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Diffusion Monte Carlo method: Numerical Analysis in 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. makrimo@cermics.enpc.fr; jourdain@cermics.enpc.fr; lelievre@cermics.enpc.fr
Benjamin Jourdain
Affiliation:
ENPC-CERMICS, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-Vallée Cedex 2, France. makrimo@cermics.enpc.fr; jourdain@cermics.enpc.fr; lelievre@cermics.enpc.fr
Tony Lelièvre
Affiliation:
ENPC-CERMICS, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-Vallée Cedex 2, France. makrimo@cermics.enpc.fr; jourdain@cermics.enpc.fr; lelievre@cermics.enpc.fr
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Abstract


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.


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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References

Alfonsi, A., On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11 (2005) 355–384. CrossRef
Assaraf, R., Caffarel, M. and Khelif, A., Diffusion Monte Carlo with a fixed number of walkers. Phys. Rev. E 61 (2000) 4566–4575. CrossRef
E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational Quantum Chemistry: a Primer, in Handbook of Numerical Analysis, Special volume, Computational Chemistry, volume X, Ph.G. Ciarlet and C. Le Bris Eds., North-Holland (2003) 3–270.
Cancès, E., Jourdain, B. and Lelièvre, T., Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation. Math. Mod. Methods Appl. Sci. 16 (2006) 1403–1440. CrossRef
O. Cappé, R. Douc and E. Moulines, Comparison of Resampling Schemes for Particle Filtering, in 4th International Symposium on Image and Signal Processing and Analysis (ISPA), Zagreb, Croatia (2005).
Chopin, N., Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 (2004) 2385–2411. CrossRef
P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer-Verlag (2004).
Del Moral, P. and Doucet, A., Particle motions in absorbing medium with hard and soft obstacles. Stochastic Anal. Appl. 22 (2004) 1175–1207. CrossRef
P. Del Moral and L. Miclo, Branching and Interacting Particle Systems. Approximation of Feynman-Kac Formulae with Applications to Non-Linear Filtering, in Séminaire de Probabilités XXXIV, Lecture Notes in Mathematics 1729, Springer-Verlag (2000) 1–145.
Del Moral, P. and Miclo, L., Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171–208. CrossRef
P. Glasserman, Monte Carlo methods in financial engineering. Springer-Verlag (2004).
Hetherington, J.H., Observations on the statistical iteration of matrices. Phys. Rev. A 30 (1984) 2713–2719. CrossRef
Reynolds, P.J., Ceperley, D.M., Alder, B.J. and Lester, W.A., Fixed-node quantum Monte Carlo for molecules. J. Chem. Phys. 77 (1982) 5593–5603. CrossRef
Rousset, M., On the control of an interacting particle approximation of Schrödinger groundstates. SIAM J. Math. Anal. 38 (2006) 824–844. CrossRef
Sorella, S., Green Function Monte Carlo with Stochastic Reconfiguration. Phys. Rev. Lett. 80 (1998) 4558–4561. CrossRef
Talay, D. and Tubaro, L., Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 (1990) 94–120. CrossRef
Umrigar, C.J., Nightingale, M.P. and Runge, K.J., Diffusion Monte Carlo, A algorithm with very small time-step errors. J. Chem. Phys. 99 (1993) 2865–2890. CrossRef