Une approache probabiliste de resolution
                    d’equations non lineaires

Bezandry, Paul H.; Ferland, René; Giroux, Gaston; Roberge, Jean-Claude

doi:10.1090/crmp/005/02

Cited by 7 publications

(4 citation statements)

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“…For simplicity, we now assume that (E, E) = (R d , B R d ). In such a setting, we can deduce, from this sequence of processes, an associated system of ODE's using the same techniques as in Ferland and Giroux [11] (see also Bezandry et al [1]). This implies that, for each time t, the laws of the sequence (X N 1 (t)) N ≥m converge to the probability law µ t where µ t is the solution of the Cauchy problem for the associated system of ODE's: We can think of µ •m t as the law at the root of the m-ary tree with only one interaction.…”

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Some New Results on Information Percolation

Bélanger

Giroux

2013

Stochastic Systems

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Some New Results on Information Percolation

Bélanger

Giroux

2013

Stochastic Systems

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“…Further references concerning the spatially inhomogeneous case are [66], [6], [149], [126], [116], [17], [205], [207], [72]. Developing the stochastic approach to the Boltzmann equation, systems with a general binary interaction between particles and a general (Markovian) single particle evolution (including spatial motion) were considered.…”

Section: Spatially Inhomogeneous Casementioning

confidence: 99%

Stochastic Numerics for the Boltzmann Equation

Rjasanow¹,

Wagner²

2005

Springer Series in Computational Mathematics

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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Cover design: design&production, Heidelberg Typeset by the authors using a Springer L A T E X macro package Printed on acid-free paper 46/3142sz-5 4 3 2 1 0 PrefaceStochastic numerical methods play an important role in large scale computations in the applied sciences. Such algorithms are convenient, since inherent stochastic components of complex phenomena can easily be incorporated. However, even if the real phenomenon is described by a deterministic equation, the high dimensionality often makes deterministic numerical methods intractable. A stochastic procedure, called direct simulation Monte Carlo (DSMC) method, has been developed in the physics and engineering community since the sixties. This method turned out to be a powerful tool for numerical studies of complex rarefied gas flows. It was successfully applied to problems ranging from aerospace engineering to material processing and nanotechnology. In many situations, DSMC can be considered as a stochastic algorithm for solving some macroscopic kinetic equation. An important example is the classical Boltzmann equation, which describes the time evolution of large systems of gas molecules in the rarefied regime, when the mean free path (distance between subsequent collisions of molecules) is not negligible compared to the characteristic length scale of the problem. This means that either the mean free path is big (space-shuttle design, vacuum technology), or the characteristic length is small (micro-device engineering). As the dimensionality of this nonlinear integro-differential equation is high (time, position, velocity), its numerical treatment is a typical application field of Monte Carlo algorithms.Intensive mathematical research on stochastic algorithms for the Boltzmann equation started in the eighties, when techniques for studying the convergence of interacting particle systems became available. Since that time much progress has been made in the justification and further development of these numerical methods.The purpose of this book is twofold. The first goal is to give a mathematical description of various classical DSMC procedures, using the theory of Markov processes (in particular, stochastic interacting particle systems) as a unifying framework. The second goal is a systematic treatment of an extension of DSMC, called stochastic weighted particle method (SWPM). This VI Preface method has been developed by the authors during the last decade. SWPM includes several new features, which are introduced for the purpose of variance reduction (rare event simulation). Rigorous results concerning the approximation of solutions to the Boltzmann equation by particle systems are given, covering both DSMC and SWPM. Thorough numerical experiments are performed, illustrating the behavior of systematic...

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“…In such a setting, we can deduce, from this sequence of processes, an associated system of ODE's using the same techniques as in Ferland and Giroux [11] (see also Bezandry et al [1]). This implies that, for each time t, the laws of the sequence (X N 1 (t)) N ≥m converge to the probability law µ t where µ t is the solution of the Cauchy problem for the associated system of ODE's:…”

mentioning

confidence: 99%

“…Now we will show that our Cauchy problem has a unique solution which can be expressed, by conditioning on the number of interactions up to time t, and then by the investor's history. Such conditioning give us (1) µ…”

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confidence: 99%