An offline-online strategy for multiscale problems with random defects

TL;DR

This paper introduces an offline-online computational strategy using Localized Orthogonal Decomposition to efficiently solve multiscale elliptic problems with random defects, enabling fast approximations in Monte Carlo simulations.

Contribution

It develops a novel offline-online approach leveraging periodicity and defect probability to efficiently approximate solutions with reduced computational cost.

Findings

Good coarse-scale approximation for small defect probabilities

Efficient online computation via pre-computed matrices

Numerical experiments confirm accuracy and efficiency

Abstract

In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good coarse-scale approximation of the solution for small $p$, which is illustrated by extensive numerical experiments. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.