Noname manuscript No.
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On parallel implementation of Sequential Monte Carlo methods: the island particle model
Christelle Verg´ e · Cyrille Dubarry · Pierre
arXiv:1306.3911v1 [math.PR] 17 Jun 2013
Del Moral · Eric Moulines
the date of receipt and acceptance should be inserted later
Abstract The approximation of the Feynman-Kac semigroups by systems of inter-
acting 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 Lib´ eration, 33405 Talence Cedex, E-mail:
[email protected] Cyrille Dubarry SAMOVAR, CNRS UMR 5157 - Institut T´ el´ ecom/T´ el´ ecom SudParis, 9 rue Charles Fourier, 91000 Evry Eric Moulines LTCI, CNRS UMR 8151 - Institut T´ el´ ecom/T´ el´ ecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France, E-mail:
[email protected] Christelle Verg´ e ONERA - The French Aerospace Lab, F-91761 Palaiseau, CNES - 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, E-mail:
[email protected]
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C. Verg´ e 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 Bb (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
Island models
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Markov chain on the sequence of state-spaces (En )n∈N with initial distribution η0 on (E0 , E0 ) and Markov kernels (Mn )n∈N∗ 1 . We associate to the sequences of potential functions (gn )n∈N and Markov kernels (Mn )n∈N∗ the sequence of Feynman-Kac measures, defined for all n ≥ 1 and for any fn ∈ Bb (En ) by def
ηn (fn ) = γn (fn )/γn (1) ,
(1)
def
γn (fn ) = E fn (Xn )
Y
gp (Xp )
(2)
0≤p
