Stochastic Density Functional Theory Through the Lens of Multilevel Monte Carlo Method
Xue Quan, Huajie Chen
TL;DR
The paper addresses the high cost of traditional Kohn–Sham DFT by employing stochastic density functional theory (sDFT) with random orbitals and Chebyshev expansions. It introduces a multilevel Monte Carlo (MLMC) framework to reduce variance and decouple cost from discretization size and temperature, via two hierarchical decompositions: energy cutoffs and polynomial orders. The authors provide rigorous variance bounds and show, through numerical experiments on graphene-like systems, that MLMC can achieve significant cost reductions while preserving accuracy, with the total work scaling as $O(n^{(0)} N M)$ or $O(n^{(0)} N M^{(0)})$, independent of the finest discretization or smearing parameter. This work therefore offers a theoretically grounded, scalable route for large-scale electronic structure calculations using sDFT.
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.