Numerical Methods And Stochastics
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Numerical Methods For Stochastic Partial Differential Equations With White Noise
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Author : Zhongqiang Zhang
language : en
Publisher: Springer
Release Date : 2017-09-12
Numerical Methods For Stochastic Partial Differential Equations With White Noise written by Zhongqiang Zhang and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-12 with Mathematics categories.
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical comparison with other integration methods in random space is made. Part III covers spatial white noise. Here the authors discuss numerical methods for nonlinear elliptic equations as well as other equations with additive noise. Numerical methods for SPDEs with multiplicative noise are also discussed using the Wiener chaos expansion method. In addition, some SPDEs driven by non-Gaussian white noise are discussed and some model reduction methods (based on Wick-Malliavin calculus) are presented for generalized polynomial chaos expansion methods. Powerful techniques are provided for solving stochastic partial differential equations. This book can be considered as self-contained. Necessary background knowledge is presented in the appendices. Basic knowledge of probability theory and stochastic calculus is presented in Appendix A. In Appendix B some semi-analytical methods for SPDEs are presented. In Appendix C an introduction to Gauss quadrature is provided. In Appendix D, all the conclusions which are needed for proofs are presented, and in Appendix E a method to compute the convergence rate empirically is included. In addition, the authors provide a thorough review of the topics, both theoretical and computational exercises in the book with practical discussion of the effectiveness of the methods. Supporting Matlab files are made available to help illustrate some of the concepts further. Bibliographic notes are included at the end of each chapter. This book serves as a reference for graduate students and researchers in the mathematical sciences who would like to understand state-of-the-art numerical methods for stochastic partial differential equations with white noise.
Numerical Methods And Stochastics
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Author : T. J. Lyons
language : en
Publisher: American Mathematical Soc.
Release Date : 2002
Numerical Methods And Stochastics written by T. J. Lyons and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.
This volume represents the proceedings of the Workshop on Numerical Methods and Stochastics held at The Fields Institute in April 1999. The goal of the workshop was to identify emerging ideas in probability theory that influence future work in both probability and numerical computation. The book focuses on up-to-date results and gives novel approaches to computational problems based on cutting-edge techniques from the theory of probability and stochastic processes. Three papers discuss particle system approximations to solutions of the stochastic filtering problem. Two papers treat particle system equations. The paper on rough paths describes how to generate good approximations to stochastic integrals. An expository paper discusses a long-standing conjecture: the stochastic fast dynamo effect. A final paper gives an analysis of the error in binomial and trinomial approximations to solutions of the Black-Scholes stochastic differential equations. The book is intended for graduate students and research mathematicians interested in probability theory.
Numerical Methods For Stochastic Processes
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Author : Nicolas Bouleau
language : en
Publisher: John Wiley & Sons
Release Date : 1994-01-14
Numerical Methods For Stochastic Processes written by Nicolas Bouleau and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994-01-14 with Mathematics categories.
Gives greater rigor to numerical treatments of stochastic models. Contains Monte Carlo and quasi-Monte Carlo techniques, simulation of major stochastic procedures, deterministic methods adapted to Markovian problems and special problems related to stochastic integral and differential equations. Simulation methods are given throughout the text as well as numerous exercises.
Numerical Methods For Stochastic Control Problems In Continuous Time
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Author : Harold Kushner
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Numerical Methods For Stochastic Control Problems In Continuous Time written by Harold Kushner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.
This book is concerned with numerical methods for stochastic control and optimal stochastic control problems. The random process models of the controlled or uncontrolled stochastic systems are either diffusions or jump diffusions. Stochastic control is a very active area of research and new prob lem formulations and sometimes surprising applications appear regularly. We have chosen forms of the models which cover the great bulk of the for mulations of the continuous time stochastic control problems which have appeared to date. The standard formats are covered, but much emphasis is given to the newer and less well known formulations. The controlled process might be either stopped or absorbed on leaving a constraint set or upon first hitting a target set, or it might be reflected or "projected" from the boundary of a constraining set. In some of the more recent applications of the reflecting boundary problem, for example the so-called heavy traffic approximation problems, the directions of reflection are actually discontin uous. In general, the control might be representable as a bounded function or it might be of the so-called impulsive or singular control types. Both the "drift" and the "variance" might be controlled. The cost functions might be any of the standard types: Discounted, stopped on first exit from a set, finite time, optimal stopping, average cost per unit time over the infinite time interval, and so forth.
Numerical Methods And Stochastics
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Author : T. J. Lyons
language : en
Publisher:
Release Date : 2002
Numerical Methods And Stochastics written by T. J. Lyons and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Numerical analysis categories.
This volume represents the proceedings of the Workshop on Numerical Methods and Stochastics held at The Fields Institute in April 1999. The goal of the workshop was to identify emerging ideas in probability theory that influence future work in both probability and numerical computation. The book focuses on new results and gives novel approaches to computational problems based on the latest techniques from the theory of probability and stochastic processes. Three papers discuss particle system approximations to solutions of the stochastic filtering problem. Two papers treat particle system equa.
Stochastic Simulation And Monte Carlo Methods
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Author : Carl Graham
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-07-16
Stochastic Simulation And Monte Carlo Methods written by Carl Graham and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-07-16 with Mathematics categories.
In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view. The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.
Stochastic Numerics For The Boltzmann Equation
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Author : Sergej Rjasanow
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-11-04
Stochastic Numerics For The Boltzmann Equation written by Sergej Rjasanow and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-11-04 with Mathematics categories.
Stochastic numerical methods play an important role in large scale computations in the applied sciences. The first goal of this book is to give a mathematical description of classical direct simulation Monte Carlo (DSMC) procedures for rarefied gases, using the theory of Markov processes as a unifying framework. The second goal is a systematic treatment of an extension of DSMC, called stochastic weighted particle method. This method includes several new features, which are introduced for the purpose of variance reduction (rare event simulation). Rigorous convergence results as well as detailed numerical studies are presented.
Numerical Methods And Stochastics
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Author : T. J. Lyons
language : en
Publisher: American Mathematical Soc.
Release Date :
Numerical Methods And Stochastics written by T. J. Lyons and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
This volume represents the proceedings of the Workshop on Numerical Methods and Stochastics held at The Fields Institute in April 1999. The goal of the workshop was to identify emerging ideas in probability theory that influence future work in both probability and numerical computation. The book focuses on new results and gives novel approaches to computational problems based on the latest techniques from the theory of probability and stochastic processes. Three papers discussparticle system approximations to solutions of the stochastic filtering problem. Two papers treat particle system equations. The paper on ''rough paths'' describes how to generate good approximations to stochastic integrals. An expository paper discusses a long-standing conjecture: the stochastic fastdynamo effect. A final paper gives an analysis of the error in binomial and trinomial approximations to solutions of the Black-Scholes stochastic differential equations. The book is intended for graduate students and research mathematicians interested in probability theory.
Numerical Methods For Stochastic Partial Differential Equations With White Noise
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Author : Zhongqiang Zhang
language : en
Publisher: Springer
Release Date : 2017-09-01
Numerical Methods For Stochastic Partial Differential Equations With White Noise written by Zhongqiang Zhang and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-01 with Mathematics categories.
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical comparison with other integration methods in random space is made. Part III covers spatial white noise. Here the authors discuss numerical methods for nonlinear elliptic equations as well as other equations with additive noise. Numerical methods for SPDEs with multiplicative noise are also discussed using the Wiener chaos expansion method. In addition, some SPDEs driven by non-Gaussian white noise are discussed and some model reduction methods (based on Wick-Malliavin calculus) are presented for generalized polynomial chaos expansion methods. Powerful techniques are provided for solving stochastic partial differential equations. This book can be considered as self-contained. Necessary background knowledge is presented in the appendices. Basic knowledge of probability theory and stochastic calculus is presented in Appendix A. In Appendix B some semi-analytical methods for SPDEs are presented. In Appendix C an introduction to Gauss quadrature is provided. In Appendix D, all the conclusions which are needed for proofs are presented, and in Appendix E a method to compute the convergence rate empirically is included. In addition, the authors provide a thorough review of the topics, both theoretical and computational exercises in the book with practical discussion of the effectiveness of the methods. Supporting Matlab files are made available to help illustrate some of the concepts further. Bibliographic notes are included at the end of each chapter. This book serves as a reference for graduate students and researchers in the mathematical sciences who would like to understand state-of-the-art numerical methods for stochastic partial differential equations with white noise.
Stochastic Dynamical Systems
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Author : Josef Honerkamp
language : de
Publisher: John Wiley & Sons
Release Date : 1996-12-17
Stochastic Dynamical Systems written by Josef Honerkamp and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-12-17 with Mathematics categories.
Dieser einzigartige Band führt den Leser in die mathematische Begriffsbildung für komplexe Systeme ein. Er ist ideal für Studenten der Mathematik, Physik, Chemie und Medizin, die sich in ihrem Studium erstmals mit stochastischen dynamischen Systemen beschäftigen. Das Buch stellt praktische Methoden zur Verfügung, um mit solchen Systemen umgehen zu können, und stellt die zugundeliegenden Definitionen und theoretischen Annahmen, wo erforderlich, klar heraus. Im Gegensatz zu anderen Büchern über dieses Gebiet, die oft einen bestimmten Zugang bevorzugen, deckt Stochastical Dynamical Systems eine Vielzahl von stochastischen und statistischen Methoden ab, die für die Untersuchung von komplexen Systemen wie Polymerschmelzen, dem menschlichen Körper und der Atmosphäre absolut notwendig sind. Das Buch behandelt die Datenanalyse ebenso wie Simulationsmethoden für gegebene Modelle. Die ganze Vielfalt der klassischen und neuartigen Begriffe der mathematischen Stochastik wird in einem leicht verständlichen Stil erklärt, so daß die Leser diese Konzepte leicht für die Untersuchung ihrer Daten anwenden können.