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Mean field simulation for Monte Carlo integration. (English) Zbl 1282.65011

Monographs on Statistics and Applied Probability 126. Boca Raton, FL: CRC Press (ISBN 978-1-4665-0405-9/hbk; 978-1-138-19873-9/pbk; 978-1-4665-0417-2/ebook). xlvii, 578 p. (2013).
Monte Carlo simulation is one of the largest and most important numerical technique for the computer simulation of mathematical models with random ingredients. These simulation methods are of current use in computational physics, physical chemistry and computational biology for simulating the complex behavior of systems in high dimension.
This book deals with the theoretical foundations and the applications of mean field simulation models for Monte Carlo integration. One central theme of this book is the applications of mean field simulation theory of the mathematical and numerical analysis of nonlinear evolution equations in distribution spaces.
Chapter 1 provides a brief overview of some traditional Monte Carlo simulation of linear evolution equations associated with discrete and continuous time Marcov processes. The Markov chain models are developed, also continuous and discrete time models are discussed. The details of Fokker-Planck differential equations are presented. Nonlinear evolutions models in the space of probability measures are presented, and McKean type Markov chain models are realized. Feynman-Kac distribution flows and models are used to solve certain classes of probabilistic differential equations. The time discretizations scheme, especially Euler type approximations, are developed. The abstract nonlinear Markov chain models are illustrated through stochastic kinetic models, McKean-Vlasov diffusion models and Feynman-Kac models.
In Chapter 2, a brief exposition of the mathematical theory that is useful for the analysis of the asymptotic behavior of mean field particle models is given. An original stochastic perturbation analysis of mean field particle models in terms of local sampling error propagation and backward semigroup techniques are discussed. Some exponential concentration inequalities that apply to general McKean particle models are presented. The mean field Feynman-Kac models, including concentration inequalities for particle density profiles, genealogical measures, particle free energy models and particle backward Markov chain models are considered. The Feynman-Kac models are extended to noncommutatine models. Boltzmann-Gibbs measures are introduced and applied to the traditional Ising model and Sherrington-Kirkpatrick model. The microscopic evolution of a many-body system is described in terms of the Hamiltonian functional. A nonintrusive mean field IPS technique for the simulation of importance sampling distributions is developed and a multilevel splitting simulation is given.
In Chapter 3, an introduction to discrete time Feynman-Kac models and their application domains, including spatial branching processes, particle absorption evolutions, nonlinear filtering, hidden Markov chain models as well as sensitivity measure models arising in risk analysis, is realized. Some discussions on path space models, including the terminal time conditioning principle and Markov chain bridge models are provided. An introduction to two evolution operators – the Boltzmann-Gibbs transformation and the Feynman-Kac transformation – is given. An alternative description of the Feynman-Kac measures is presented and applied to the evolution equations. The abstract Feynman-Kac models are illustrated through special branching processes, particle absorption models, quenched and annealed models and historical processes. Feynman-Kac sensivity measures are designed.
Chapter 4 is dedicated to present four equivalent particle interpretations of the Feynman-Kac IPS mean field models – the branching process, the sequential Monte Carlo methodology, the interacting Markov chain Monte Carlo sampler and the mean field interacting particle system interpretation. Also, a Markov-McKean interpretation of nonlinear evolution equations in distribution spaces is given.
In Chapter 5, some connections between discrete generating Feynman-Kac models and their continuous time version are discussed. Some basic facts about processes and their infinitesimal generators are recalled. These stochastic modeling techniques are used to desing McKean jump type interpretations of Feynman-Kac models and the mean field particle interpretations of continuous time McKean models, in terms of infinitesimal generators. Some operator aspects of Markov processes and infinitesimal generators are considered. The stochastic modeling and the numerical analysis of continuous time Feynman-Kac models are developed. These models are related with the nonlinear Markov transport interpretation of the Boltzmann-Gibbs transformations. The mean field \(N\)-particle model associated with a given collection of generators is designed.
Chapter 6 focuses on nonlinear evolution of intensity measure models arising in spatial branching processes and the multiple object filtering theory. A McKean interpretation of the measure valued processes is given. Applications of mean field simulation theory to the analysis of nonlinear evolution equations are realized. The multiple-objects nonlinear filtering problems are discussed in more details. Spatial Poisson point processes, including restriction techniques and conditioning principles for partially observed models, are recalled. Association tree based measures and their mean field approximation are presented.
In Chapter 7, two applications as particle absorption type models and filtering and optimal control problems are considered. A mean field simulation of nonabsorbed particle motions in terms of genealogical tree based occupation measures is considered. Particle absorption models in random environments are discussed. A particle Feynman-Kac model that combines the evolution of the environment parameter with the mean field sampling schemes is presented. Time homogeneous absorption models and quasi-invariant measures are considered.
Chapter 8 is dedicated to signal processing and control systems. The probabilistic description of filtering problems in terms of signal-observation Markov chains in general state spaces is presented. A Feynman-Kac description of the conditional distributions of the signals is provided. The linear-Gaussian filtering models and the derivation of the traditional forward-backward Kalman filters are considered. A mean field interacting Kalman filter is presented. The partial linear-Gaussian models, the quenched and annealed filtering models are developed. The problem how to fixe parameter estimation in hidden Markov chain models and their mean field IPS interpretations is solved. An original and powerful perturbation analysis of nonlinear semigroups is given. A mean field IPS solving of Snell envelope evolution equations is realized.
In Chapter 9, some basic definitions on Feynman-Kac models, their normalized and unnormalized distributions, and their evolution equations on general state space models are reviewed. A short reminder of dome notations, concepts and definitions, related to the general class of McKean-Markov chain models is provided. A perfect sampling interpretation of the McKean models is presented. A Monte Carlo model, based on independent copies of the McKean model is discussed. The fluctuations of the perfect sampling model in terms of the local sampling errors, using the backward decomposition formula are analyzed. Some of the simplest convergence estimates are introduced. The variance analysis of the local sampling random field models, associated with a given mean field simulation scheme is realized. The \(L_m\)-mean error estimates as well as a functional central limit theorem describing the limiting fluctuations in terms of a sequence of independent Gaussian random fields with covariance functions that depend on the choice of the McKean interpretation model are shown. Also, some rather crude nonasymptotic \(L_m\)-mean error estimates are obtained. A particle estimation of the essential supremum of some functions and the limiting Feynman-Kac measures are discussed.
In Chapter 10, the convergence analysis of a general and abstract class of mean field IPS models is investigated. A formula which describes the mean field IPS models associated with Markov-McKean interpretation models of nonlinear measure valued equations in the space of probability measures is presented. Some rather weak regularity properties that allow the development of a stochastic perturbation technique and a first-order fluctuation analysis of mean field IPS models are presented. Several illustrations, including Feynman-Kac models, integrating jump processes, simplified McKean models of gases, Gaussian mean field models as well as generalized McKean-Vlasov type diffusion models with integrating jumps are given. A differential calculus of semigroups on measure spaces equipped with the total variation norm is developed. This framework provides an alternative description of the regularity properties in terms of Fréchet and Gâteaux differentials. An alternative stochastic coupling technique to analyze the deviations of the mean field IPS models is provided. The central limit theorem is proved in details.
Chapter 11 is dedicated to the theory of empirical processes and measure concentration theory. Some more or less well-known stochastic techniques for analyzing the concentration properties of empirical processes, associated with independent random sequences on a measurable state spaces are collected. Stochastic perturbation techniques for analyzing exponential concentration properties of functional empirical processes are presented. This stochastic analysis combines Orlicz’ s norm techniques, Kintchine type inequalities, maximal inequalities as well as Laplace-Cramèr-Chernov estimation models to derive quantitative concentration inequalities. The exponential concentration analysis of sequences of empirical processes associated with conditionally independent random variables is developed. A general class of interaction particle processes with non-necessarily mean field type dependency is considered. A stochastic perturbation technique to analyze the second-order type decompositions is presented.
Chapters 12 and 13 consider the semigroup structure and the weak regularity properties of Feynman-Kac distribution flows and their extended version, discussed in Chapter 6. A quantitative construction estimate for the normalized Feynman-Kac semigroups is presented. Semigroups of nonlinear Markov chain models are presented. The semigroup analysis of the backward Markov chain models is developed. In Chapter 13, the semigroup analysis of the intensity measures of spation branching processes, discussed in Section 6.1, is further developed, also results obtained in Section 6.2 are further investigated. An essential weak Lipschitz type property of the semigroup associated with the one step transformations of the flow is presented.
In Chapter 14, the analysis of the different classes of mean field models, presented in Section 1.5 and Chapter 2, are further developed. The convergence analysis of the particle density profiles is realized. The Feynman-Kac particle models and their extended version, discussed in Chapter 6, as well as more general classes of mean field particle models are developed. Some key first-order Taylor’ s type decompositions in distribution spaces, which are the progenitors, are presented. The fluctuations of the particle density profiles are analyzed. A general class of IPS models, introduced in Chapter 10, is considered. Nonlinear evolution models are considered and analyzed.
In Chapter 15, some key equivalence principles that allow applying without further work most of the results presented in Chapter 14 dedicated to the stochastic analysis of particle density profiles, are reviewed. Some non-asymptotic theorems, including Orlicz norms and \(L_p\)-mean error estimates, are presented. Also, some non-asymptotic theorems for the occupation measures of the ancestral tree, are presented.
Chapters 16 and 17 are dedicated to the convergence analysis of particle free energy models and backward particle Markov chains.
The bibliography of the book contains 576 items.

MSC:

65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
65C35 Stochastic particle methods
65C30 Numerical solutions to stochastic differential and integral equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
68U20 Simulation (MSC2010)
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
60G35 Signal detection and filtering (aspects of stochastic processes)
81V70 Many-body theory; quantum Hall effect
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