Multi-Level Monte Carlo approaches for numerical homogenization

TL;DR

This paper applies Multi-Level Monte Carlo methods to numerical homogenization, efficiently estimating expectations of homogenized coefficients and solutions by combining computations at various levels of RVE sizes and mesh resolutions, resulting in computational speed-ups.

Contribution

It introduces a multi-level Monte Carlo framework for numerical homogenization, optimizing the balance between computational cost and accuracy across different levels.

Findings

Achieves computational speed-up over standard Monte Carlo methods.

Demonstrates effectiveness through 1D and 2D numerical tests.

Provides guidelines for selecting levels and sample sizes.

Abstract

In this article, we study the application of Multi-Level Monte Carlo (MLMC) approaches to numerical random homogenization. Our objective is to compute the expectation of some functionals of the homogenized coefficients, or of the homogenized solutions. This is accomplished within MLMC by considering different levels of representative volumes (RVE), and, when it comes to homogenized solutions, different levels of coarse-grid meshes. Many inexpensive computations with the smallest RVE size and the largest coarse mesh are combined with fewer expensive computations performed on larger RVEs and smaller coarse meshes. We show that, by carefully selecting the number of realizations at each level, we can achieve a speed-up in the computations in comparison to a standard Monte Carlo method. Numerical results are presented both for one-dimensional and two-dimensional test-cases.