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Some variance reduction methods for numerical stochastic homogenization.

X Blanc1, C Le Bris2, F Legoll3

  • 1Université Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, 75205 Paris, France.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 23, 2016
PubMed
Summary
This summary is machine-generated.

This study explores variance reduction techniques to improve the accuracy and efficiency of numerical stochastic homogenization. These methods are crucial for solving complex microscale corrector problems in random environments, reducing computational costs.

Keywords:
elliptic partial differential equationsmathematical modelling in materials sciencestochastic homogenizationvariance reduction

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Area of Science:

  • Computational mathematics
  • Materials science
  • Scientific computing

Background:

  • Numerical stochastic homogenization is essential for determining macroscopic material behavior from microscale properties.
  • Solving corrector problems in random environments requires repeated simulations, leading to high computational costs.
  • Variance in Monte Carlo methods significantly impacts the accuracy and efficiency of these computations.

Purpose of the Study:

  • To provide an overview of recent variance reduction techniques applicable to numerical stochastic homogenization.
  • To analyze and test the effectiveness of these techniques in addressing computational challenges.
  • To enhance the accuracy and reduce the cost of approximating effective material coefficients.

Main Methods:

  • Review and presentation of various variance reduction techniques adapted from engineering sciences.
  • Application and testing of selected techniques on stochastic homogenization problems.
  • Utilizing Monte Carlo methods for empirical averaging over multiple random configurations.

Main Results:

  • Demonstration of how variance reduction techniques can mitigate accuracy and cost issues in numerical homogenization.
  • Empirical evidence supporting the utility of these methods in solving stochastic corrector problems.
  • Identification of effective strategies for improving computational efficiency.

Conclusions:

  • Variance reduction techniques offer significant benefits for numerical stochastic homogenization.
  • These methods are crucial for accurate and cost-effective computation of effective material properties.
  • Further research and application of these techniques are recommended for complex random media problems.