Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. II. Interior eigenvalues

TL;DR

This paper introduces the PLMR method, a preconditioned eigensolver designed for efficiently computing interior eigenvalues of large-scale nonlinear Hermitian problems with variational characterizations, outperforming standard methods.

Contribution

The paper develops a novel PLMR method with refined Rayleigh-Ritz projection, block extension, and soft deflation for large-scale nonlinear Hermitian eigenproblems, improving efficiency and robustness.

Findings

PLMR converges rapidly to interior eigenvalues.

PLMR outperforms standard preconditioned conjugate gradient methods.

The method is effective for computing many extreme eigenvalues.

Abstract

We consider the solution of large-scale nonlinear algebraic Hermitian eigenproblems of the form $T(\lambda)v=0$ that admit a variational characterization of eigenvalues. These problems arise in a variety of applications and are generalizations of linear Hermitian eigenproblems $Av\!=\!\lambda Bv$. In this paper, we propose a Preconditioned Locally Minimal Residual (PLMR) method for efficiently computing interior eigenvalues of problems of this type. We discuss the development of search subspaces, preconditioning, and eigenpair extraction procedure based on the refined Rayleigh-Ritz projection. Extension to the block methods is presented, and a moving-window style soft deflation is described. Numerical experiments demonstrate that PLMR methods provide a rapid and robust convergence towards interior eigenvalues. The approach is also shown to be efficient and reliable for computing a large number of extreme eigenvalues, dramatically outperforming standard preconditioned conjugate gradient methods.