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Diffusion Monte Carlo method: Numerical Analysisin a Simple Case

Published online by Cambridge University Press:  16 June 2007

Mohamed El Makrini ,
Benjamin Jourdain  and
Tony Lelièvre
Mohamed El Makrini
Affiliation:
ENPC-CERMICS, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-Vallée Cedex 2, France. [email protected]; [email protected]; [email protected]
Benjamin Jourdain
Affiliation:
ENPC-CERMICS, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-Vallée Cedex 2, France. [email protected]; [email protected]; [email protected]
Tony Lelièvre
Affiliation:
ENPC-CERMICS, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-Vallée Cedex 2, France. [email protected]; [email protected]; [email protected]
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Abstract


The Diffusion Monte Carlo method is devoted to the computation ofelectronic ground-state energies of molecules. In this paper, we focus onimplementations of this method which consist in exploring theconfiguration space with a fixed number of random walkers evolvingaccording to a stochastic differential equation discretized in time. Weallow stochastic reconfigurations of the walkers to reduce thediscrepancy between the weights that they carry. On a simpleone-dimensional example, we prove the convergence of the method for afixed number of reconfigurations when the number of walkers tends to+∞ while the timestep tends to 0. We confirm our theoreticalrates of convergence by numerical experiments. Various resamplingalgorithms are investigated, both theoretically and numerically.


Keywords

Diffusion Monte Carlo methodinteracting particle systemsground stateSchrödinger operatorFeynman-Kac formula.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007 , pp. 189 - 213
Copyright
© EDP Sciences, SMAI, 2007

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