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

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.

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Cited by in corpus (8)

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• Fast and robust all-electron density functional theory calculations in solids using orthogonalized enriched finite elements
• Preconditioned steepest descent-like methods for symmetric indefinite systems
• Partition of unity finite element method for quantum mechanical materials calculations
• The virtual element method for eigenvalue problems with potential terms on polytopal meshes
• Simulation of non-linear structural elastodynamic and impact problems using minimum energy and simultaneous diagonalization high-order bases
• Convergence analysis of a locally accelerated preconditioned steepest descent method for Hermitian-definite generalized eigenvalue problems