Efficient Krylov Methods for Linear Response in Plane-Wave Electronic Structure Calculations

Abstract

Abstract

Abstract. We propose a novel algorithm based on inexact GMRES methods for linear response calculations in density functional theory. Such calculations require iteratively solving a nested linear problem [Formula: see text] to obtain the variation of the electron density [Formula: see text]. Notably each application of the dielectric operator [Formula: see text] in turn requires the iterative solution of multiple linear systems, the Sternheimer equations. We develop computable bounds to estimate the accuracy of the density variation given the tolerances to which the Sternheimer equations have been solved. Based on this result we suggest reliable strategies for adaptively selecting the convergence tolerances of the Sternheimer equations, such that each application of [Formula: see text] is no more accurate than needed. Experiments on challenging materials systems of practical relevance demonstrate our strategies to achieve superlinear convergence as well as a reduction of computational time by about 40% while preserving the accuracy of the returned response solution. Our algorithm seamlessly combines with standard preconditioning approaches known from the context of self-consistent field problems, making it a promising framework for efficient response solvers based on Krylov subspace techniques. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/bonans/inexact_Krylov_response and in the supplementary materials ( inexact_Krylov_response-master.zip [815KB]). [Formula: see text]