Stochastic Algorithms for Self-consistent Calculations of Electronic Structures

TL;DR

This paper introduces a stochastic algorithm for electronic structure calculations that uses random sampling and Krylov subspace approximation, demonstrating convergence properties under various conditions.

Contribution

It presents a novel stochastic SCF algorithm that reduces computational cost and proves its convergence in mean-square and probability senses under different assumptions.

Findings

Algorithm converges in mean-square with bounded stochastic error.

Convergence in probability is established under weaker assumptions.

Sampling one random vector per iteration reduces computational effort.

Abstract

The convergence property of a stochastic algorithm for the self-consistent field (SCF) calculations of electron structures is studied. The algorithm is formulated by rewriting the electron charges as a trace/diagonal of a matrix function, which is subsequently expressed as a statistical average. The function is further approximated by using a Krylov subspace approximation. As a result, each SCF iteration only samples one random vector without having to compute all the orbitals. We consider the common practice of SCF iterations with damping and mixing. We prove with appropriate assumptions that the iterations converge in the mean-square sense, when the stochastic error has an almost sure bound. We also consider the scenario when such an assumption is weakened to a second moment condition, and prove the convergence in probability.