Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems

TL;DR

This paper demonstrates that successive eigenproblems in Density Functional Theory simulations are strongly correlated, and exploiting this correlation with block iterative eigensolvers significantly speeds up computations.

Contribution

It introduces the use of block iterative eigensolvers that leverage eigenvector reuse, improving efficiency in solving correlated eigenproblems in LAPW-based DFT simulations.

Findings

Eigenvector reuse leads to substantial speed-up.

Successive eigenproblems show strong eigenvector correlation.

Block iterative eigensolvers are effective in this context.

Abstract

In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear. This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach.