A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem

TL;DR

This paper introduces a stable, efficient, and parallel scalable eigensolver for large-scale linear response eigenvalue problems, leveraging biorthogonal structure preservation and nullspace properties to handle zero eigenvalues effectively.

Contribution

It proposes a novel biorthogonal structure-preserving iterative eigensolver that deflates converged eigenvectors without artificial parameters and efficiently computes large eigenpair sets in parallel.

Findings

The method is stable and efficient for large-scale problems.

It demonstrates excellent parallel scalability.

Numerical examples confirm robustness and performance.

Abstract

The linear response eigenvalue problem, which arises from many scientific and engineering fields, is quite challenging numerically for large-scale sparse/dense system, especially when it has zero eigenvalues. Based on a direct sum decomposition of biorthogonal invariant subspaces and the minimization principles in the biorthogonal complement, using the structure of generalized nullspace, we propose a Bi-Orthogonal Structure-Preserving subspace iterative solver, which is stable, efficient, and of excellent parallel scalability. The biorthogonality is of essential importance and created by a modified Gram-Schmidt biorthogonalization (MGS-Biorth) algorithm. We naturally deflate out converged eigenvectors by computing the rest eigenpairs in the biorthogonal complementary subspace without introducing any artificial parameters. When the number of requested eigenpairs is large, we propose a moving mechanism to compute them batch by batch such that the projection matrix size is small and independent of the requested eigenpair number. For large-scale problems, one only needs to provide the matrix-vector product, thus waiving explicit matrix storage. The numerical performance is further improved when the matrix-vector product is implemented using parallel computing. Ample numerical examples are provided to demonstrate the stability, efficiency, and parallel scalability.