ANALYSIS OF ADAPTIVE MULTILEVEL SPLITTING ALGORITHMS IN AN IDEALIZED CASE

The Adaptive Multilevel Splitting algorithm [F. Cerou 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 k less well-adapted particles among n are killed while k 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 k . We also obtain a precise asymptotic expansion for the variance of the estimator and the cost of the algorithm in the large n limit, for a fixed k .

[1]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[2]  P. Prescott,et al.  Monte Carlo Methods , 1964, Computational Statistical Physics.

[3]  M. Freidlin,et al.  Functional Integration and Partial Differential Equations. (AM-109), Volume 109 , 1985 .

[4]  M. Freidlin.  Functional Integration And Partial Differential Equations , 1985 .

[5]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[6]  Paul Glasserman,et al.  Multilevel Splitting for Estimating Rare Event Probabilities , 1999, Oper. Res..

[7]  J. Beck,et al.  Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation , 2001 .

[8]  J. Skilling.  Nested sampling for general Bayesian computation , 2006 .

[9]  Arnaud Guyader,et al.  ADAPTIVE PARTICLE TECHNIQUES AND RARE EVENT ESTIMATION , 2007 .

[10]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[11]  F. Cérou,et al.  Adaptive Multilevel Splitting for Rare Event Analysis , 2007 .

[12]  Gerardo Rubino,et al.  Rare Event Simulation using Monte Carlo Methods , 2009 .

[13]  Arnaud Guyader,et al.  A multiple replica approach to simulate reactive trajectories. , 2010, The Journal of chemical physics.

[14]  N. Hengartner,et al.  Simulation and Estimation of Extreme Quantiles and Extreme Probabilities , 2011 .

[15]  Pierre Del Moral,et al.  Sequential Monte Carlo for rare event estimation , 2012, Stat. Comput..

[16]  B. Charles-Edouard,et al.  Unbiasedness of some generalized Adaptive Multilevel Splitting algorithms , 2015, 1505.02674.

[17]  Ludovic Goudenège,et al.  Unbiasedness of some generalized Adaptive Multilevel Splitting algorithms , 2016 .