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Effective Dynamics Of Stochastic Partial Differential Equations

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Effective Dynamics of Stochastic Partial Differential Equations

Effective Dynamics of Stochastic Partial Differential Equations Book
Author : Jinqiao Duan,Wei WANG
Publisher : Elsevier
Release : 2014-03-06
ISBN : 0128012692
Language : En, Es, Fr & De

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Book Description :

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

An Introduction to Stochastic Dynamics

An Introduction to Stochastic Dynamics Book
Author : Jinqiao Duan
Publisher : Cambridge University Press
Release : 2015-04-13
ISBN : 1107075394
Language : En, Es, Fr & De

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Book Description :

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

Stochastic Partial Differential Equations and Related Fields

Stochastic Partial Differential Equations and Related Fields Book
Author : Andreas Eberle,Martin Grothaus,Walter Hoh,Moritz Kassmann,Wilhelm Stannat,Gerald Trutnau
Publisher : Springer
Release : 2018-07-03
ISBN : 3319749293
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : Zhongqiang Zhang,George Em Karniadakis
Publisher : Springer
Release : 2017-09-01
ISBN : 3319575112
Language : En, Es, Fr & De

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Book Description :

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 Pdes And Modelling Of Multiscale Complex System

Stochastic Pdes And Modelling Of Multiscale Complex System Book
Author : Wang Wei,Chen Xiaopeng,Lv Yan
Publisher : World Scientific
Release : 2019-05-07
ISBN : 981120036X
Language : En, Es, Fr & De

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Book Description :

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.

Stochastic Ordinary and Stochastic Partial Differential Equations

Stochastic Ordinary and Stochastic Partial Differential Equations Book
Author : Peter Kotelenez
Publisher : Springer Science & Business Media
Release : 2007-12-05
ISBN : 0387743170
Language : En, Es, Fr & De

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Book Description :

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.

Approximation of Stochastic Invariant Manifolds

Approximation of Stochastic Invariant Manifolds Book
Author : Mickaël D. Chekroun,Honghu Liu,Shouhong Wang
Publisher : Springer
Release : 2014-12-20
ISBN : 331912496X
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : Mickaël D. Chekroun,Honghu Liu,Shouhong Wang
Publisher : Springer
Release : 2014-12-23
ISBN : 3319125206
Language : En, Es, Fr & De

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Book Description :

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.

Infinite Dimensional And Finite Dimensional Stochastic Equations And Applications In Physics

Infinite Dimensional And Finite Dimensional Stochastic Equations And Applications In Physics Book
Author : Grecksch Wilfried,Lisei Hannelore
Publisher : World Scientific
Release : 2020-04-22
ISBN : 9811209804
Language : En, Es, Fr & De

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Book Description :

Download Infinite Dimensional And Finite Dimensional Stochastic Equations And Applications In Physics book written by Grecksch Wilfried,Lisei Hannelore, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

New Trends in Stochastic Analysis and Related Topics

New Trends in Stochastic Analysis and Related Topics Book
Author : Huaizhong Zhao,Aubrey Truman
Publisher : World Scientific
Release : 2012
ISBN : 9814360910
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : Volker Schulz,Diaraf Seck
Publisher : Springer
Release : 2018-09-05
ISBN : 3319904698
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : Christian Kuehn
Publisher : Springer
Release : 2015-02-25
ISBN : 3319123165
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : Grigoris Pavliotis,Andrew Stuart
Publisher : Springer Science & Business Media
Release : 2008-01-18
ISBN : 0387738290
Language : En, Es, Fr & De

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Book Description :

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.

Stochastic Dynamics Out of Equilibrium

Stochastic Dynamics Out of Equilibrium Book
Author : Giambattista Giacomin,Stefano Olla,Ellen Saada,Herbert Spohn,Gabriel Stoltz
Publisher : Springer
Release : 2019-06-30
ISBN : 3030150968
Language : En, Es, Fr & De

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Book Description :

Stemming from the IHP trimester "Stochastic Dynamics Out of Equilibrium", this collection of contributions focuses on aspects of nonequilibrium dynamics and its ongoing developments. It is common practice in statistical mechanics to use models of large interacting assemblies governed by stochastic dynamics. In this context "equilibrium" is understood as stochastically (time) reversible dynamics with respect to a prescribed Gibbs measure. Nonequilibrium dynamics correspond on the other hand to irreversible evolutions, where fluxes appear in physical systems, and steady-state measures are unknown. The trimester, held at the Institut Henri Poincaré (IHP) in Paris from April to July 2017, comprised various events relating to three domains (i) transport in non-equilibrium statistical mechanics; (ii) the design of more efficient simulation methods; (iii) life sciences. It brought together physicists, mathematicians from many domains, computer scientists, as well as researchers working at the interface between biology, physics and mathematics. The present volume is indispensable reading for researchers and Ph.D. students working in such areas.

Hamiltonian Partial Differential Equations and Applications

Hamiltonian Partial Differential Equations and Applications Book
Author : Philippe Guyenne,David Nicholls,Catherine Sulem
Publisher : Springer
Release : 2015-09-11
ISBN : 149392950X
Language : En, Es, Fr & De

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Book Description :

This book is a unique selection of work by world-class experts exploring the latest developments in Hamiltonian partial differential equations and their applications. Topics covered within are representative of the field’s wide scope, including KAM and normal form theories, perturbation and variational methods, integrable systems, stability of nonlinear solutions as well as applications to cosmology, fluid mechanics and water waves. The volume contains both surveys and original research papers and gives a concise overview of the above topics, with results ranging from mathematical modeling to rigorous analysis and numerical simulation. It will be of particular interest to graduate students as well as researchers in mathematics and physics, who wish to learn more about the powerful and elegant analytical techniques for Hamiltonian partial differential equations.

Metastability

Metastability Book
Author : Anton Bovier,Frank den Hollander
Publisher : Springer
Release : 2016-02-11
ISBN : 3319247778
Language : En, Es, Fr & De

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Book Description :

This monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes. The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets. The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour. The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability.

Nonlinear Dynamics and Stochastic Mechanics

Nonlinear Dynamics and Stochastic Mechanics Book
Author : Wolfgang Kliemann
Publisher : CRC Press
Release : 2018-05-04
ISBN : 1351091956
Language : En, Es, Fr & De

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Book Description :

Engineering systems have played a crucial role in stimulating many of the modern developments in nonlinear and stochastic dynamics. After 20 years of rapid progress in these areas, this book provides an overview of the current state of nonlinear modeling and analysis for mechanical and structural systems. This volume is a coherent compendium written by leading experts from the United States, Canada, Western and Eastern Europe, and Australia. The 22 articles describe the background, recent developments, applications, and future directions in bifurcation theory, chaos, perturbation methods, stochastic stability, stochastic flows, random vibrations, reliability, disordered systems, earthquake engineering, and numerics. The book gives readers a sophisticated toolbox that will allow them to tackle modeling problems in mechanical systems that use stochastic and nonlinear dynamics ideas. An extensive bibliography and index ensure this volume will remain a reference standard for years to come.

Stochastic Processes Multiscale Modeling and Numerical Methods for Computational Cellular Biology

Stochastic Processes  Multiscale Modeling  and Numerical Methods for Computational Cellular Biology Book
Author : David Holcman
Publisher : Springer
Release : 2017-10-04
ISBN : 3319626272
Language : En, Es, Fr & De

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Book Description :

This book focuses on the modeling and mathematical analysis of stochastic dynamical systems along with their simulations. The collected chapters will review fundamental and current topics and approaches to dynamical systems in cellular biology. This text aims to develop improved mathematical and computational methods with which to study biological processes. At the scale of a single cell, stochasticity becomes important due to low copy numbers of biological molecules, such as mRNA and proteins that take part in biochemical reactions driving cellular processes. When trying to describe such biological processes, the traditional deterministic models are often inadequate, precisely because of these low copy numbers. This book presents stochastic models, which are necessary to account for small particle numbers and extrinsic noise sources. The complexity of these models depend upon whether the biochemical reactions are diffusion-limited or reaction-limited. In the former case, one needs to adopt the framework of stochastic reaction-diffusion models, while in the latter, one can describe the processes by adopting the framework of Markov jump processes and stochastic differential equations. Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology will appeal to graduate students and researchers in the fields of applied mathematics, biophysics, and cellular biology.

Stochastic Partial Differential Equations and Applications VII

Stochastic Partial Differential Equations and Applications   VII Book
Author : Giuseppe Da Prato,Luciano Tubaro
Publisher : CRC Press
Release : 2005-10-12
ISBN : 9781420028720
Language : En, Es, Fr & De

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Book Description :

Stochastic Partial Differential Equations and Applications gives an overview of current state-of-the-art stochastic PDEs in several fields, such as filtering theory, stochastic quantization, quantum probability, and mathematical finance. Featuring contributions from leading expert participants at an international conference on the subject, this book presents valuable information for PhD students in probability and PDEs as well as for researchers in pure and applied mathematics. Coverage includes Navier-Stokes equations, Ornstein-Uhlenbeck semigroups, quantum stochastic differential equations, applications of SPDE, 3D stochastic Navier-Stokes equations, and nonlinear filtering.

Amplitude Equations for Stochastic Partial Differential Equations

Amplitude Equations for Stochastic Partial Differential Equations Book
Author : Dirk Blomker
Publisher : World Scientific
Release : 2007
ISBN : 9812770607
Language : En, Es, Fr & De

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Book Description :

Rigorous error estimates for amplitude equations are well known for deterministic PDEs, and there is a large body of literature over the past two decades. However, there seems to be a lack of literature for stochastic equations, although the theory is being successfully used in the applied community, such as for convective instabilities, without reliable error estimates at hand. This book is the first step in closing this gap. The author provides details about the reduction of dynamics to more simpler equations via amplitude or modulation equations, which relies on the natural separation of time-scales present near a change of stability. For students, the book provides a lucid introduction to the subject highlighting the new tools necessary for stochastic equations, while serving as an excellent guide to recent research.