An improved Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves

TL;DR

This paper introduces IFEAST, an inexact Krylov-based variation of the FEAST eigenvalue algorithm, which efficiently solves large eigenvalue problems by combining Krylov subspace methods with parallel contour integration.

Contribution

The paper presents IFEAST, a novel inexact Krylov eigenvalue solver that leverages iterative solutions of shifted systems within FEAST, enabling large Krylov subspaces without storing bases.

Findings

IFEAST is mathematically equivalent to a block Krylov method.

It can solve large eigenproblems using only matrix-vector multiplications.

Numerical examples demonstrate its effectiveness and parallelism.

Abstract

The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration in the complex plane to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user-defined search interval. By computing small numbers of eigenvalues in specific regions of the complex plane, FEAST is able to naturally parallelize the solution of eigenvalue problems by solving for multiple eigenpairs simultaneously. The traditional FEAST algorithm is implemented by directly solving collections of shifted linear systems of equations; in this paper, we describe a variation of the FEAST algorithm that uses iterative Krylov subspace algorithms for solving the shifted linear systems inexactly. We show that this iterative FEAST algorithm (which we call IFEAST) is mathematically equivalent to a block Krylov subspace method for solving eigenvalue problems. By using Krylov subspaces indirectly through solving shifted linear systems, rather than directly for projecting the eigenvalue problem, IFEAST is able to solve eigenvalue problems using very large dimension Krylov subspaces, without ever having to store a basis for those subspaces. IFEAST thus combines the flexibility and power of Krylov methods, requiring only matrix-vector multiplication for solving eigenvalue problems, with the natural parallelism of the traditional FEAST algorithm. We discuss the relationship between IFEAST and more traditional Krylov methods, and provide numerical examples illustrating its behavior.