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

Abstract

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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.


How to cite

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El 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

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