Stochastic Density Functional Theory Through the Lens of Multilevel Monte Carlo Method

TL;DR

This paper introduces a multilevel Monte Carlo approach to stochastic density functional theory, reducing computational costs and improving efficiency for large-scale electronic structure calculations.

Contribution

It develops a multilevel Monte Carlo framework for sDFT, enabling variance reduction and cost independence from discretization size or temperature.

Findings

Cost of density matrix evaluation becomes independent of discretization size.

Variance reduction algorithms improve computational efficiency.

Numerical experiments confirm the effectiveness of the proposed method.

Abstract

The stochastic density functional theory (sDFT) has exhibited advantages over the standard Kohn-Sham DFT method and has become an attractive approach for large-scale electronic structure calculations. The sDFT method avoids the expensive matrix diagonalization by introducing a set of random orbitals and approximating the density matrix via Chebyshev expansion of a matrix-valued function. In this work, we study the sDFT with a plane-wave discretization, and discuss variance reduction algorithms in the framework of multilevel Monte Carlo (MLMC) methods. In particular, we show that the density matrix evaluation in sDFT can be decomposed into many levels by increasing the plane-wave cutoffs or the Chebyshev polynomial orders. This decomposition renders the computational cost independent of the discretization size or temperature. To demonstrate the efficiency of the algorithm, we provide rigorous analysis of the statistical errors and present numerical experiments on some material systems.