A Parallel Augmented Subspace Method for Eigenvalue Problems

TL;DR

This paper introduces a parallel augmented subspace method leveraging multigrid techniques to efficiently solve eigenvalue problems by decomposing them into simpler subproblems, enabling parallel computation and improved scalability.

Contribution

The paper proposes a novel parallel augmented subspace scheme that avoids orthogonalization in high dimensions, enhancing scalability and computational efficiency for eigenvalue problems.

Findings

Significant reduction in computational time demonstrated.

Parallelization enables independent solving of subproblems.

Validation through numerical examples confirms effectiveness.

Abstract

A type of parallel augmented subspace scheme for eigenvalue problems is proposed by using coarse space in the multigrid method. With the help of coarse space in multigrid method, solving the eigenvalue problem in the finest space is decomposed into solving the standard linear boundary value problems and very low dimensional eigenvalue problems. The computational efficiency can be improved since there is no direct eigenvalue solving in the finest space and the multigrid method can act as the solver for the deduced linear boundary value problems. Furthermore, for different eigenvalues, the corresponding boundary value problem and low dimensional eigenvalue problem can be solved in the parallel way since they are independent of each other and there exists no data exchanging. This property means that we do not need to do the orthogonalization in the highest dimensional spaces. This is the main aim of this paper since avoiding orthogonalization can improve the scalability of the proposed numerical method. Some numerical examples are provided to validate the proposed parallel augmented subspace method.