Basis adaptation and domain decomposition for steady partial differential equations with random coefficients

TL;DR

This paper introduces a combined domain decomposition and basis adaptation method to efficiently solve high-dimensional stochastic PDEs, significantly reducing computational costs while maintaining accuracy.

Contribution

It proposes a novel approach that integrates spatial domain decomposition with local basis adaptation to handle high-dimensional randomness in steady PDEs.

Findings

Accurate global solutions achieved with reduced computational effort.

Effective basis adaptation captures local stochastic behavior.

Numerical experiments validate the method's efficiency and accuracy.

Abstract

We present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate low-dimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support our construction with numerical experiments for a steady-state diffusion equation with a random spatially dependent coefficient. Our results show that highly accurate global solutions can be obtained with significantly reduced computational costs.