SurrogateModel Accelerated Random Search SMARS Algorithm for Global Optimization with Applications t

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Surrogate-Model Accelerated Random Search (SMARS)
Algorithm for Global Optimization with
Applications to Inverse Material Identification

• Wilkins Aquino
• Assistant Professor
• School of Civil and Environmental Engineering
• Cornell University, Ithaca, NY
• July 23rd, 2007

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Outline

• Background and Motivation
• SMARS Overview
• Example Procedure
• Simulated Examples
• Rastrigins Function
• Vibroacoustic
• Functionally Graded Material
• Concluding Remarks

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Background and Motivation

• Nondestructive (ND)/noninvasive (NI) material
  characterization is of paramount importance in
  all fields of science and engineering

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Background and Motivation (2)

• Analytical solutions of ND/NI inverse problems
  are often not possible
• Cast inverse problems as optimization problems

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Background and Motivation (3)

• Problems associated with model-updating
  optimization problems for material
  characterization
• Large parameter ranges (e.g. biological
  structures)
• Computationally expensive numerical modeling
• Typically non-convex error surfaces

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The Surrogate-Model Accelerated Random Search
(SMARS) Algorithm

• Combines random search algorithm with surrogate
  model method of optimization
• Random Search Stochastic Global Search
• Surrogate-Model Efficient Local Search
• Locate global solutions with limited function
  evaluations
• General applicability and ease of implementation
• Easy parallelization

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SMARS Algorithm(Example Procedure)
Unknown Error Surface
Optimization Error
Error Tolerance
Global Minimum
Optimization Parameter
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Initial Estimates(Random Search)
Initial Search Range
Initial Uniform Distribution of Parameter
Estimates
Optimization Error
Error Tolerance
Current Best Estimate
Optimization Parameter
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Surrogate-Model Method(Local Application)
SM Window
SM Representation
SM Estimate
Optimization Error
Error Tolerance
Optimization Parameter
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Search Poles(Random Search)
Search Pole 2 (For Diversity)
Search Pole 1 (Current Best)
Optimization Error
Error Tolerance
Optimization Parameter
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Random Search (Cont.)
Current Best Estimate
Random Estimates Centered on Search Poles
Optimization Error
Error Tolerance
Optimization Parameter
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Surrogate-Model Method (Cont.)
Suitable Solution Found
SM Window
SM Representation
SM Estimate
Optimization Error
Error Tolerance
Optimization Parameter
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Examples

• 3 simulated optimization problems
• Minimization of Rastrigins function
• Inverse characterization of viscoelasticity
  through a ND/NI testing procedure
• Inverse characterization of a functionally graded
  diffusivity through temperature measurements
• SMARS performance compared to a genetic algorithm
  and a pure random search algorithm
• All trials were repeated 10 times due to the
  stochastic nature of algorithms

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Example 1 Rastrigins Function

• Known non-convex error surface

• Global Minimum at (100,9000)
• Fixed number of function evaluations for
  performance comparisons

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Example 1 Results (1)

• Solution error (mean and standard deviation)

Solution Error
GA
SMARS
Pure RS
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Example 1 Results (2)

• Optimization error vs. function evaluations
  (mean)

Optimization Error (x105)
Function Evaluations
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Example 2 Viscoelasticity

• Non-unique parameter values
• Parameter ranges are several orders of magnitude

Vibroacoustic-Based Experiment
Plane-Strain Finite Element Model

• Fixed number of function evaluations for
  performance comparisons

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Example 2 Results (1)

• Optimization error (mean and standard deviation)

Optimization Error
GA
SMARS
Pure RS
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Example 2 Results (2)

• Optimization error vs. function evaluations
  (mean)

Optimization Error
Function Evaluations
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Example 3 Functionally Graded Diffusivity

• High-dimensional parameter space

Experiment Schematic
Functionally Graded Diffusivity
Diffusivity (m2/s)
(Interpolated with 20 linear elements)
Position (m)

• Fixed error tolerance for performance comparisons

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Example 3 Results (1)

• Number of function evaluations (mean and standard
  deviation)

Function Evaluations
GA
SMARS
Pure RS
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Example 3 Results (2)

• Optimization error vs. function evaluations
  (mean)

Optimization Error
Function Evaluations
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Conclusions

• The SMARS algorithm was found to be an efficient
  and consistent solution method to optimization
  problems with
• Multiple local minima
• Large search ranges
• Non-unique parameter values
• High-dimensional parameter spaces
• In addition the SMARS algorithm was found to
  outperform two traditional optimization
  algorithms (GA and RS) for certain problems.

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Acknowledgements

• Collaborators
• John C. Brigham, PhD Student (Cornell University)
• Sponsor
• National Institute of Biomedical Imaging and
  Bioengineering
• Cornell University

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