Schwarz iteration method for elliptic equation with rough media based on random sampling

TL;DR

This paper introduces a Schwarz iteration method for elliptic equations with rough media, utilizing random sampling to achieve efficient low-rank approximations for fast online computations.

Contribution

The paper presents a novel combination of Schwarz method with random sampling for low-rank approximation, improving efficiency in solving elliptic equations with rough media.

Findings

Demonstrates accuracy of the method through numerical examples

Shows robustness of the approach in various scenarios

Achieves computational efficiency in online stages

Abstract

We propose a computationally efficient Schwarz method for elliptic equations with rough media. A random sampling strategy is used to find low-rank approximations of all local solution maps in the offline stage; these maps are used to perform fast Schwarz iterations in the online stage. Numerical examples demonstrate accuracy and robustness of our approach.