Numerical Methods For Stochastic Partial Differential Equations With White Noise
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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 : 2017-09-01
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.
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.
An Introduction To Computational Stochastic Pdes
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Author : Gabriel J. Lord
language : en
Publisher: Cambridge University Press
Release Date : 2014-08-11
An Introduction To Computational Stochastic Pdes written by Gabriel J. Lord and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-08-11 with Business & Economics categories.
This book offers a practical presentation of stochastic partial differential equations arising in physical applications and their numerical approximation.
Applied Stochastic Differential Equations
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Author : Simo Särkkä
language : en
Publisher: Cambridge University Press
Release Date : 2019-05-02
Applied Stochastic Differential Equations written by Simo Särkkä and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-02 with Business & Economics categories.
With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
Stochastic Processes
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Author : Don Kulasiri
language : en
Publisher: BoD – Books on Demand
Release Date : 2024-07-31
Stochastic Processes written by Don Kulasiri and has been published by BoD – Books on Demand this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-07-31 with categories.
Numerical And Symbolic Scientific Computing
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Author : Ulrich Langer
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-11-19
Numerical And Symbolic Scientific Computing written by Ulrich Langer 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 2011-11-19 with Mathematics categories.
The book presents the state of the art and results and also includes articles pointing to future developments. Most of the articles center around the theme of linear partial differential equations. Major aspects are fast solvers in elastoplasticity, symbolic analysis for boundary problems, symbolic treatment of operators, computer algebra, and finite element methods, a symbolic approach to finite difference schemes, cylindrical algebraic decomposition and local Fourier analysis, and white noise analysis for stochastic partial differential equations. Further numerical-symbolic topics range from applied and computational geometry to computer algebra methods used for total variation energy minimization.
Approximation Methods And Analytical Modeling Using Partial Differential Equations
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Author : Tamara Fastovska
language : en
Publisher: Frontiers Media SA
Release Date : 2025-03-28
Approximation Methods And Analytical Modeling Using Partial Differential Equations written by Tamara Fastovska and has been published by Frontiers Media SA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-03-28 with Science categories.
Adequate mathematical modeling is the key to success for many real-world projects in engineering, medicine, and other applied areas. As soon as an appropriate mathematical model is developed, it can be comprehensively analyzed by a broad spectrum of available mathematical methods. For example, compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases and in mathematical models of population genetics. While the existence of an optimal solution under certain condition can be often proved rigorously, this does not always mean that such a solution is easy to implement in practice. Finding a reasonable approximation can in itself be a challenging research problem. This Research Topic is devoted to modeling, analysis, and approximation problems whose solutions exploit and explore the theory of partial differential equations. It aims to highlight new analytical tools for use in the modeling of problems arising in applied sciences and practical areas. Researchers are invited to submit articles that investigate the qualitative behavior of weak solutions (removability conditions for singularities), the dependence of the local asymptotic property of these solutions on initial and boundary data, and also the existence of solutions. Contributors are particularly encouraged to focus on anisotropic models: analyzing the preconditions on the strength of the anisotropy, and comparing the analytical estimates for the growth behavior of the solutions near the singularities with the observed growth in numerical simulations. The qualitative analysis and analytical results should be confirmed by the numerically observed solution behavior.
Computational Science And Its Applications Iccsa 2025 Workshops
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Author : Osvaldo Gervasi
language : en
Publisher: Springer Nature
Release Date : 2025-07-08
Computational Science And Its Applications Iccsa 2025 Workshops written by Osvaldo Gervasi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-08 with Computers categories.
The fourteen-volume set LNCS 15886-15899 constitutes the papers of several workshops which were held in conjunction with the 25th International Conference on Computational Science and Its Applications, ICCSA 2025, held in Istanbul, Turkey, during June 30–July 3, 2025.. The 362 full papers, 37 short papers and 2 PHD showcase included in this book were carefully reviewed and selected from 1043 submissions. In addition, the conference consisted of 58 workshops, focusing on very topical issues of importance to science, technology and society: from new mathematical approaches for solving complex computational systems, to information and knowledge in the Internet of Things, new statistical and optimization methods, several Artificial Intelligence approaches, sustainability issues, smart cities and related technologies.
Stochastic Partial Differential Equations
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Author : Étienne Pardoux
language : en
Publisher:
Release Date : 2021
Stochastic Partial Differential Equations written by Étienne Pardoux and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with categories.
This book gives a concise introduction to the classical theory of stochastic partial differential equations (SPDEs). It begins by describing the classes of equations which are studied later in the book, together with a list of motivating examples of SPDEs which are used in physics, population dynamics, neurophysiology, finance and signal processing. The central part of the book studies SPDEs as infinite-dimensional SDEs, based on the variational approach to PDEs. This extends both the classical Itô formulation and the martingale problem approach due to Stroock and Varadhan. The final chapter considers the solution of a space-time white noise-driven SPDE as a real-valued function of time and (one-dimensional) space. The results of J. Walsh's St Flour notes on the existence, uniqueness and Hölder regularity of the solution are presented. In addition, conditions are given under which the solution remains nonnegative, and the Malliavin calculus is applied. Lastly, reflected SPDEs and their connection with super Brownian motion are considered. At a time when new sophisticated branches of the subject are being developed, this book will be a welcome reference on classical SPDEs for newcomers to the theory.
Stochastic Partial Differential Equations
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Author : Sergey V. Lototsky
language : en
Publisher: Springer
Release Date : 2017-07-06
Stochastic Partial Differential Equations written by Sergey V. Lototsky and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-07-06 with Mathematics categories.
Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions. As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected to the material discussed at a particular place in the text. The questions usually ask to verify something, so that the reader already knows the answer and, if pressed for time, can move on. Accordingly, no solutions are provided, but there are often hints on how to proceed. The book will be of interest to everybody working in the area of stochastic analysis, from beginning graduate students to experts in the field.