## Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems (1996)

Venue: | Ann. Appl. Prob |

Citations: | 18 - 6 self |

### BibTeX

@ARTICLE{Moral96measurevalued,

author = {P. Del Moral},

title = {Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems},

journal = {Ann. Appl. Prob},

year = {1996},

volume = {8},

pages = {438--495}

}

### OpenURL

### Abstract

In the paper we study interacting particle approximations of discrete time and measure valued dynamical systems. Such systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the so-called particle density profiles to converge to the desired distribution when the number of particles is growing. The strength of our approach is that is applicable to a large class of measure valued dynamical system arising in engineering and particularly in nonlinear filtering problems. Our second objective is to use these results to solve numerically the nonlinear filtering equation. Examples arising in fluid mechanics are also given. 1 Introduction 1.1 Measure valued processes Let (E; fi(E)) be a locally compact and separable metric space, endowed with a Borel oe-field, state space. Denote by P(E) be the space of all probability measures on E with the weak topology. The aim of this work is the design of a stochastic particle system approach fo...

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Citation Context ...ed to be the first convergence results for such approximations of the nonlinear filtering equations. This new treatment was influenced primarily by the development of genetic algorithms (J.H. Holland =-=[26]-=-, R. Cerf [6]) and secondarily by the papers of H.Kunita and L.Stettner ([29], [40]). Several practical problems examples which have been solved using these methods are given in [4], [20], [24], [18],... |

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Citation Context ...random particle systems to solve numerically the filtering problem. That particle algorithms are gaining popularity is attested by the recent papers of Crisan/Lyons [9] [10] [11], Gordon/Salmon/Smith =-=[24]-=-, Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del Moral [15] [16], [17]. Instead of h... |

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Citation Context ...exception of the linear-Gaussian situation or wider classes of models (B`enes filters [1]) optimal filters have no finitely recursive solution (Chaleyat-Maurel/Michel [7]). Although Kalman filtering (=-=[27]-=-,[30]) is a popular tool in handling estimation problems its optimality heavily depends on linearity. When used for non linear filtering (Extended Kalman Filter) its performance relies on and is limit... |

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Citation Context ...ion some kind of approximation is needed. On the other hand, particle methods have been developed in physics since the second world war, mainly for the need of Fluid Mechanics (M'el'eard [32], McKean =-=[33]-=-, Sznitman [43]) and Statistical Mechanics ( Dobrushin [22], Liggett [31], Spitzer [42]). During the decade their application area has grown establishing unexpected connection with number of other fie... |

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Citation Context ...s referred to the pioneering paper of Stratonovich [41] and to the more rigorous studies of Shiryaev [39] and Kallianpur-Striebel [28]. More recent developments can be found in Ocone [34] and Pardoux =-=[35]-=-. Some collateral readings such as Kunita [29], Stettner [40], Michel [7] will be helpful in appreciating the relevance of our approximations. As far as the author knows the various numerical methods ... |

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Citation Context ...ations are made, and random perturbations are present in the dynamics as well as in the sensors. With the notable exception of the linear-Gaussian situation or wider classes of models (B`enes filters =-=[1]-=-) optimal filters have no finitely recursive solution (Chaleyat-Maurel/Michel [7]). Nevertheless, guided by Bayes' rule we will see that the dynamics structure of such conditional distributions can be... |

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Citation Context ...e efficient way is to use random particle systems to solve numerically the filtering problem. That particle algorithms are gaining popularity is attested by the recent papers of Crisan/Lyons [9] [10] =-=[11]-=-, Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del Moral [15... |

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Citation Context ...a more efficient way is to use random particle systems to solve numerically the filtering problem. That particle algorithms are gaining popularity is attested by the recent papers of Crisan/Lyons [9] =-=[10]-=- [11], Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del Mora... |

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Citation Context ...which ensure the convergence to the optimal filter uniformly with respect to time. Crisan and Lyons develop in [9] a sequence of branching processes in the spirit of Dawson and Watanabe constructions =-=[12]-=-,[46] whose expectation converges to the solution of the Zakai equation. In [10] they construct a different sequence of branching particle systems converging in distribution to the solution of the Zak... |

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Citation Context ...Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del Moral [15] =-=[16]-=-, [17]. Instead of hand-crafting algorithms often on the basis of ad-hoc criteria, particle systems approaches provide powerful tools for solving a large class of non linear filtering problems. Let us... |

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Citation Context ...irst convergence results for such approximations of the nonlinear filtering equations. This new treatment was influenced primarily by the development of genetic algorithms (J.H. Holland [26], R. Cerf =-=[6]-=-) and secondarily by the papers of H.Kunita and L.Stettner ([29], [40]). Several practical problems examples which have been solved using these methods are given in [4], [20], [24], [18], [5] includin... |

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Citation Context ...sumed. For a detailed discussion of the filtering problem the reader is referred to the pioneering paper of Stratonovich [41] and to the more rigorous studies of Shiryaev [39] and Kallianpur-Striebel =-=[28]-=-. More recent developments can be found in Ocone [34] and Pardoux [35]. Some collateral readings such as Kunita [29], Stettner [40], Michel [7] will be helpful in appreciating the relevance of our app... |

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Citation Context ...bability measures on E. Such type of processes appear naturally in the study of Nonlinear Filtering problems (see for instance [9], [29] and [40]) and in Propagation of Chaos Theory (see for instance =-=[38]-=-, [43] and [44] ). In this short section we summarize the key concepts and the various forms of convergence of probability measures which are used throughout the paper. For further information the rea... |

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Citation Context ...e gaining popularity is attested by the recent papers of Crisan/Lyons [9] [10] [11], Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut =-=[4]-=-, Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del Moral [15] [16], [17]. Instead of hand-crafting algorithms often on the basis of ad-hoc criteria, particle systems approaches pro... |

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Citation Context ...hat a more efficient way is to use random particle systems to solve numerically the filtering problem. That particle algorithms are gaining popularity is attested by the recent papers of Crisan/Lyons =-=[9]-=- [10] [11], Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del... |

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Citation Context ..., particle methods have been developed in physics since the second world war, mainly for the need of Fluid Mechanics (M'el'eard [32], McKean [33], Sznitman [43]) and Statistical Mechanics ( Dobrushin =-=[22]-=-, Liggett [31], Spitzer [42]). During the decade their application area has grown establishing unexpected connection with number of other fields. Short title: Non Linear Filtering Using Interacting Pa... |

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Citation Context ...in the sensors. With the notable exception of the linear-Gaussian situation or wider classes of models (B`enes filters [1]) optimal filters have no finitely recursive solution (Chaleyat-Maurel/Michel =-=[7]-=-). Nevertheless, guided by Bayes' rule we will see that the dynamics structure of such conditional distributions can be viewed, under mild assumptions, as a special case of (1). In our formulation the... |

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Citation Context ...blem the reader is referred to the pioneering paper of Stratonovich [41] and to the more rigorous studies of Shiryaev [39] and Kallianpur-Striebel [28]. More recent developments can be found in Ocone =-=[34]-=- and Pardoux [35]. Some collateral readings such as Kunita [29], Stettner [40], Michel [7] will be helpful in appreciating the relevance of our approximations. As far as the author knows the various n... |

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Citation Context ...an/Lyons [9] [10] [11], Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut =-=[20]-=- and Del Moral [15] [16], [17]. Instead of hand-crafting algorithms often on the basis of ad-hoc criteria, particle systems approaches provide powerful tools for solving a large class of non linear fi... |

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Citation Context ...f approximation is needed. On the other hand, particle methods have been developed in physics since the second world war, mainly for the need of Fluid Mechanics (M'el'eard [32], McKean [33], Sznitman =-=[43]-=-) and Statistical Mechanics ( Dobrushin [22], Liggett [31], Spitzer [42]). During the decade their application area has grown establishing unexpected connection with number of other fields. Short titl... |

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Citation Context ...tion of the linear-Gaussian situation or wider classes of models (B`enes filters [1]) optimal filters have no finitely recursive solution (Chaleyat-Maurel/Michel [7]). Although Kalman filtering ([27],=-=[30]-=-) is a popular tool in handling estimation problems its optimality heavily depends on linearity. When used for non linear filtering (Extended Kalman Filter) its performance relies on and is limited by... |

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Citation Context ... criteria, particle systems approaches provide powerful tools for solving a large class of non linear filtering problems. Let us now briefly survey some different approaches and motivate our work. In =-=[14]-=- and [15] the author introduced particle methods where particles are independent each other and weighted by regularized exponentials and he proposes minimal conditions which ensure the convergence to ... |

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Citation Context ...by the recent papers of Crisan/Lyons [9] [10] [11], Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut =-=[18]-=-, Del Moral/Noyer/Salut [20] and Del Moral [15] [16], [17]. Instead of hand-crafting algorithms often on the basis of ad-hoc criteria, particle systems approaches provide powerful tools for solving a ... |

3 |
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Citation Context ...amma h n (z i )) P N j=1 g n (y n \Gamma h n (z j )) ffi z i (dx p ) (113) where y n is the observation data at the time n and x; z 2 S N . Equations (112) and (113) resemble a genetic algorithm [6], =-=[21], [26]. The adva-=-ntage of this formulation is that it incorporates separately the prediction b �� n\Gamma1 ; �� n and the updating �� n ; b �� n mechanisms. Thus, we see that the particles move accordi... |

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Citation Context ...ems have arisen in such diverse scientific disciplines as physics (see Sznitman [44] and the references given there), nonlinear economic modelling and signal processing ( see Kunita [29] and Stettner =-=[40]-=-). Solving (1) is in general an enormous task as it is nonlinear and usually involves integrations over the whole space E. To obtain a computionnally feasible solution some kind of approximation is ne... |

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Citation Context ... 2 [0; 1]. ffl j n\Gamma1 is the distribution of X n\Gamma1 . ffl W n is an [0; 1]-valued random variable with law \Gamma. The description of such models in continuous time may be founded in Sznitman =-=[44]. Arguing as bef-=-ore the corresponding interacting particle system is given by 8 ? ? ! ? ? : P ( b �� k 2 dx=�� k = z) = N Y p=1 1 N N X i=1 ffi z i (dx p ) �� i n = b �� i n\Gamma1 + 1 N P N j=1 a( b ... |

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Citation Context ...e. Another question we let in the dark is the study of the asymptotic behavior of these algorithms in terms of the ergodic properties of the signal semigroup. In these direction something was done in =-=[17]-=- when the state space is compact but many questions have yet no answers. 4.1 Formulation of the Non Linear Filtering Problem The object of this section is to introduce the filtering model in such a wa... |

2 |
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Citation Context ... developed in physics since the second world war, mainly for the need of Fluid Mechanics (M'el'eard [32], McKean [33], Sznitman [43]) and Statistical Mechanics ( Dobrushin [22], Liggett [31], Spitzer =-=[42]-=-). During the decade their application area has grown establishing unexpected connection with number of other fields. Short title: Non Linear Filtering Using Interacting Particles. Keywords: Non linea... |

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Citation Context ...11], Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier [45], Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del Moral =-=[15]-=- [16], [17]. Instead of hand-crafting algorithms often on the basis of ad-hoc criteria, particle systems approaches provide powerful tools for solving a large class of non linear filtering problems. L... |

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Citation Context ... the filtering problem. That particle algorithms are gaining popularity is attested by the recent papers of Crisan/Lyons [9] [10] [11], Gordon/Salmon/Smith [24], Van Dootingh/Viel/Rakotopara/Gauthier =-=[45]-=-, Carvalho [5], Carvalho/Del Moral/Monin/Salut [4], Del Moral/Rigal/Noyer/Salut [18], Del Moral/Noyer/Salut [20] and Del Moral [15] [16], [17]. Instead of hand-crafting algorithms often on the basis o... |