Hierarchical Coarse Basis by Randomised SVD: the Helmholtz Problem

TL;DR

This paper introduces a hierarchical Schwarz method with randomized SVD for the Helmholtz problem, enabling efficient coarse correction by decoupling the hierarchical coarse problem, thus addressing the large size issue.

Contribution

It proposes a novel hierarchical coarse correction approach combining randomized SVD and domain decomposition for Helmholtz problems.

Findings

Hierarchical coarse problem reduces computational complexity.

Decoupled solution at each hierarchy level improves efficiency.

Effective handling of oscillatory wave resolution in Helmholtz problems.

Abstract

The oscillatory waves require sufficient degrees of freedom to resolve. That restriction usually applies also to coarse problems for Schwarz methods. The resulting coarse problem is then too large. To address the issue, a new form of Schwarz methods with coarse correction is proposed for the Helmholtz problem. There are two components in the proposed form: randomised SVD of interface iteration, and hierarchical domain decomposition. The resulting coarse problem is hierarchical and can be solved in a decoupled way at each level of hierarchy.