Effective Dynamics of Stochastic Partial Differential Equations

Effective Dynamics of Stochastic Partial Differential Equations
Author: Jinqiao Duan,Wei WANG
Publsiher: Elsevier
Total Pages: 282
Release: 2014-03-06
Genre: Mathematics
ISBN: 9780128012697

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Effective Dynamics of Stochastic Partial Differential Equations focuses on stochastic partial differential equations with slow and fast time scales, or large and small spatial scales. The authors have developed basic techniques, such as averaging, slow manifolds, and homogenization, to extract effective dynamics from these stochastic partial differential equations. The authors’ experience both as researchers and teachers enable them to convert current research on extracting effective dynamics of stochastic partial differential equations into concise and comprehensive chapters. The book helps readers by providing an accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations. Each chapter also includes exercises and problems to enhance comprehension. New techniques for extracting effective dynamics of infinite dimensional dynamical systems under uncertainty Accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations Solutions or hints to all Exercises

Stochastic Partial Differential Equations and Related Fields

Stochastic Partial Differential Equations and Related Fields
Author: Andreas Eberle,Martin Grothaus,Walter Hoh,Moritz Kassmann,Wilhelm Stannat,Gerald Trutnau
Publsiher: Springer
Total Pages: 574
Release: 2018-07-03
Genre: Mathematics
ISBN: 9783319749297

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This Festschrift contains five research surveys and thirty-four shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10–14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Röckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker–Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art.

Numerical Methods for Stochastic Partial Differential Equations with White Noise

Numerical Methods for Stochastic Partial Differential Equations with White Noise
Author: Zhongqiang Zhang,George Em Karniadakis
Publsiher: Springer
Total Pages: 394
Release: 2017-09-01
Genre: Mathematics
ISBN: 9783319575117

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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 Ordinary and Stochastic Partial Differential Equations

Stochastic Ordinary and Stochastic Partial Differential Equations
Author: Peter Kotelenez
Publsiher: Springer Science & Business Media
Total Pages: 459
Release: 2007-12-05
Genre: Mathematics
ISBN: 9780387743172

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Stochastic Partial Differential Equations analyzes mathematical models of time-dependent physical phenomena on microscopic, macroscopic and mesoscopic levels. It provides a rigorous derivation of each level from the preceding one and examines the resulting mesoscopic equations in detail. Coverage first describes the transition from the microscopic equations to the mesoscopic equations. It then covers a general system for the positions of the large particles.

Stochastic Pdes And Modelling Of Multiscale Complex System

Stochastic Pdes And Modelling Of Multiscale Complex System
Author: Wang Wei,Chen Xiaopeng,Lv Yan
Publsiher: World Scientific
Total Pages: 240
Release: 2019-05-07
Genre: Mathematics
ISBN: 9789811200366

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This volume is devoted to original research results and survey articles reviewing recent developments in reduction for stochastic PDEs with multiscale as well as application to science and technology, and to present some future research direction. This volume includes a dozen chapters by leading experts in the area, with a broad audience in mind. It should be accessible to graduate students, junior researchers and other professionals who are interested in the subject. We also take this opportunity to celebrate the contributions of Professor Anthony J Roberts, an internationally leading figure on the occasion of his 60th years birthday in 2017.

Approximation of Stochastic Invariant Manifolds

Approximation of Stochastic Invariant Manifolds
Author: Mickaël D. Chekroun,Honghu Liu,Shouhong Wang
Publsiher: Springer
Total Pages: 127
Release: 2014-12-20
Genre: Mathematics
ISBN: 9783319124964

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This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.

Stochastic Parameterizing Manifolds and Non Markovian Reduced Equations

Stochastic Parameterizing Manifolds and Non Markovian Reduced Equations
Author: Mickaël D. Chekroun,Honghu Liu,Shouhong Wang
Publsiher: Springer
Total Pages: 129
Release: 2014-12-23
Genre: Mathematics
ISBN: 9783319125206

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In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.

An Introduction to Stochastic Dynamics

An Introduction to Stochastic Dynamics
Author: Jinqiao Duan
Publsiher: Cambridge University Press
Total Pages: 307
Release: 2015-04-13
Genre: Mathematics
ISBN: 9781107075399

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An accessible introduction for applied mathematicians to concepts and techniques for describing, quantifying, and understanding dynamics under uncertainty.

Infinite Dimensional And Finite Dimensional Stochastic Equations And Applications In Physics

Infinite Dimensional And Finite Dimensional Stochastic Equations And Applications In Physics
Author: Grecksch Wilfried,Lisei Hannelore
Publsiher: World Scientific
Total Pages: 260
Release: 2020-04-22
Genre: Science
ISBN: 9789811209802

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New Trends in Stochastic Analysis and Related Topics

New Trends in Stochastic Analysis and Related Topics
Author: Huaizhong Zhao,Aubrey Truman
Publsiher: World Scientific
Total Pages: 437
Release: 2012
Genre: Mathematics
ISBN: 9789814360913

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The volume is dedicated to Professor David Elworthy to celebrate his fundamental contribution and exceptional influence on stochastic analysis and related fields. Stochastic analysis has been profoundly developed as a vital fundamental research area in mathematics in recent decades. It has been discovered to have intrinsic connections with many other areas of mathematics such as partial differential equations, functional analysis, topology, differential geometry, dynamical systems, etc. Mathematicians developed many mathematical tools in stochastic analysis to understand and model random phenomena in physics, biology, finance, fluid, environment science, etc. This volume contains 12 comprehensive review/new articles written by world leading researchers (by invitation) and their collaborators. It covers stochastic analysis on manifolds, rough paths, Dirichlet forms, stochastic partial differential equations, stochastic dynamical systems, infinite dimensional analysis, stochastic flows, quantum stochastic analysis and stochastic Hamilton Jacobi theory. Articles contain cutting edge research methodology, results and ideas in relevant fields. They are of interest to research mathematicians and postgraduate students in stochastic analysis, probability, partial differential equations, dynamical systems, mathematical physics, as well as to physicists, financial mathematicians, engineers, etc.

Shape Optimization Homogenization and Optimal Control

Shape Optimization  Homogenization and Optimal Control
Author: Volker Schulz,Diaraf Seck
Publsiher: Springer
Total Pages: 275
Release: 2018-09-05
Genre: Mathematics
ISBN: 9783319904696

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The contributions in this volume give an insight into current research activities in Shape Optimization, Homogenization and Optimal Control performed in Africa, Germany and internationally. Seeds for collaboration can be found in the first four papers in the field of homogenization. Modelling and optimal control in partial differential equations is the topic of the next six papers, again mixed from Africa and Germany. Finally, new results in the field of shape optimization are discussed in the final international three papers. This workshop, held at the AIMS Center Senegal, March 13-16, 2017, has been supported by the Deutsche Forschungsgemeinschaft (DFG) and by the African Institute for Mathematical Sciences (AIMS) in Senegal, which is one of six centres of a pan-African network of centres of excellence for postgraduate education, research and outreach in mathematical sciences.

Multiple Time Scale Dynamics

Multiple Time Scale Dynamics
Author: Christian Kuehn
Publsiher: Springer
Total Pages: 814
Release: 2015-02-25
Genre: Mathematics
ISBN: 9783319123165

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This book provides an introduction to dynamical systems with multiple time scales. The approach it takes is to provide an overview of key areas, particularly topics that are less available in the introductory form. The broad range of topics included makes it accessible for students and researchers new to the field to gain a quick and thorough overview. The first of its kind, this book merges a wide variety of different mathematical techniques into a more unified framework. The book is highly illustrated with many examples and exercises and an extensive bibliography. The target audience of this book are senior undergraduates, graduate students as well as researchers interested in using the multiple time scale dynamics theory in nonlinear science, either from a theoretical or a mathematical modeling perspective.

Multiscale Methods

Multiscale Methods
Author: Grigoris Pavliotis,Andrew Stuart
Publsiher: Springer Science & Business Media
Total Pages: 310
Release: 2008-01-18
Genre: Mathematics
ISBN: 9780387738291

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This introduction to multiscale methods gives you a broad overview of the methods’ many uses and applications. The book begins by setting the theoretical foundations of the methods and then moves on to develop models and prove theorems. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable you to build your own skills and put them into practice. Extensions and generalizations of the results presented in the book, as well as references to the literature, are provided in the Discussion and Bibliography section at the end of each chapter.With the exception of Chapter One, all chapters are supplemented with exercises.

A Minicourse on Stochastic Partial Differential Equations

A Minicourse on Stochastic Partial Differential Equations
Author: Robert C. Dalang,Carl Mueller,Yimin Xiao,David Nualart
Publsiher: Springer Science & Business Media
Total Pages: 216
Release: 2009
Genre: Mathematics
ISBN: 9783540859932

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This title contains lectures that offer an introduction to modern topics in stochastic partial differential equations and bring together experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic PDEs.

Stochastic Partial Differential Equations

Stochastic Partial Differential Equations
Author: René Carmona
Publsiher: American Mathematical Soc.
Total Pages: 334
Release: 1999
Genre: Mathematics
ISBN: 9780821821008

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The field of Stochastic Partial Differential Equations (SPDEs) is one of the most dynamically developing areas of mathematics. It lies at the cross section of probability, partial differential equations, population biology, and mathematical physics. The field is especially attractive because of its interdisciplinary nature and the enormous richness of current and potential future applications. This volume is a collection of six important topics in SPDEs presented from the viewpoint of distinguished scientists working in the field and related areas. Emphasized are the genesis and applications of SPDEs as well as mathematical theory and numerical methods.