Work

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Year: 2021

Type: article

Abstract: This article is concerned with the numerical solution of subspace optimization problems, consisting of minimizing a smooth functional over the set of orthogonal projectors of fixed rank. Such problems are encountered in particular in electronic structure calculation (Hartree--Fock and Kohn--Sham density functional theory models). We compare from a numerical analysis perspective two simple representatives, the damped self-consistent field (SCF) iterations and the gradient descent algorithm, of the two classes of methods competing in the field: SCF and direct minimization methods. We derive asymptotic rates of convergence for these algorithms and analyze their dependence on the spectral gap and other properties of the problem. Our theoretical results are complemented by numerical simulations on a variety of examples, from toy models with tunable parameters to realistic Kohn--Sham computations. We also provide an example of chaotic behavior of the simple SCF iterations for a nonquadratic functional.

Source: SIAM Journal on Matrix Analysis and Applications

DOI: 10.1137/20m1332864

Authors: Éric Cancès, Gaspard Kemlin, Antoine Levitt

Institutions: CERMICS, MATHERIALS: MATHematics for MatERIALS

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funder: H2020 European Research Council

Award: Extreme-scale Mathematically-based Computational Chemistry