Distributed solution of Laplacian eigenvalue problems

TL;DR

This paper introduces a new distributed domain decomposition Ritz method for efficiently approximating Laplacian eigenvalues in large-scale, distributed computing environments, reducing communication costs while maintaining accuracy.

Contribution

A novel domain decomposition Ritz method that enables distributed eigenvalue computation with minimal inter-node communication and independent local subspace construction.

Findings

Method achieves accurate eigenvalue approximations.

Numerical validation confirms error bounds and efficiency.

Suitable for cloud and networked cluster environments.

Abstract

The purpose of this article is to approximately compute the eigenvalues of the symmetric Dirichlet Laplacian within an interval $(0,\Lambda)$. A novel domain decomposition Ritz method, partition of unity condensed pole interpolation method, is proposed. This method can be used in distributed computing environments where communication is expensive, e.g., in clusters running on cloud computing services or networked workstations. The Ritz space is obtained from local subspaces consistent with a decomposition of the domain into subdomains. These local subspaces are constructed independently of each other, using data only related to the corresponding subdomain. Relative eigenvalue error is analysed. Numerical examples on a cluster of workstations validate the error analysis and the performance of the method.