Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

TL;DR

This paper introduces Zolotarev-based rational approximants to enhance the FEAST eigensolver, resulting in faster convergence, better load balancing, and computational savings for large sparse eigenproblems.

Contribution

It develops new rational approximants based on Zolotarev's work, improving FEAST's convergence speed and load balancing for parallel eigenproblem solutions.

Findings

Faster convergence of FEAST with Zolotarev approximants.

Improved load balancing in parallel FEAST runs.

Significant computational savings for challenging eigenproblems.

Abstract

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.