On a shrink-and-expand technique for symmetric block eigensolvers

TL;DR

This paper introduces a dynamic, non-deflation-based block size adjustment technique for symmetric block eigensolvers, significantly reducing computational costs while maintaining convergence speed, demonstrated through theoretical analysis and numerical experiments.

Contribution

It proposes a novel adaptive block size adjustment method applicable to various eigensolvers, enhancing efficiency without sacrificing convergence.

Findings

Achieves 20-30% acceleration in eigensolver performance

Applicable to multiple well-known eigensolvers

Provides theoretical and numerical validation

Abstract

In symmetric block eigenvalue algorithms, such as the subspace iteration algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm, a large block size is often employed to achieve robustness and rapid convergence. However, using a large block size also increases the computational cost. Traditionally, the block size is typically reduced after convergence of some eigenpairs, known as deflation. In this work, we propose a non-deflation-based, more aggressive technique, where the block size is adjusted dynamically during the algorithm. This technique can be applied to a wide range of block eigensolvers, reducing computational cost without compromising convergence speed. We present three adaptive strategies for adjusting the block size, and apply them to four well-known eigensolvers as examples. Detailed theoretical analysis and numerical experiments are provided to illustrate the efficiency of the proposed technique. In practice, an overall acceleration of 20% to 30% is observed.