Joseph P. Castro Jr. *, Genetha Gray , Anthony Giunta , Patricia Hough , and Paul Demmie

Sandia National Laboratories: * Computational Sciences, Computational Physics/Simulation Frameworks, Validation & Uncertainty Quantification

Processes, Computational Sciences & Math

2005 SIAM Conference on Computational Science and EngineeringFebruary 13, 2005

Orlando, FL

*Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.

A New Scheme for Multifidelity A New Scheme for Multifidelity Optimization Incorporating Pattern Optimization Incorporating Pattern

Search and Space Mapping Search and Space Mapping

Low Fidelity30,000 DOF

High Fidelity800,000 DOF

Finite ElementModels of the

Same Component

Multifidelity Surrogate ModelsMultifidelity Surrogate Models

• The low-fidelity surrogate model retains many of the important features of the high-fidelity “truth” model, but is simplified in some way.– decreased physical resolution– decreased FE mesh resolution– simplified physics

• An MFO approach optimizes an inexpensive, low fidelity model while making periodic corrections using the expensive, high fidelity model.

• Works well when low-fidelity trends match high-fidelity trends.

parent nodechild node

pruned nodedislocated node

APPS Allows Us To Integrate a Low-Fidelity APPS Allows Us To Integrate a Low-Fidelity Response For Multifidelity OptimizationResponse For Multifidelity Optimization

• Pattern search is a non-gradient optimization search with pre-determined patterns.

• Asynchronous Parallel Pattern Search (APPS)*, takes advantage of non-dependent responses with very different compute times– Ideal fit for use with multifidelity optimization

• APPSPACK is open source software that implements the APPS algorithm– Does not assume homogeneous processors

(MPI implementation)

*Developed by Patricia Hough, Tamara Kolda, Virginia Torczon

• Space mapping* is a technique that maps the design space of a low fidelity model to the design space of high fidelity model such that both models result in approximately the same response.

The parameters within xH need not match the parameters within xL

Space MappingSpace Mapping** Provides a Conduit Between The Provides a Conduit Between The Design Spaces of the Low and High Fidelity Models Design Spaces of the Low and High Fidelity Models

x – design variablesR - responseP - mapping

xH

RH(xH)

high-fimodel

xL

RL(xL)

low-fimodel

*Developed by John Bandler, et. al.

xH

RL(P(xH))

mappedlow-fi model

P(xH)

xL=P(xH) RL(P(xH))RH(xH)such that

We’re using the mapping ( )H HP x x

?

The APPS/Space Mapping Scheme The APPS/Space Mapping Scheme

Outer LoopOuter Loop Inner LoopInner Loop

Low Fidelity ModelOptimizationxH

High Fidelity ModeOptimization

viaAPPSPACK

Space MappingVia NonlinearLeast Squares

Calculation

multiple

xH,f(xH

)

xH

trial

A Simple Polynomial was Used to Study A Simple Polynomial was Used to Study Space Mapping SensitivitiesSpace Mapping Sensitivities

• Ideally * , *, etc... (fH = fmapped)

• Studied the space mapping sensitivities to various inputs– # high fidelity responses used for the mapping– scaling of the mapping parameters (size of offset between the low and

high fidelity models)– starting point

• Compare the optimum found and the number of high fidelity runs required to reach the optimum

* *0 1

2 2* * * *

0 1 0 0 0 1 1 1( , )mappedf x x x x ( )H HP x x

0 12 2

0 1 0 0 0 1 1 1( , )Hf x x x x High Fidelity Model:

2 2

0 1 0 1( , )Lf x x x x Low Fidelity Model:

Mapped Space (*,*,* calculated via Least Squares):

# responses(x0,x1)

(-0.5,0.83)

Objective

Value

0.0

# Hi-Fi

Calculations

X Speed

Up

2 (-0.50, 0.83) 3.12e-10 13 3.31

4 (-0.50, 0.83) 3.12e-10 17 2.52

6 (-0.49, 0.81) 4.27e-4 27 2.52

8 (-0.48, 0.81) 1.03e-3 27 1.59

Apps only (-0.40, 0.80) 1.09e-2 43 -

~;=1 Starting Point = (-2.0,-2.0)

2 2 2 20 1 0 1 0 0 1 1( , ) ( ) ( )Lf x x x x x x 2 2

0 1 0 1( , ) ( 0.5) ( 0.83)Hf x x x x

The APPS/Space Mapping Scheme Improved The APPS/Space Mapping Scheme Improved Optimization Performance and ValueOptimization Performance and Value

Plot of Best Points Found With APPS/Space Mapping Scheme Plot of Best Points Found With APPS/Space Mapping Scheme

Polynomial Model with Polynomial Model with ~~O(1);O(1);=1=1, starting point = (-2,-2), starting point = (-2,-2)

1317272743

• In all cases the inner loop In all cases the inner loop call finds a best point with call finds a best point with the first callthe first call

• All inner loop calls beyond All inner loop calls beyond this do not find a best point this do not find a best point (APPS dominates at this (APPS dominates at this point)point)

2 20 1 0 1( , ) (0.8 0.5) (0.5 0.83)Hf x x x x 2 2

0 1 0 1( , )Lf x x x x

View of High Fidelity Design Space View of Unmapped Low Fidelity Design Space

Comparison of Design Space of High and Low Comparison of Design Space of High and Low Fidelity Polynomial Models with Fidelity Polynomial Models with , , ~~O(1);O(1);=1=1

Plot of Best Points Found With APPS/Space Mapping Scheme Plot of Best Points Found With APPS/Space Mapping Scheme

Polynomial Model with Polynomial Model with ~~O(1);O(1);=1=1, starting point = (-2,-2) starting point = (-2,-2)

47513453

• Though there is an Though there is an improvement with the inner improvement with the inner loop, the performance is not loop, the performance is not as great as with the as great as with the previous caseprevious case

•The APPS only case had The APPS only case had the best optimal value as the best optimal value as wellwell

1

Best Case: # response points = 82 calls to inner loop

2 20 1 0 1( , ) (0.8 0.5) (0.5 0.83)Hf x x x x

Approximate Inner Loop Call Locations within Hi-Fi Model

(-0.76,2.0)

(-0.8,-1.2)

1

2

•The numbered white boxes show approximately where the inner loop was called•The point in red brackets is where APPSPACK is before the inner loop call•The point in green was found by the inner loop

2

(-0.56,1.6)

(-0.61,1.25)

• A computational cost ratio of 1:240 • The low-fidelity model gives the same

general response trends as the high-fidelity model

• These factors makes these models prime candidates for multifidelity optimization

V

p

c

N

Penetrator Case: 3-D Model of Steel Earth Penetrator Striking a Penetrator Case: 3-D Model of Steel Earth Penetrator Striking a Concrete TargetConcrete Target

• Steel Penetrator composed of multiple materials entering a concrete target

• High Fidelity Model = elastic EP body– ~40 minute calculation time

• Low Fidelity Model = rigid EP body– ~10 second calculation time

Target-y

Parameter Study - Acceleration dependence on Density (Log Scale)

1.00E+05

1.00E+06

1.00E+07

1.00E+08

1.00E+09

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Density

Acc

eler

atio

n

acceleration - coarse (elastic)

acceleration-fine (rigid)

acceleration-fine (elastic)

Parameter Study - Acceleration dependence on Density

0.00E+00

2.00E+07

4.00E+07

6.00E+07

8.00E+07

1.00E+08

1.20E+08

1.40E+08

1.60E+08

1.80E+08

2.00E+08

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Density

Acc

eler

atio

n

acceleration - coarse (elastic)

acceleration-fine (rigid)

acceleration-fine (elastic)

Acceleration Comparison with Varying MeshAcceleration Comparison with Varying Mesh

• Rigid body response follows the trend of elastic body response

Fine Fine MeshMesh

Coarse Coarse MeshMesh

# Elements# Elements 78487848 11521152

Elastic Elastic Calc. Time Calc. Time

(s)(s)24002400 180180

Minimize Acceleration with Varying DensityMinimize Acceleration with Varying DensityHi-Fidelity Model = Elastic Model (Fine Mesh)Hi-Fidelity Model = Elastic Model (Fine Mesh)Low-Fidelity Model = Rigid Model (Fine Mesh)Low-Fidelity Model = Rigid Model (Fine Mesh)

• A series of calculations were done minimizing acceleration and maximizing displacement

displacement 2-3x speed up acceleration 1-2x speed up

• For this case, the APPS/Space Mapping scheme took longer to converge but a better optimum was found

• provides a type of global search capability to get past the local “noise”

Ongoing and Future WorkOngoing and Future Work

• Study spaces defined using different constraints.• Implement a generic oracle in APPSPACK.• Include a space mapping that does not require

domain spaces to be defined by the same numbers of parameters.

• Apply our multifidelity optimization schemes to some real world problems:Earth penetrator analysisGroundwater problems including well field

design & hydraulic captureCircuit system design

References and Contact InformationReferences and Contact Information

APPSPACK: Software WebsiteAPPSPACK 4.0  http://software.sandia.gov/appspack/version4.0This website includes the software itself (open-source) and instructions for

downloading, installing, and using it.  It also has a complete list of references to papers on the software development and convergence analysis. 

  DAKOTA: Software Websitehttp://endo.sandia.gov/DAKOTA ORACLE:  Overview Paper Kolda, Lewis, and Torczon, "Optimization by Direct Search: New Perspectives

on Some Classical and Modern Methods," SIAM Review, 45(3):385-482, 2003.

 SPACE MAPPING:  Bakr, Bandler, Madsen, and Sondergard, "An Introduction to Space Mapping

Technique, " Optimization & Engineering, 2:369-384, 2001.

Contact Info:Joseph Castro: jpcastr@sandia.gov