteGrowth Rates Coupled with the Environment. . . . . . . . . . . . . . . . . 7

H.T. Banks - Inverse Problems: Aggregate Data and Approximations. . . . . . 8

J.E. Banks - Surrogate Species and Risk Assessment in Applied Ecology: APopulation Model Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

J. Bollati - Existence and Uniqueness of the p-Generalized Modified Error Func-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

J.P. Borgna - Optical Solitons in Nematic Liquid Crystals. . . . . . . . . . . . . 11

J.A. Burns - High Order Approximations for Model Reduction in PDE ControlSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

J. Cebeiro - On Novel Geometries in Compton Scattering Tomography. . . . . . 13

A.G. Flesia - Boosting Confidence in Detecting Dependent Ultradian RhythmsUsing Complex Wavelet Analysis. . . . . . . . . . . . . . . . . . . . . . . . 14

M.L. Joyner - Exploration of Model Selection Methods for Stochastic Models. . 15

F. Kojima - Inverse Problem for Production Process of a Steel Casting UsingEMAT under Extremely High Temperature. . . . . . . . . . . . . . . . . . 16

CONTENTS iii

P. Lotito - A Non Linear Filter for Traffic State Estimation. . . . . . . . . . . 17

J. Martinez - Branching Brownian Particles with Spatial Selection and the F-KPP Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

M. Perez-Millan - Algebraic Methods in Biochemical Reaction Networks. . . . . 19

J. Pinasco - How to Aggregate Experts Opinions. . . . . . . . . . . . . . . . . . 20

C.N. Rautenberg - Evolutionary Quasi-Variational Inequalities: Applications,Theory, and Numerics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

D. Rubio - Source Image Reconstruction from a Compton Camera Data. . . . . 22

E. Sachs - Recent Trends in Algorithmic Optimization. . . . . . . . . . . . . . . 23

N. Saintier - Measure-valued Solutions for Some Problems Arising in Biologyand Social Sciences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

K.L.Sutton - Modeling the Role of Social Distance in TB Transmission: Impli-cations for Interrupting Transmission in Communities. . . . . . . . . . . . 25

D. Tarzia - A Non-Classical Heat Conduction Problem with Source Dependingof the Total Heat Flux on the Boundary. . . . . . . . . . . . . . . . . . . . 26

H. Tran - An Ensemble Kalman Filtering Approach for Discrete-Time InverseOptimal Control Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

M.I. Troparevsky - Approximate Solutions to Fractional Differential Equations 28

R. White - A Physics-Based Modeling Approach to Quantifying Porosity inCortical Bone Using Ultrasonic Attenuation. . . . . . . . . . . . . . . . . 29

G. Yin - Quantifying Effects of Noise in Two Stochastic Models . . . . . . . . . 30

iv CONTENTS

1

Administrative informations

2 Administrative informations

Organizing committee

• Tom Banks - CRSC, North Carolina Sate University (USA)

• Marcela Morvidone - CEDEMA, Universidad Nacional San Martin (Argentina)

• Diana Rubio - CEDEMA, Universidad Nacional San Martin (Argentina)

• Nicolas Saintier - Facultad de Ciencias Exactas y Naturales, Universidad deBuenos Aires (Argentina)

• Maria Ines Tropravesky - Facultad de Ingenieria, Universidad de Buenos Aires(Argentina)

Web page of the conference

http://www.unsam.edu.ar/icamdc/

Lunch on Wednesdy 7th and Thursday 8th

All speakers are invited without charge (free lunch) to have a lunch at

ValientesTte Juan Domingo Peron 901

GoogleMap

Social event

We will celebrate the end of the meeting with a lunch Friday 9 at

Rodizio Puerto MaderoAv. Alicia M. de Justo 838 - Buenos Aires

GoogleMap

All speakers are invited without charge (free lunch).

Any non-speaker willing to join will have to pay.

Administrative informations 3

Location of the meeting

The meeting will take place in

Edificio VOLTA - UNSAM

Av. Pres. Roque Saenz Pena 8324to Piso, Aula 413 - 4th Floor, Room 413

(2 blocks from the obelisk)

GoogleMap

17/10/2018 Av. Pres. Roque Sáenz Peña 832 - Google Maps

https://www.google.com.ar/maps/place/Av.+Pres.+Roque+S%C3%A1enz+Pe%C3%B1a+832,+C1035AAQ+CABA/@-34.604946,-58.3790062,17z/data=!4m5!3m4!1s0x95bccace13e1d0cd:0x484923f69… 1/1

Datos del mapa ©2018 Google 100 m

Av. Pres. Roque Sáenz Peña 832

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2

Schedule of the conference

Schedule of the conference 5

Mathematical Methods and Modeling in Engineering and Life Sciences

Universidad Nacional de San Martín Edificio Volta, Av. Pres. Roque Sáenz Peña 832, C.A.B.A., Argentina

November 7-9, 2018

WEDNESDAY 7 THURSDAY 8 FRIDAY 9

8:30 – 9:00 REGISTRATION REGISTRATION

9:00 – 9:30

Tom Banks Julián Martinez Ekkehard Sachs

Inverse Problems: Aggregate Data

and Approximations

Branching Brownian particles with spatial selection and the F-KPP

equation.

Recent Trends in Algorithmic Optimization

9:30-10:00

Pablo Lotito Michael Joyner Carlos Rautenberg

A Non Linear Filter for Traffic State Estimation

Exploration of Model Selection Methods for Stochastic Models

Evolutionary Quasi-Variational

Inequalities: Applications, Theory,

and Numerics

10:00-10:30

John Banks George Yin Karen Sutton

Surrogate species and risk assessment in applied ecology: A

population model approach

Quantifying Effects of Noise in Two Stochastic Models

Modeling the Role of Social Distance in TB Transmission: Implications for Interrupting Transmission in Communities

10:30-11:00 Coffee Break Coffee Break Coffee Break

11:00-11:30

Julieta Bollati Hien Tran Javier Cebeiro

Existence and uniqueness of the p-generalized modified error function

An Ensemble Kalman Filtering Approach for Discrete-Time Inverse

Optimal Control Problems

On novel geometries in Compton Scattering Tomography

11:30-12:00

John Burns Fumio Kojima Domingo Tarzia

High Order Approximations for Model Reduction in PDE Control

Systems

Inverse problem for production process of a steel casting using EMAT

under extremely high temperature

A Non-classical Heat Conduction Problem with a Source Depending

of the Total Heat Flux on the Boundary

noon-1:00 pm Lunch Lunch LUNCH

Rodizio Puerto Madero

Av. Alicia Moreau de Justo 838

1:30-2:00

María Inés Troparevsky Diana Rubio

Approximate Solutions to Fractional Differential Equations.

Source Image Reconstruction from a Compton Camera data.

2:00-2:30

Georgina Flesia Juan Pablo Pinasco

Boosting Confidence in detecting dependent ultradian rhythms using

complex wavelet analysis How to aggregate experts opinions

Juan Pablo Borgna Azmy Ackleh

2:30-3:00 Optical solitons in nematic liquid

crystals

A Model for Structured Population

Dynamics with Indefinite Growth

Rates Coupled with the Environment

3:00-3:30 Coffee Break Coffee Break

3:30-4:00

Rebekah White Nicolas Saintier

A physics-based modeling approach to quantifying porosity in cortical bone using ultrasonic attenuation

Measure-valued solutions for some

problems arising in biology and social

sciences.

4:00-4:30

Mercedes Pérez Millán

Algebraic methods in biochemical reaction networks

3

Abstracts of the talks

Abstracts of the talks 7

A Model for Structured Population Dynamics with Indefi-nite Growth Rates Coupled with the Environment.

Azmy S. Ackleh, Robert L. Miller

University of Louisiana at Lafayette - USA

http://www.ucs.louisiana.edu/~asa5773/

A model is presented for the nonlinear dynamics of a structured population coupledwith the environment. It features a growth rate with indefinite sign, i.e., a growth ratethat may be positive or negative depending on density levels and other environmentalfactors. The model consists of a nonlinear partial differential equation for the populationdensity with nonlocal coupling to a system of ordinary differential equations for theenvironment. A novel finite difference scheme is developed to approximate weak solutionsof this model and convergence is proved. Numerical results are presented to verify theperformance of the scheme and to illustrate a broad range of potential applications.

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8 Abstracts of the talks

Inverse Problems: Aggregate Data and Approximations.

H.T. Banks

Center for Research in Scientific Computation - North Carolina State University - USA.

https://projects.ncsu.edu/crsc/htbanks/

We discuss numerous numerical approximations (state, parameter distributions, data,etc) routinely employed in using aggregate data with the Prohorov metric in inverseproblems.

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Abstracts of the talks 9

Surrogate Species and Risk Assessment in Applied Ecology:A Population Model Approach.

J.E. BanksCalifornia State University, Monterey Bay - USA

https://sites.google.com/a/csumb.edu/john-e-banks-ph-d/home

Surrogate species are often used in conservation science in order to understand risksfaced by endangered species or species that are important providers of ecosystem ser-vices. Surrogates are often chosen on the basis of convenience or similarities in physiologyor life history to species of concern, but few formal protocols exist for evaluating thechoice of surrogates. Furthermore, our ability to predict how species of concern will farewhen subjected to anthropogenic disturbances such as environmental contaminants ortoxicants is often based on potentially misleading comparisons of static toxicity tests(e.g., the LC50). Here I present an alternative approach that features matrix models pa-rameterized with life history data, applied to different assemblages of species. I describeseveral case studies using data from diverse taxa including endangered salmonids and asuite of parasitoid wasps important for biological control in agroecosystems to illustratehow we can incorporate life history traits into models in order to better understand andpredict population outcomes. The results demonstrate that we cannot always reliablyuse the response of designated surrogate species to predict the fate of similar – evenclosely related – species exposed to the same disturbances. This modelling approachreveals implications for how we assess risk and set conservation policy in both managedand natural/semi-natural ecosystems.

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10 Abstracts of the talks

Existence and Uniqueness of the p-Generalized ModifiedError Function.

J. Bollati*, M.F. Natale*, J.A. Semitiel*, D.A. Tarzia*.*Departamento de Matematica, Universidad Austral, Rosario, Argentina y CONICET.

http://www.austral.edu.ar/cienciasempresariales/conocimiento-e-impacto/

investigacion-y-transferencia/matematica/

In [1], it was studied a Stefan problem with variable thermal conductivity and a Robinboundary condition. It was obtained an explicit similarity solution in terms of what theauthors called a generalized modified error function (GME). Motivated by this article, wedefine the p-generalized modified error function (p-GME) as the solution to the followingnonlinear differential problem:

[(1 + δyp(x))y′(x)]′ + 2xy′(x) = 0 0 < x < +∞(1 + δyp(0))y′(0)γy(0) = 0

y(+∞) = 1

where δ ≥ 0, γ > 0 and p ≥ 1.

Existence and uniqueness of solution is proved by using a fixed point strategy. It isshown that the p-generalized modified error function converges to the p-modified errorfunction defined as the solution to a similar problem with a Dirichlet condition at x = 0.In both problems, for p = 1, the generalized modified error function [1] and the modifiederror function [2], studied recently in literature, are recovered.

References

[1] A.N. Ceretani, N. N. Salva, D.A. Tarzia, An exact solution to a Stefan problem withvariable thermal conductivity and a Robin boundary condition, Nonlinear Analysis: RealWorld Applications, 40 (2018), 243-259.

[2] A.N. Ceretani, N. N. Salva, D.A. Tarzia, Existence and uniqueness of the modifiederror function, Applied Mathematics Letters, 70 (2017), 14-17.

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Abstracts of the talks 11

Optical Solitons in Nematic Liquid Crystals.

Juan Pablo Borgna*, P. Panoyotaros**, D. Rial***, C. Sanchez de la Vega***.

* Centro de Matematica Aplicada - Universidad Nacional de San Martın - Argentina.** Universidad Nacional Autonoma de Mexico - Mexico*** Universidad de Buenos Aires FCEyN - Argentina

jpborg@gmail.com

We present a 2-D system that couples a Schrdinger evolution equation with a nonlinearelliptic equation, both equations model the propagation of a laser beam in a nematicliquid crystal layer under the presence of a preconditioning transverse electric field:

∂zu =1

2i∇2u+ iγ(sin2(θ0 + θ)− sin2(θ0))u,

ν∇2ψ =1

2E2

0 sin(2θ0)−1

2(E2

0 + |u|2) sin(2(θ0 + θ)).

(3.1)

This system arises in experimental devices designed by G. Assanto and collaborators.The nonlinear elliptic equation describes the response of the desviation director angleθ0+θ of the molecules after applying a laser beam of intensity u. Angle θ0 is, in average,the director angle of the molecules under only the action of the preconditioning electricfield.

As a first approach, this problem was replaced by the simpliest system

i∂zu+1

2∇2u+ 2θu = 0,

ν∇2ψ − 2qθ = −2|u|2.

From it, the authors were able to explain the physical eect that the presence of a well-located field u produces a deformation in the long-distance director angle and it avoidsthe occurrence of the expected blow up in a cubic Schrdinger equation in R2. However,from this simple system it was not possible to explain the experimentally observeddirector angle saturation effect: no matter how much the laser power u is increased,the deviation angle θ can not be aligned with the preconditioning electric field.

In this work we present an analysis of the nonlinear elliptic system (3.1), by useof nonlinear techniques. We show the existence of the local and global solutions of thesystem (3.1) and that the deviation θ of the director field, remains bounded in [0;π/2−θ0)(saturation effect). And in addition, we never require the smallness of θ.

At the end, for sufficiently large L2-norm for incident laser u we show the existenceof energy minimizing optical solitons with radial, positive and monotone profiles.

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12 Abstracts of the talks

High Order Approximations for Model Reduction in PDEControl Systems.

John A. Burns, James Cheung

Interdisciplinary Center for Applied Mathematics Virginia Tech - USA.

http://www.math.vt.edu/people/burns/

In this talk we consider hp - refinement finite element methods for approximatingcontrol systems governed by convection diffusion equations. These adaptive high ordermethods can be employed to construct low order approximations of the Riccati partialdifferential equations that arise in LQR and LQG control. We present convergence resultsand error estimates for this method. Employing high order methods one can obtainaccuracy with fewer degrees of freedom than can be achieved with mesh refinementalone. Thus, this approach may be viewed as a model reduction method for which thereare rigorous error bounds. Numerical examples are provided to illustrate the advantagesof higher order methods for optimal control and model reduction.

This work was supported in part by a gift from the Climate, Controls and Se-curity Division of the United Technologies Corporation and DARPA under contractN660011824030

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Abstracts of the talks 13

On Novel Geometries in Compton Scattering Tomography.

Javier Cebeiro

Universidad Nacional San Martin - Argentina.

In conventional imaging systems like CT or SPECT scattering is a source of degrada-tion. Consequently, second an higher order photons are considered as noise and must berejected. Compton imaging is an emerging imaging concept that exploits the connectionbetween energy and scattering angle in Compton interaction in order to obtain infor-mation of the inner structure of an object. This systems employ most of the radiationleaving the object and exhibit higher signal-to-noise ratios.

In transmission Compton Scattering Tomography, the object is irradiated using anexternal source and radiation is collected by a detector in order to reconstruct a function,i.e. a map of its electronic density. Data collected at detectors is modeled using integralson different manifolds including: broken lines, circles,arcs, etc. In order to recover thefunction of interest, an inversion strategy is used. Exact solutions for some of thisproblems have been obtained using filtered back-projection in Fourier domain, expansionin circular components, geometric inversion, Hilbert transform, etc.There are also matrixbased approximated methods like Algebraic Reconstruction Technique widely used inconventional CT.

A Compton scattering modality is defined by the geometry of its setup: relativeposition of source, object and detector, the shape of the detector array, the scanningprotocol, etc. A given setup with its associated manifold is related to a specific directmodel. The introduction of a new setup with different geometry leads to a new inverseproblem and requires finding the exact solution for it. Up to now, the exact inversionfor some proposed configurations is still unknown.

Recently, several geometries of Compton Scattering Tomography have been proposedand some strategies have been attempted in order to recover the object with an exactreconstruction formula. In this talk, we explore some of them and introduce someinversion strategies leading to exact reconstruction.

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14 Abstracts of the talks

Boosting Confidence in Detecting Dependent UltradianRhythms Using Complex Wavelet Analysis.

Ana Georgina Flesia

CIEM-Conicet and FaMAF- Universidad Nacional de Cordoba - Argentina

http://www.famaf.unc.edu.ar/~flesia/

Recently, biologists have shown self similar and oscillatory characteristics in animalbehavior time series. Aspects so different can be explained by a model with addedcomponents that include deterministic cycles (ultradian and circadian rhythms), poly-nomial tendencies and an underlying self similar process with stationary increments.Such components can be extracted from the data using wavelet analysis by selecting thetransformation appropriately. In this talk we address the issue of confidence in detectionof such components in real data. Often, estimated ultradian rhythms are reported asmean value of point estimator outputs with error bars computed with all the time seriesindependently, which do not report the behavior of periods as a function of time, nor thesynchronicity between animals rhythms. In Guzman-Flesia et al. (2017) we proposed awavelet-correlation plot to assess the synchronicity of rhythms from replicate data. Now,we discuss the wavelet cross spectrum (WCS), the normalized wavelet coherency (WCO)and wavelet based Spearman correlation of the temporal pattern of locomotor activityof Japanese quail (Coturnix coturnix) as tools for detecting circadian and ultraradianrhythms with confidence.

References

[1] D. Guzman, A.G. Flesia, P. Pellegrini, M. Aon, R. Marin,J. Kembro. (2017) Thefractal organization of circadian and ultradian rhythms in animal behavior. ScientificReports, 7: 684, 1-13.

[2] F. Kurz, J. Kembro, A.G. Flesia, A. Armoundas, S. Cortassa, M. Aon, D. Lloyd.(2016) Network dynamics: quantitative analysis of complex behavior in metabolism,organelles and cells, from experiments to models and back. Wiley InterdisciplinaryReviews: Systems Biology and Medicine.

[3] D. Guzman, P. Pellegrini, A.G. Flesia, M. Aon, R. Marin, J. Kembro. (2016) Highresolution, week-long, locomotion time series from Japanese quail in a home-box envi-ronment. Scientific Data. 3:160036. DOI: 10.1038/sdata.2016.36.

[4] J. Kembro, A.G. Flesia, R. Gleiser, M. Perillo, R. Marin. (2013) Assessment of long-range correlation in animal behavior time series: The temporal pattern of locomotoractivity of Japanese quail (Coturnix coturnix) and mosquito larva (Culex quinquefascia-tus). Physica A: Statistical Mechanics and its Applications, 392 (24).

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Abstracts of the talks 15

Exploration of Model Selection Methods for StochasticModels.

Michele L. Joyner

East Tennessee State University - USA

http://faculty.etsu.edu/joynerm/

Mathematical modeling has become an important tool in the study of physical systems.In practice, however, there may be a set of models which could describe the system. Inthe case of deterministic models, one can compare the candidate models and their fitto a given dataset using well-established methods such as the residual sum of squares(RSS), Akaike Information Criterion (AIC), etc. to determine which model is the bestamong the candidate models. Although model comparison techniques are well developedfor deterministic models, techniques are more limited when dealing with stochastic mod-els. In this talk, we extend deterministic model comparison techniques to three differentclasses of stochastic models: continuous time Markov chains (CTMC), stochastic dif-ferential equations (SDE), and random differential equations (RDE). The effectivenessand limitations of the model comparison techniques are explored using simulated dataas well as algae longitudinal growth data.

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16 Abstracts of the talks

Inverse Problem for Production Process of a Steel CastingUsing EMAT under Extremely High Temperature.

Fumio Kojima

Kobe University and Tohoku University - Japan

http://www.kojimalab.com/

Motivated by ultrasonic test using electromagnetic acoustic transducer (EMAT), anelectromagnetic-elastic coupling model is considered under extremely high temperature.Since the mechanism of the producing test signal is quite sensitive with respect to tem-perature inside the target material, it is very crucial to analyze numerical treatmentsin such coupling procedures. First, the mechanisms of transmitter-receiver EMAT aredescribed by hybrid use of a transient magnetic potential equation in the subsurface ofthe target material and a boundary integral equation on the outside material. Secondly,the dynamics of ultrasonic wave propagation is given by an elastic equation with temper-ature dependent physical parameters. The sensitivities of EMAT based nondestructivetesting method are investigated through the computational experiments. Finally, aninversion methodology for profiling the solid-liquid coexisting zone during solidificationis considered using the proposed model.

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Abstracts of the talks 17

A Non Linear Filter for Traffic State Estimation.

Pablo Lotito

Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia deBuenos Aires - CONICET - Argentina.

pablo.lotito@gmail.com

For the real-time traffic state estimation problem for motorways, we consider theunscented extension of the traditional Kalman filter. In a previous work, we considerednatural constraints on the variables, showing that estimations can be improved. Here,we propose two methods to deal with constraints and we compare the performance ofthose methods when loop detectors provide density and velocities measurements andthere are also connected vehicles providing speed measurements. The performance ofthe methods is compared in a realistic case taken from previous work on the subjectunder different scenarios.

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18 Abstracts of the talks

Branching Brownian Particles with Spatial Selection andthe F-KPP Equation.

P. Groisman, M. Jonckheere, Julian Martınez

Universidad de Buenos Aires - Argentina

mate.dm.uba.ar/~jmartine/

The F-KPP equation was introduced in 1937 as a model for the evolution of a genetictrait. This equation admits an infinite number of travelling wave solutions but onlyone of them has a physical meaning, the one with minimal velocity. In the same lineof [2]-[3]-[4], we consider a system of N interacting Branching Brownian particles andshow that the empirical cumulative distribution associated to this process converges tothe solution of the F-KPP equation. Additionally, for each N, we prove existence of avelocity for the cloud of particles. These velocities turns out to converge to the minimalone for the F-KPP, namely, a microscopic selection principle holds.

References

[1] J. Berard and J.B. Gouere. Brunet-Derrida behavior of branching-selection particlesystems on the line. Comm. Math. Phys., 298(2):323342, 2010.

[2] M. Bramson, P. Calderoni, A. De Masi, P. Ferrari, J. Lebowitz, and R. H. Schonmann.Microscopic selection principle for a diffusion-reaction equation. J. Statist. Phys., 45(5-6):905920, 1986.

[3] E. Brunet and B. Derrida. Effect of microscopic noise on front propagation. J.Statist. Phys., 103(1-2):269282, 2001.

[4] R. Durrett and D. Remenik. Brunet-derrida particle systems, free boundary problemsand Wiener-Hopf equations. Ann. Probab., 39(6):20432078, 2011.

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Abstracts of the talks 19

Algebraic Methods in Biochemical Reaction Networks.

Mercedes Perez-Millan

Facultad de Ciencias Exactas y Naturales - Universidad de Buenos Aires - CONICET -Argentina.

http://cms.dm.uba.ar/Members/mpmillan/es

Biochemical reaction networks give rise, through mass-action kinetics, to polynomialdynamical systems. Algebraic methods have proved to be a useful tool to obtain re-sults by exploiting the network structure, regardless of specific values for the kineticequations parameters. One important aspect to address are the steady states of bio-chemical systems. I will show how algebraic tools can help to determine whether certainwell structured biochemical systems have the capacity for multistationarity, and whenthen do, it is possible to explicitly find witnesses to multistationarity. This means thatwe can find parameter values and two different positive solutions for the correspondingpolynomial system of equations. I will also focus on a different aspect that can also beapproached by algebraic tools: the identifiability of the parameters by measuring a re-duced set of biologically relevant species. I will apply these results to a specific biologicalsystem: the phosphorylation signaling cascade.

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20 Abstracts of the talks

How to Aggregate Experts Opinions.

Juan Pablo Pinasco*, Lucia Pedraza*, N. Saintier** Facultad de Ciencias Exactas y Naturales - Universidad de Buenos Aires - CONICET

- Argentina.

mate.dm.uba.ar/~jpinasco/

In this talk we analyze a method of opinion pooling based on pairwise interactions.We assume that each expert has a probability measure on the possible outcomes of somesituations, and they try to find a single measure aggregating their individual estimates.

We describe different models depending on the type of probability measure involved,we show the associate kinetic equations which describe the dynamics, and we prove theconvergence to (quasi) consensus.

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Abstracts of the talks 21

Evolutionary Quasi-Variational Inequalities: Applications,Theory, and Numerics.

Carlos N. Rautenberg

Humboldt-Universitat zu Berlin — Mathematisch-Naturwissenschaftliche Fakultat —Institut fur Mathematik - Germany

http://www2.mathematik.hu-berlin.de/~rautenca/

We consider quasi-variational inequalities (QVIs), with pointwise constraints on func-tion values and its gradient, arising in recent mathematical models from physics andbiology. A semi-discretization in time is employed for the study of the problem classand the derivation of a solution algorithm. We prove convergence of the discretizationscheme, and establish existence, regularity and non-decreasing properties of the solu-tion. The talk is finalized with a report on numerical tests obtained involving differentnonlinearities and types of constraints.

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22 Abstracts of the talks

Source Image Reconstruction from a Compton CameraData.

Marcela Morvidone, Diana Rubio

Centro de Matematica Aplicada - Universidad Nacional de San Martın - Argentina.

diana.rubio@unsam.edu.ar

In this work we consider the Compton camera which proposes the use of electronic col-limation that registers the radiation emitted by the object under study using two planardetectors: a dispersion detector that deflects the radiation and an absorption detector.The detected signal is the amount of gamma photons of given energy arriving along theline joining the scattering site to the detection site, and it contains the contribution ofall the emitting sites of the radiating object situated on a circular cone sheet. We focuson a 2D version of this imaging technique that is mathematically modeled by integralsover two lines in the shape of a V. This integral transform permits an anlytic inversion,however, numerical reconstruction techniques usually yield lower image noise and higherspatial resolution than the analytical methods. For this reason, in addition to the al-gorithms of inversion based on analytical formulas, equipments use numerical methodsfor the reconstruction of the source from the acquired data. We propose a numericalalgorithm based on the finite element methods and least square errors to reconstructthe source from noisy data obtained by means of the transformation which models aCompton camera. Although a more complete analysis is required, the results presentedhere are encouraging as they show a good performance for a variety of examples.

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Abstracts of the talks 23

Recent Trends in Algorithmic Optimization.

Ekkehard Sachs

Trier University (Germany) - Lawrence Livermore National Laboratory (California,USA)

https://www.uni-trier.de/index.php?id=53766

In the past 5-10 years, the interest in algorithms for optimization problems has shiftedsignificantly. Methods that are based on first order information have become the focusof current research.

In this talk we give some motivation why this trend is taking place. Furthermore,we highlight some of the current research direction. Among those applications machinelearning represents an important area and for the analysis of algorithms we considercomplexity estimates for optimization problems.

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24 Abstracts of the talks

Measure-valued Solutions for Some Problems Arising in Bi-ology and Social Sciences.

Nicolas Saintier

Facultad de Ciencias Exactas y Naturales - Universidad de Buenos Aires - CONICET -Argentina.

mate.dm.uba.ar/~nsaintie/

In this talk I will present some results concerning the well-posedness and asymptoticbehaviour of some evolution equations arising in biology and social sciences.

In the context of social sciences I will present a model of opinion formation. Imaginea group of individuals debating over some question. As a result of interactions amongindividuals, the distribution of opinon iis not static but evolves in time. The problemis then to study the long-time behaviour of this distibution from the knowledge of theinteraction rules. This amounts to study the aymptotic behaviour of some transportequation.

In a biolgical perspective, one is interested in the evolution of the mass density ofanimals, cells ... whose movement results from internal interactions and also possiblyfrom interactions with the environment like e.g. a chemical (that can be produced by theagents themselves). This leads naturally to a system of transport/diffusion equations.

In both settings, the density of opinion or agents can be (asymptotically) singular (e.g.due to the presence of a Dirac mass). It is then natural to study the resulting (systemsof) equations in the framework of measure-valued solutions.

The results I will present were obtained in collaboration with 1) A. Acklez (Univ. ofLouisiana at Lafayette - USA) and 2) J.P. Pinasco, M. Perez (Univ. Buenos Aires -Argentina)) and A. Silva (Univ. San Luis - Argentina).

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Abstracts of the talks 25

Modeling the Role of Social Distance in TB Transmission:Implications for Interrupting Transmission in Communities.

Karyn L. Sutton

Institute for Disease Modeling, Bellevue, WA, USA.

http://idmod.org/team/bio/168

TB transmission is determined by an individual’s infectiousness throughout the courseof disease, combined with the exposure of other susceptible individuals. The course ofan individuals infectiousness remains poorly understood and is likely highly variable.Efforts to accelerate the worldwide decline in TB, the deadliest infectious disease today,are primarily focused on early diagnosis and effective chemotherapies. Mathematicalmodels, which are used to inform intervention strategies, largely assume uniform infec-tiousness and equal exposure between susceptible and infected individuals, neglectingthe potentially powerful effects of social structure. We extend an individual-based TBtransmission model to incorporate social distance, with differential exposure rates repre-senting social structures (e.g., peers or familial relationships) or physical distances (e.g.,household members, neighbors, community members). We present the re-calibrationof this model under multiple hypothesized social structures and patterns of individualinfectiousness, to produce the same TB prevalence trends in equal mixing settings. Wediscuss the compensating changes needed in TB transmission parameters to reproducethe same prevalence trends, along with implications for further recommended interven-tion strategies

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26 Abstracts of the talks

A Non-Classical Heat Conduction Problem with Source De-pending of the Total Heat Flux on the Boundary.

Mahdi Boukrouche*, Domingo Tarzia**

* Institut Camile Jordan, Universite de Lyon, France

** CONICET y Departamento de Matemtica, Univ. Austral, Rosario, Argentina.

DTarzia@austral.edu.ar

http://www.austral.edu.ar/cienciasempresariales/conocimiento-e-impacto/

investigacion-y-transferencia/matematica/

Motivated by the modeling of the temperature regulation within a medium we considerthe non-classical heat conduction equation in a semi-space n-dimensional domain D =R+×Rn−1 for which the internal energy supply depends on the total heat flux in the timevariable on the boundary S = ∂D = {0}×DRn−1, with homogeneous Dirichlet boundarycondition and an initial condition. The problem consists in finding the temperatureu = u(x, t) such that the following conditions are satisfied:

ut −∆u = −F(∫ t

0ux(0, y, s) ds

)x > 0, y ∈ Rn−1, t > 0,

u(0, y, t) = 0, y ∈ Rn−1, t > 0,

u(x, y, 0) = h(x, y), x > 0, y ∈ Rn−1.

By using a Volterra integral equation of second kind in the time variable with aparameter y ∈ Rn−1 the solution to this problem is obtained. The solution to thatVolterra integral equation is the heat flux on S , which is an additional unknown of theconsidered nonlinear problem. We show that a unique local solution exists, which canbe extended globally in time.

Finally, a one-dimensional case is studied and we obtain the explicit solution by usingthe Adomian method and we derive its properties. We must use a double inductionprinciple in order to obtain that explicit solution which is also related to the MittagLefflerfunction. Moreover, we obtain a relationship between this solution with a third orderordinary differential equation with a singular second member, with two initial conditionsat the fixed boundary and an integral boundary condition within the domain.

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Abstracts of the talks 27

An Ensemble Kalman Filtering Approach for Discrete-TimeInverse Optimal Control Problems.

Hien Tran

Department of Mathematics and Center for Research in Scientific Computation -North Carolina State University - USA.

http://www4.ncsu.edu/~tran/

https://math.sciences.ncsu.edu/people/tran/

Solving the inverse optimal control problem for discrete-time nonlinear systems re-quires the construction of a stabilizing feedback control law based on a control Lyapunovfunction (CLF). However, there are few systematic approaches available for defining ap-propriate CLFs. We propose a method that utilizes nonlinear Bayesian filtering toparameterize a quadratic CLF. In particular, we use the ensemble Kalman filter (EnKF)to estimate parameters used in defining the CLF within the control loop of the inverseoptimal control problem formulation. Using the EnKF in this setting provides a naturallink between uncertainty quantification and optimal design and control, as well as anovel and intuitive way to find the one control out of an ensemble that stabilizes thesystem the fastest. Results of the EnKF CLF procedure are demonstrated on both alinear and nonlinear test problem.

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28 Abstracts of the talks

Approximate Solutions to Fractional Differential Equations.

Maria Ines Troparevsky*, Marcela A. Fabio**, and Silvia A. Seminara*

* Facultad de Ingenieria - Universidad de Buenos Aires - Argentina.

** Centro de Matematica Aplicada, Universidad Nacional de San Martın, Argentina

mariainestro@gmail.com

Fractional calculus has achieved a great interest in the last decades since many phys-ical problems are modeled with differential equations involving fractional derivatives.Recently, the importance and usefulness of fractional differential equations involvingRiemann-Liouville, Caputo and Caputo-Fabrizio derivatives, were shown in various ar-ticles (see [1]-[3]). Fractional derivatives are nonlocal. Their definition involves integraloperators, some of them having singular kernel, and calculations related to them maynot be easy. For this reason, in addition to theoretical developments, it is important tolook for accurate numerical approximations to these operators.

In this work we propose approximate solutions to Initial Value Problems involvingCaputo and Caputo Fabrizio Fractional Derivatives. Following some previous works([4]-[5]) we combine the wavelet transform with the fractional derivatives of the waveletbasis by means of a Galerkin type scheme. The properties of the chosen basis guaranteethat the numerical scheme is simple, stable and its accuracy can be easily improved. Wepresent some numerical examples to show its performance.

References

[1] N. Al Sati, E. Karimov and S. Kerbal, Boundary value Problems for fractional hetaequation involving Caputo-Fabrizio derivative, NTMSCI 4, No. 4, 79-89, (2016).

[2] A. Atangana and A. H. Cloot, Stability and convergence of the space fractionalvariable-order Schrodinger equation, Adv. Difference Equ., 2013-80, Vol.10, (2013).

[3] A. N. Ceretani, D. A. Tarzia, Determination of two unknown thermal coefficientsthrough an inverse one-phase fractional Stefan problem. Fractional Calculus and AppliedAnalysis. Vol. 20, No. 2 (2017).

[4] M. A. Fabio and M. I. Troparevsky, An Inverse Problem for the Caputo FractionalDerivative by Means of the Wavelet Transform. Progr. Fract. Differ. Appl. AnInternational Journal. Vol. 4, No. 1,pp. 15-26, (2018).

[5] E. Serrano , M. I. Troparevsky and M. Fabio, Wavelet Projection Methods for solvingPseudodifferential Inverse Problems, International Journal of Wavelets Multiresolutionand Information Processing, Vol. 12, No. 2, (2014).

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Abstracts of the talks 29

A Physics-Based Modeling Approach to Quantifying Poros-ity in Cortical Bone Using Ultrasonic Attenuation.

Rebekah White

Department of Mathematics and Center for Research in Scientific Computation -North Carolina State University - USA.

https://math.sciences.ncsu.edu/people/rdwhite/

Osteoporosis affects the micro-architecture of cortical bone. However, quantifyingdegradation levels using ultrasonic waves has yet to be done. Here, a physics-basedmodel, known as the independent scattering approximation (ISA), is used to model fre-quency dependent ultrasonic attenuation in cortical bone. Furthermore, Bessel functionsare used to estimate the forward scattering pressure of longitudinally propagating elasticwaves. We numerically generate attenuation data for 3D samples using a finite-differencetime domain (FDTD) package. The model is then validated by fitting it to the gener-ated data under an ordinary least squares (OLS) framework. With this we establish theability to infer pore diameter and density from attenuation data. Applying this to datafrom human bone samples could allow one to quantify osteoporosis using non-invasiveultrasonic waves.

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30 Abstracts of the talks

Quantifying Effects of Noise in Two Stochastic Models.

George Yin

Department of Mathematics - Wayne State University - USA.

http://www.math.wayne.edu/~gyin/

In this talk, we reveal the noise effects of two stochastic models. The first one isconcerned with an SIR epidemic model, whereas the second one deals with safeguardingbiodiversity using protection zones. For the SIR model, we provide sharp conditionsthat are sufficient and close to necessary for the permanence, develop ergodicity of theunderlying system, and prove the transition probabilities converge in total variationnorm to the invariant measure together with a precise characterization of the supportof the invariant measure. For the biodiversity model, we answer the question of howlarge a protection zone should be so that the species in both the protection sub-regionand natural environment are able to survive. Along with deriving sufficient and almostnecessary conditions for surviving, convergence to the invariant distribution and ratesof convergence are also obtained.

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