# Diffusion Monte Carlo method: Numerical Analysis in a Simple Case

Mohamed El Makrini; Benjamin Jourdain; Tony Lelièvre

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 2, page 189-213
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topEl Makrini, Mohamed, Jourdain, Benjamin, and Lelièvre, Tony. "Diffusion Monte Carlo method: Numerical Analysis in a Simple Case." ESAIM: Mathematical Modelling and Numerical Analysis 41.2 (2007): 189-213. <http://eudml.org/doc/250054>.

@article{ElMakrini2007,

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.
},

author = {El Makrini, Mohamed, Jourdain, Benjamin, Lelièvre, Tony},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Diffusion Monte Carlo method; interacting particle systems;
ground state; Schrödinger operator; Feynman-Kac formula.},

language = {eng},

month = {6},

number = {2},

pages = {189-213},

publisher = {EDP Sciences},

title = {Diffusion Monte Carlo method: Numerical Analysis in a Simple Case},

url = {http://eudml.org/doc/250054},

volume = {41},

year = {2007},

}

TY - JOUR

AU - El Makrini, Mohamed

AU - Jourdain, Benjamin

AU - Lelièvre, Tony

TI - Diffusion Monte Carlo method: Numerical Analysis in a Simple Case

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/6//

PB - EDP Sciences

VL - 41

IS - 2

SP - 189

EP - 213

AB -
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.

LA - eng

KW - Diffusion Monte Carlo method; interacting particle systems;
ground state; Schrödinger operator; Feynman-Kac formula.

UR - http://eudml.org/doc/250054

ER -

## References

top- A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl.11 (2005) 355–384. Zbl1100.65007
- R. Assaraf, M. Caffarel and A. Khelif, Diffusion Monte Carlo with a fixed number of walkers. Phys. Rev. E61 (2000) 4566–4575.
- 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. Zbl1070.81534
- E. Cancès, B. Jourdain and T. Lelièvre, Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation. Math. Mod. Methods Appl. Sci.16 (2006) 1403–1440. Zbl1098.81095
- 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).
- N. Chopin, Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist.32 (2004) 2385–2411. Zbl1079.65006
- P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer-Verlag (2004). Zbl1130.60003
- P. Del Moral and A. Doucet, Particle motions in absorbing medium with hard and soft obstacles. Stochastic Anal. Appl.22 (2004) 1175–1207. Zbl1071.60100
- 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 Mathematics1729, Springer-Verlag (2000) 1–145. Zbl0963.60040
- P. Del Moral and L. Miclo, Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS7 (2003) 171–208. Zbl1040.81009
- P. Glasserman, Monte Carlo methods in financial engineering. Springer-Verlag (2004). Zbl1038.91045
- J.H. Hetherington, Observations on the statistical iteration of matrices. Phys. Rev. A30 (1984) 2713–2719.
- P.J. Reynolds, D.M. Ceperley, B.J. Alder and W.A. Lester, Fixed-node quantum Monte Carlo for molecules. J. Chem. Phys.77 (1982) 5593–5603.
- M. Rousset, On the control of an interacting particle approximation of Schrödinger groundstates. SIAM J. Math. Anal.38 (2006) 824–844. Zbl1174.60045
- S. Sorella, Green Function Monte Carlo with Stochastic Reconfiguration. Phys. Rev. Lett.80 (1998) 4558–4561.
- D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl.8 (1990) 94–120. Zbl0718.60058
- C.J. Umrigar, M.P. Nightingale and K.J. Runge, A Diffusion Monte Carlo algorithm with very small time-step errors. J. Chem. Phys.99 (1993) 2865–2890.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.