Multiscale Algorithms for Eigenvalue Problems

TL;DR

This paper explores a multiscale iterative method for solving large eigenvalue problems in electronic structure calculations, demonstrating efficiency for small problems but facing challenges with larger molecules.

Contribution

It introduces a nonlinear multigrid approach for eigenvalue problems and analyzes its performance on large molecular systems, highlighting current limitations.

Findings

Efficient for small eigenvalue problems

Stalls for large molecules with many levels

Work ongoing to improve scalability

Abstract

Iterative multiscale methods for electronic structure calculations offer several advantages for large-scale problems. Here we examine a nonlinear full approximation scheme (FAS) multigrid method for solving fixed potential and self-consistent eigenvalue problems. In principle, the expensive orthogonalization and Ritz projection operations can be moved to coarse levels, thus substantially reducing the overall computational expense. Results of the nonlinear multiscale approach are presented for simple fixed potential problems and for self-consistent pseudopotential calculations on large molecules. It is shown that, while excellent efficiencies can be obtained for problems with small numbers of states or well-defined eigenvalue cluster structure, the algorithm in its original form stalls for large-molecule problems with tens of occupied levels. Work is in progress to attempt to alleviate those difficulties.