The Adaptive Multilevel Splitting algorithm [F. Cérou and A. Guyader, Stoch. Anal. Appl. 25 (2007) 417–443] is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the less well-adapted particles among are killed while new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever . We also obtain a precise asymptotic expansion for the variance of the estimator and the cost of the algorithm in the large limit, for a fixed .
DOI : 10.1051/ps/2014029
Mots-clés : Monte-Carlo simulation, rare events, multilevel splitting
@article{PS_2015__19__361_0,
author = {Br\'ehier, Charles-Edouard and Leli\`evre, Tony and Rousset, Mathias},
title = {Analysis of adaptive multilevel splitting algorithms in an idealized case},
journal = {ESAIM: Probability and Statistics},
pages = {361--394},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2014029},
zbl = {1348.65017},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2014029/}
}
TY - JOUR AU - Bréhier, Charles-Edouard AU - Lelièvre, Tony AU - Rousset, Mathias TI - Analysis of adaptive multilevel splitting algorithms in an idealized case JO - ESAIM: Probability and Statistics PY - 2015 SP - 361 EP - 394 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014029/ DO - 10.1051/ps/2014029 LA - en ID - PS_2015__19__361_0 ER -
%0 Journal Article %A Bréhier, Charles-Edouard %A Lelièvre, Tony %A Rousset, Mathias %T Analysis of adaptive multilevel splitting algorithms in an idealized case %J ESAIM: Probability and Statistics %D 2015 %P 361-394 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014029/ %R 10.1051/ps/2014029 %G en %F PS_2015__19__361_0
Bréhier, Charles-Edouard; Lelièvre, Tony; Rousset, Mathias. Analysis of adaptive multilevel splitting algorithms in an idealized case. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 361-394. doi : 10.1051/ps/2014029. http://www.numdam.org/articles/10.1051/ps/2014029/
S. Asmussen and P.W. Glynn, Stochastic Simulation: Algorithms and Analysis. Springer (2007). | Zbl
and , Estimation of small failure probabilities in high dimensions by subset simulation. J. Probab. Engrg. Mech. 16 (2001) 263–277.
C.E. Bréhier, M. Gazeau, L. Goudenège, T. Lelièvre and M. Rousset, Unbiasedness of some generalized Adaptive Multilevel Splitting algorithms. Preprint (2015). | arXiv
and , Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 25 (2007) 417–443. | Zbl
F. Cérou and A. Guyader, Adaptive particle techniques and rare event estimation. In Conf. Oxford sur les méthodes de Monte Carlo séquentielles. Vol. 19 of ESAIM Proc. EDP Sciences, Les Ulis (2007) 65–72. | Zbl
, , and , A multiple replica approach to simulate reactive trajectories. J. Chem. Phys. 134 (2011) 054108.
, P. Del Moral, T. Furon and A. Guyader, Sequential Monte Carlo for rare event estimation. Stat. Comput. 22 (2012) 795–808. | Zbl
M. Freidlin, Functional integration and partial differential equations. In vol. 109 of Ann. Math. Stud. Princeton University Press, Princeton, NJ (1985). | Zbl
, , and , Multilevel splitting for estimating rare event probabilities. Oper. Res. 47 (1999) 585–600. | Zbl
, and , Simulation and estimation of extreme quantiles and extreme probabilities. Appl. Math. Optim. 64 (2011) 171–196. | Zbl
J.M. Hammersley and D.C. Handscomb, Monte Carlo methods. Methuen and Co. Ltd. (1965). | Zbl
and , Estimation of Particle Transmission by Random Sampling. National Bureau of Standards, Appl. Math. Ser. 12 (1951) 27–30.
G. Rubino and B. Tuffin, Rare Event Simulation using Monte Carlo Methods. Wiley (2009). | Zbl
, Nested Sampling for General Bayesian Computation. Bayesian Anal. 1 (2006) 833–859. | Zbl
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