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

TL;DR

This work develops an efficient solver for linear response in plane-wave DFT by solving the Dyson equation with an inexact, preconditioned GMRES framework. A rigorous error analysis shows how inexact Sternheimer solves propagate to the Dyson solution, enabling adaptive selection of Sternheimer tolerances that preserve accuracy while reducing cost; Kerker preconditioning is used to accelerate convergence in metals. The proposed balanced (bal) and related strategies consistently reduce computational effort by about 40% (and up to 4× when combined with preconditioning) across challenging materials such as Aluminium, Heusler alloys, and Silicon, without sacrificing the final accuracy. The approach integrates seamlessly with standard preconditioning and Krylov techniques, offering a scalable, black-box solver for DFPT that can be extended to other mean-field theories.

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 $\mathcal{E} δρ= b$ to obtain the variation of the electron density $δρ$. 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.

Figures (7)

• Figure 1: Discretisation of continuous objects in the plane-wave basis sets.
• Figure 1: True residuals $\|\bm r_i\|$ (solid lines) and estimated residuals $\|\tilde{\bm r}_i\|$ (dashed lines) v.s. GMRES iteration for system in Section \ref{['sec:structure_dyson']}. The baseline strategies $\tau^{\mathrm{CG}}_{i,n} = \tau/10,\ \tau/100$ (D10, D100) are used for the CG solvers for the SEs.
• Figure 1: Aluminium supercell: True residual norms $\|\bm r_i\|$ v.s. (A) number of GMRES iterations, (B) accumulated number of Hamiltonian applications. The shaded areas indicate the strategies with Kerker preconditioning.
• Figure 1: Residual norms $\| \bm{r}_{n,k}^{(i)} \|$, $\ell_2$ errors $\| \bm z_{n,k}^{(i)} \|$ and error bounds $\| \bm{r}_{n,k}^{(i)} \|/(\epsilon_{N_{\mathrm{occ}} + 1} - \epsilon_{N_{\mathrm{occ}}})$ v.s. CG iteration number $k$ for the worst conditioned Sternheimer equation of (A) the aluminium system ($\ell = 10$) and (B) the Heusler system described in Section \ref{['sec:experiments']}.
• Figure 1: Values of $\| \bm K \bm v_i \|$ (circles) during the GMRES iterations for Aluminium supercell systems of different sizes (A) without and (B) with Kerker preconditioning. We employ the agr strategy to set the CG tolerances (see Table \ref{['tab:strategies']}) and use $\ell$ to denote the factor by which $\bm a_1$ grows relative to the smallest Aluminium system, see Section \ref{['sec:Al']} for details on the computational setup.

Theorems & Definitions (16)

• Lemma 3.1
• Lemma 3.2: simoncini2003theory, Lemma 5.1
• Theorem 3.3
• Remark 3.4
• Lemma 3.5
• Proof 1: Proof of Lemma \ref{['lem:cgtol_E']}
• Remark 3.6
• Theorem 3.7
• Remark 3.8: Superlinear convergence
• Lemma 3.9