An efficient multi-modes Monte Carlo homogenization method for random materials

TL;DR

This paper introduces a two-stage stochastic homogenization method for diffusion equations with random, oscillatory coefficients, combining spatial homogenization and multi-modes Monte Carlo to improve efficiency and accuracy.

Contribution

It presents a novel two-stage approach that separates spatial oscillation and randomness, enhancing computational efficiency in stochastic homogenization.

Findings

Proves convergence of the spatially homogenized solution to the original solution.

Establishes the optimal convergence rate of the multi-modes Monte Carlo method.

Demonstrates efficiency and accuracy through numerical experiments.

Abstract

In this paper, we propose and analyze a new stochastic homogenization method for diffusion equations with random and fast oscillatory coefficients. In the proposed method, the homogenized solutions are sought through a two-stage procedure. In the first stage, the original oscillatory diffusion equation is approximated, for each fixed random sample w, by a spatially homogenized diffusion equation with piecewise constant coefficients, resulting a random diffusion equation. In the second stage, the resulted random diffusion equation is approximated and computed by using an efficient multi-modes Monte Carlo method which only requires to solve a diffusion equation with a constant diffusion coefficient and a random right-hand side. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial fast oscillation of the solution and that caused by the randomness of the solution, so they can be overcome separately using different strategies. The convergence of the solution of the spatially homogenized equation (from the first stage) to the solution of the original random diffusion equation is established and the optimal rate of convergence is also obtained for the proposed multi-modes Monte Carlo method. Numerical experiments on some benchmark test problems for random composite materials are also presented to gauge the efficiency and accuracy of the proposed two-stage stochastic homogenization method.