The approximation of the Feynman-Kac semigroups by systems of interacting particles is a very active research field, with applications in many different areas. In this paper, we study the parallelization of such approximations. The total population of particles is divided into sub-populations, referred to as islands. The particles within each island follow the usual selection / mutation dynamics. We Pierre Del Moral Centre INRIA Bordeaux Sud Ouest -351 Cours de la Libération, 33405 Talence Cedex, 2 C. Vergé and al.show that the evolution of each island is also driven by a Feynman-Kac semigroup, whose transition and potential can be explicitly related to ones of the original problem. Therefore, the same genetic type approximation of the Feynman-Kac semi-group may be used at the island level; each island might undergo selection / mutation algorithm. We investigate the impact of the population size within each island and the number of islands, and study different type of interactions. We find conditions under which introducing interactions between islands is beneficial. The theoretical results are supported by some Monte Carlo experiments. Keywords Particle approximation of Feynman-Kac flow, Island models, parallel implementation 1 Introduction Numerical approximation of Feynman-Kac semigroups by systems of interacting particles is a very active field of researchs. Interacting particle systems are increasingly used to sample complex high dimensional distributions in a wide range of applications including nonlinear filtering, data assimilation problems, rare event sampling, hidden Markov chain parameter estimation, stochastic control problems, financial mathematics; see for example [8], [2], [4], [1], [6] and the references therein. Let (En, En) n≥0 be a sequence of measurable spaces. Denote by B b (En) the Banach space of all bounded and measurable real valued functions f on En, equipped with the uniform norm. Let (gn) n∈N be a sequence of measurable potential functions, gn : En → R + . Let (Ω, F, P) be a probability space. In the sequel, all the processes are defined on this probability space. Let (Xn) n∈N be a non-homogenous i=1which are generated recursively. Typically, the update of the particles may be decomposed into a mutation and a selection step. For example, the bootstrap algorithm proceeds as follows. In the selection step the particles are first sampled with weights proportional to the potential functions. In the mutation step, a new generation of particles (X i n+1 ) N1 i=1 is generated from the selected particles using the kernel M n+1 . The asymptotic behavior of such particle approximation is now well understood (see [4] and [6]). Feynman-Kac measures appear naturally in the filtering problem for HiddenMarkov Model (HMM). Recall that a HMM is a pair of discrete time randomC. Vergé and al.processes (X, Y ) = (Xn, Yn) n∈N , where (Xn) n≥0 is the hidden state process (often called signal) and (Yn) n≥0 are the observations. To fix the ideas, Xn and Yn take values in X ⊂ R k and Y ⊂ R l . The state sequen...

International audienceCrude Monte-Carlo or quasi Monte-Carlo methods are well suited to characterize events of which associated probabilities are not too low with respect to the simulation budget. For very seldom observed events, such as the collision probability between two aircraft in airspace, these approaches do not lead to accurate results. Indeed, the number of available samples is often insufficient to estimate such low probabilities (at least 10^6 samples are needed to estimate a probability of order 10^-4with 10% relative error with Monte-Carlo simulations). In this article, one reviewed different appropriate techniques to estimate rare event probabilities that require a fewer number of samples. These methods can be divided into four main categories: parameterization techniques of probability density function tails, simulation techniques such as importance sampling or importance splitting, geometric methods to approximate input failure space and finally, surrogate modeling. Each technique is detailed, its advantages and drawbacks are described and a synthesis that aims at giving some clues to the following question is given: “which technique to use for which problem?”