Subspace decomposition with defect diffusion coefficient

TL;DR

This paper introduces an offline-online subspace decomposition preconditioner for multiscale elliptic diffusion problems with random defects, significantly reducing computational costs in uncertainty quantification tasks.

Contribution

It proposes a novel offline-online approach that precomputes local subspace solves for reference configurations, enabling efficient online construction for arbitrary realizations.

Findings

The preconditioner is robust across different random defect realizations.

Numerical experiments confirm the efficiency and spectral robustness of the method.

The approach significantly reduces computational costs in Monte Carlo simulations.

Abstract

Elliptic diffusion problems with multiscale heterogeneous coefficients lead to poorly conditioned discrete systems and therefore require effective preconditioning strategies. While subspace decomposition preconditioners perform well for fixed realizations of the coefficient, their repeated construction becomes prohibitively expensive in uncertainty quantification settings, particularly in Monte-Carlo simulations, where a large number of fine-scale realizations must be treated. In this study, we propose an offline-online approximation of a subspace decomposition preconditioner that exploits the localized structure of the random defects. The preconditioner is constructed from local subspace solves that are precomputed offline for a small set of reference configurations and efficiently combined online for arbitrary realizations. We analyze the spectral properties of the resulting offline-online approximation operator and confirm its robustness and efficiency through numerical experiments.