Diffusion Monte Carlo method: numerical analysis in a simple case

Tony Lelièvre (1) , Mohamed El Makrini (1) , Benjamin Jourdain (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

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 +1 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.

Mots clés: Diffusion Monte Carlo method; interacting particle systems; ground state; Schrödinger operator; Feynman-Kac formula

Domaines: Analyse numérique [math.NA]

Tony Lelièvre, Mohamed El Makrini, Benjamin Jourdain. Diffusion Monte Carlo method: numerical analysis in a simple case. 2007. ⟨hal-00136428⟩