Diffusion Monte Carlo method : numerical analysis in a simple case
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 2, pp. 189-213.

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 : https://doi.org/10.1051/m2an:2007017
Classification : 81Q05,  65C35,  60K35,  35P15
Mots clés : diffusion Monte Carlo method, interacting particle systems, ground state, Schrödinger operator, Feynman-Kac formula
     author = {Makrini, Mohamed El and Jourdain, Benjamin and Leli\`evre, Tony},
     title = {Diffusion {Monte} {Carlo} method : numerical analysis in a simple case},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {189--213},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {2},
     year = {2007},
     doi = {10.1051/m2an:2007017},
     zbl = {1135.81379},
     mrnumber = {2339625},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2007017/}
AU  - Makrini, Mohamed El
AU  - Jourdain, Benjamin
AU  - Lelièvre, Tony
TI  - Diffusion Monte Carlo method : numerical analysis in a simple case
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2007
DA  - 2007///
SP  - 189
EP  - 213
VL  - 41
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2007017/
UR  - https://zbmath.org/?q=an%3A1135.81379
UR  - https://www.ams.org/mathscinet-getitem?mr=2339625
UR  - https://doi.org/10.1051/m2an:2007017
DO  - 10.1051/m2an:2007017
LA  - en
ID  - M2AN_2007__41_2_189_0
ER  - 
Makrini, Mohamed El; Jourdain, Benjamin; Lelièvre, Tony. Diffusion Monte Carlo method : numerical analysis in a simple case. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 2, pp. 189-213. doi : 10.1051/m2an:2007017. http://www.numdam.org/articles/10.1051/m2an:2007017/

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