Efficient Krylov methods for linear response in plane-wave electronic structure calculations

Michael F. Herbst, Bonan Sun

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 $\mathcal{E} \delta\rho = b$
to obtain the variation of the electron density $\delta \rho$.
Notably each application of the dielectric operator $\mathcal{E}$
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 $\mathcal{E}$ 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.