TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

TL;DR

The paper introduces TRPL+K, a new thick-restart preconditioned Lanczos method that improves convergence speed and efficiency for computing eigenvalues of large symmetric matrices, especially with clustered eigenvalues.

Contribution

It combines locally optimal restarting, preconditioning, and thick-restart techniques into a novel method with proven asymptotic quasi-optimality and superior performance.

Findings

TRPL+K outperforms existing methods in matrix-vector multiplications.

TRPL+K reduces computational time compared to state-of-the-art eigenmethods.

The method demonstrates robust convergence even with clustered eigenvalues.

Abstract

The Lanczos method is one of the standard approaches for computing a few eigenpairs of a large, sparse, symmetric matrix. It is typically used with restarting to avoid unbounded growth of memory and computational requirements. Thick-restart Lanczos is a popular restarted variant because of its simplicity and numerically robustness. However, convergence can be slow for highly clustered eigenvalues so more effective restarting techniques and the use of preconditioning is needed. In this paper, we present a thick-restart preconditioned Lanczos method, TRPL+K, that combines the power of locally optimal restarting (+K) and preconditioning techniques with the efficiency of the thick-restart Lanczos method. TRPL+K employs an inner-outer scheme where the inner loop applies Lanczos on a preconditioned operator while the outer loop augments the resulting Lanczos subspace with certain vectors from the previous restart cycle to obtain eigenvector approximations with which it thick restarts the outer subspace. We first identify the differences from various relevant methods in the literature. Then, based on an optimization perspective, we show an asymptotic global quasi-optimality of a simplified TRPL+K method compared to an unrestarted global optimal method. Finally, we present extensive experiments showing that TRPL+K either outperforms or matches other state-of-the-art eigenmethods in both matrix-vector multiplications and computational time.