Deflated Hermitian Lanczos Methods for Multiple Right-Hand Sides

TL;DR

This paper introduces deflated and restarted Lanczos algorithms, Lan-DR and MinRes-DR, for efficiently solving hermitian linear systems with multiple right-hand sides, improving convergence speed especially in challenging cases.

Contribution

It presents novel deflated Lanczos-based algorithms, Lan-DR and MinRes-DR, tailored for hermitian systems, with specific implementations for positive definite and indefinite cases, and demonstrates their effectiveness.

Findings

Significant speed-up in convergence for hermitian systems.

Effective eigenvector computation during linear system solving.

Improved performance for Wilson fermions at kappa critical.

Abstract

A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple right-hand sides, is described. For the first right-hand side, eigenvectors with small eigenvalues are computed while simultaneously solving the linear system. Two versions of this algorithm are given. The first is called Lan-DR and is based on conjugate gradient (CG) implementation of the Lanczos algorithm. This version will be optimal for the hermitian positive definite case. The second version is called MinRes-DR and is based on the minimum residual (MinRes) implementation of Lanczos algorithm. This version is optimal for indefinite hermitian systems where the CG algorithm is subject to instabilities. For additional right-hand sides, we project over the calculated eigenvectors to speed up convergence. The algorithms used for subsequent right-hand sides are called D-CG and D-MinRes respectively. After some introductory examples are given, we show tests for the case of Wilson fermions at kappa critical. A considerable speed up in the convergence is observed compared to unmodified CG and MinRes.