Momentum accelerated power iterations and the restarted Lanczos method

TL;DR

This paper compares the convergence properties of the restarted Lanczos method and momentum accelerated power iterations for eigenvalue problems, introduces a preconditioning technique, and validates findings with numerical tests.

Contribution

It provides a theoretical comparison of two eigenvalue algorithms, introduces a new preconditioning approach, and demonstrates its effectiveness through numerical experiments.

Findings

Momentum accelerated power iterations have faster convergence in certain regimes.

The preconditioning technique improves the efficiency of the restarted Lanczos method.

Numerical tests confirm the theoretical advantages of the proposed methods.

Abstract

In this paper we compare two methods for finding extremal eigenvalues and eigenvectors: the restarted Lanczos method and momentum accelerated power iterations. The convergence of both methods is based on ratios of Chebyshev polynomials evaluated at subdominant and dominant eigenvalues; however, the convergence is not the same. Here we compare the theoretical convergence properties of both methods, and determine the relative regimes where each is more efficient. We further introduce a preconditioning technique for the restarted Lanczos method using momentum accelerated power iterations, and demonstrate its effectiveness. The theoretical results are backed up by numerical tests on benchmark problems.