Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

TL;DR

This paper introduces a hybrid preconditioning approach that combines global and local techniques to efficiently solve large, ill-conditioned generalized eigenvalue problems in electronic structure calculations, improving computational performance.

Contribution

The paper presents a novel hybrid preconditioning scheme that enhances iterative diagonalization efficiency for ill-conditioned problems in electronic structure methods using nonorthogonal bases.

Findings

Outperforms existing global preconditioning methods in PUFE calculations.

Achieves efficiency comparable to locally optimal methods for well-conditioned problems.

Reduces computational cost in large-scale electronic structure simulations.

Abstract

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods.