Riemannian Newton methods for energy minimization problems of Kohn-Sham type

TL;DR

This paper introduces Riemannian Newton methods on infinite-dimensional manifolds for solving constrained energy minimization problems in physics and chemistry, demonstrating superior performance over traditional schemes.

Contribution

It develops Riemannian Newton algorithms on Stiefel and Grassmann manifolds tailored for infinite-dimensional problems in quantum physics.

Findings

Methods outperform self-consistent field iteration

Algorithms show faster convergence

Numerical experiments validate effectiveness

Abstract

This paper is devoted to the numerical solution of constrained energy minimization problems arising in computational physics and chemistry such as the Gross-Pitaevskii and Kohn-Sham models. In particular, we introduce the Riemannian Newton methods on the infinite-dimensional Stiefel and Grassmann manifolds. We study the geometry of these two manifolds, its impact on the Newton algorithms, and present expressions of the Riemannian Hessians in the infinite-dimensional setting, which are suitable for variational spatial discretizations. A series of numerical experiments illustrates the performance of the methods and demonstrates its supremacy compared to other well-established schemes such as the self-consistent field iteration and gradient descent schemes.