Diffusion Monte Carlo method : numerical analysis in a simple case

Makrini, Mohamed El; Jourdain, Benjamin; Lelièvre, Tony

ESAIM: Modélisation mathématique et analyse numérique, Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 189-213

The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a stochastic differential equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove the convergence of the method for a fixed number of reconfigurations when the number of walkers tends to $+\infty$ while the timestep tends to $0$. We confirm our theoretical rates of convergence by numerical experiments. Various resampling algorithms are investigated, both theoretically and numerically.

Classification : 81Q05, 65C35, 60K35, 35P15
Keywords: diffusion Monte Carlo method, interacting particle systems, ground state, Schrödinger operator, Feynman-Kac formula

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