P. Del Moral

On the Convergence and Applications of Generalized Simulated Annealing

The convergence of the generalized simulated annealing with time-inhomogeneous communication cost functions is discussed. This study is based on the use of log-Sobolev inequalities and semigroup techniques in the spirit of a previous article by one of the authors. We also propose a natural test set approach to study the global minima of the virtual energy. The second part of the paper is devoted to the application of these results. We propose two general Markovian models of genetic algorithms and we give a simple proof of the convergence toward the global minima of the fitness function. Finally we introduce a stochastic algorithm that converges to the set of the global minima of a given mean cost optimization problem.

Limit theorems for the multilevel splitting algorithm in the simulation of rare events

In this article, a genetic-type algorithm based on interacting particle systems is presented, together with a genealogical model, for estimating a class of rare events arising for instance in telecommunication networks, nuclear engineering, etc. The distribution of a Markov process hitting a rare but critical set is represented in terms of a Feynman-Kac model in path space. Approximation results obtained previously for these models are applied here to estimate the probability of the rare events as well as the probability distribution of the critical trajectories.

A Moran particle system approximation of Feynman-Kac formulae

We present a weighted sampling Moran particle system model for the numerical solving of a class of Feynman-Kac formulae which arise in different fields. Our major motivation was from nonlinear filtering, but our approach is context free. We will show that under certain regularity conditions the resulting interacting particle scheme converges to the considered nonlinear equations. In the setting of nonlinear filtering, the -convergence exponent resulting from our proof also improves recent results on other particle interpretations of these equations.

On the stability and the uniform propagation of chaos properties of Ensemble Kalman–Bucy filters

The Ensemble Kalman filter is a sophisticated and powerful data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences. This Monte Carlo method can be interpreted as a mean-field McKean-Vlasov type particle interpretation of the Kalman-Bucy diffusions. In contrast to more conventional particle filters and nonlinear Markov processes these models are designed in terms of a diffusion process with a diffusion matrix that depends on particle covariance matrices. Besides some recent advances on the stability of nonlinear Langevin type diffusions with drift interactions, the long-time behaviour of models with interacting diffusion matrices and conditional distribution interaction functions has never been discussed in the literature. One of the main contributions of the article is to initiate the study of this new class of models The article presents a series of new functional inequalities to quantify the stability of these nonlinear diffusion processes. In the same vein, despite some recent contributions on the convergence of the Ensemble Kalman filter when the number of sample tends to infinity very little is known on stability and the long-time behaviour of these mean-field interacting type particle filters. The second contribution of this article is to provide uniform propagation of chaos properties as well as Lp-mean error estimates w.r.t. to the time horizon. Our regularity condition is also shown to be sufficient and necessary for the uniform convergence of the Ensemble Kalman filter. The stochastic analysis developed in this article is based on an original combination of functional inequalities and Foster-Lyapunov techniques with coupling, martingale techniques, random matrices and spectral analysis theory.

Self-Interacting Markov Chains

Abstract In this article we study a class of self-interacting Markov chain models. We propose a novel theoretical basis based on measure-valued processes and semigroup techniques to analyze its asymptotic behavior as the time parameter tends to infinity. We exhibit different types of decays to equilibrium, depending on the level of interaction. We illustrate these results in a variety of examples, including Gaussian or Poisson self-interacting models. We analyze the long-time behavior of a new class of evolutionary self-interacting chain models. These genetic type algorithms can also be regarded as reinforced stochastic explorations of an environment with obstacles related to a potential function.

On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering

We analyze the exponential stability properties of a class of measure-valued equations arising in nonlinear multi-target filtering problems. We also prove the uniform convergence properties w.r.t. the time parameter of a rather general class of stochastic filtering algorithms, including sequential Monte Carlo type models and mean field particle interpretation models. We illustrate these results in the context of the Bernoulli and the Probability Hypothesis Density filter, yielding what seems to be the first results of this kind in this subject.

Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering

In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The Glivenko–Cantelli and Donsker theorems presented in this work extend the corresponding statements in the classical theory and apply to a class of genetic type particle numerical schemes of the nonlinear filtering equation.

Open-loop regulation and tracking control based on a genealogical decision tree

The goal of this paper is to design a new control algorithm for open-loop control of complex systems. This control approach is based on a genealogical decision tree for both regulation and tracking control problems. The idea behind this control strategy consists of associating Gaussian distributions to both the norms of the control actions and the tracking errors. This stochastic search model can be interpreted as a simple genetic particle evolution model with a natural birth and death interpretation. It converges on probability. A numerical example dealing with the control of a fluidized bed combustion power plant illustrates the feasibility and the performance of this control algorithm.

Asymptotic results for genetic algorithms with applications to nonlinear estimation

Genetic algorithms (GAs) are stochastic search methods based on natural evolution processes. They are defined as a system of particles (or individuals) evolving randomly and undergoing adaptation in a time non-necessarily homogeneous environment represented by a collection of fitness functions. The purpose of this work is to study the long-time behavior as well as large population asymptotic of GAs. Another side topic is to discuss the applications of GAs in numerical function analysis, Feynman—Kac formulae approximations, and in nonlinear filtering problems. Several variations and refinements will also be presented including continuous-time and branching particle models with random population size.

Advanced interacting sequential Monte Carlo sampling for inverse scattering

The following electromagnetism (EM) inverse problem is addressed. It consists in estimating local radioelectric properties of materials recovering an object from global EM scattering measurements, at various incidences and wave frequencies. This large scale ill-posed inverse problem is explored by an intensive exploitation of an efficient 2D Maxwell solver, distributed on high performance computing machines. Applied to a large training data set, a statistical analysis reduces the problem to a simpler probabilistic metamodel, on which Bayesian inference can be performed. Considering the radioelectric properties as a hidden dynamic stochastic process, that evolves in function of the frequency, it is shown how advanced Markov Chain Monte Carlo methods, called Sequential Monte Carlo (SMC) or interacting particles, can take benefit of the structure and provide local EM property estimates.