Geodesic
Diffusion Monte Carlo method : numerical analysis in a simple case
ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 189-213

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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 +∞ 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.

DOI : 10.1051/m2an:2007017
Classification : 81Q05, 65C35, 60K35, 35P15
Keywords: diffusion Monte Carlo method, interacting particle systems, ground state, Schrödinger operator, Feynman-Kac formula 
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     title = {Diffusion {Monte} {Carlo} method : numerical analysis in a simple case},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {189--213},
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Makrini, Mohamed El; Jourdain, Benjamin; Lelièvre, Tony. Diffusion Monte Carlo method : numerical analysis in a simple case. ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 189-213. doi: 10.1051/m2an:2007017

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