Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem

TL;DR

This paper introduces an antithetic variable technique to reduce variance in the stochastic homogenization of nonlinear convex problems, improving computational efficiency in approximating homogenized energy densities.

Contribution

It demonstrates the application of antithetic variables for variance reduction in nonlinear convex stochastic homogenization, with numerical validation and analytical insights.

Findings

Variance reduction achieved in homogenization computations

Decreased computational cost at fixed accuracy

Numerical tests confirm efficiency improvements

Abstract

We consider a nonlinear convex stochastic homogenization problem, in a stationary setting. In practice, the deterministic homogenized energy density can only be approximated by a random apparent energy density, obtained by solving the corrector problem on a truncated domain. We show that the technique of antithetic variables can be used to reduce the variance of the computed quantities, and thereby decrease the computational cost at equal accuracy. This leads to an efficient approach for approximating expectations of the apparent homogenized energy density and of related quantities. The efficiency of the approach is numerically illustrated on several test cases. Some elements of analysis are also provided.