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Numerical Methods for Stochastic Partial Differential Equations with White Noise de Zhongqiang Zhang

Numerical Methods for Stochastic Partial Differential Equations with White Noise: Applied Mathematical Sciences, cartea 196

Autor Zhongqiang Zhang, George Em Karniadakis
en Limba Engleză Paperback – 10 aug 2018

Subliniem relevanța critică a volumului Numerical Methods for Stochastic Partial Differential Equations with White Noise pentru cercetătorii și studenții la nivel de masterat sau doctorat care se confruntă cu modelarea incertitudinii în sisteme complexe. Într-un domeniu în care rigoarea matematică trebuie să întâlnească eficiența computațională, autorii Zhongqiang Zhang și George Em Karniadakis propun o abordare sistematică bazată pe aproximarea Wong-Zakai. Această metodă permite transformarea SPDE-urilor conduse de zgomot alb în ecuații cu zgomot neted, facilitând astfel aplicarea metodelor numerice clasice.

Notăm cu interes structura progresivă a lucrării, care ghidează cititorul de la bazele mișcării browniene către probleme complexe de zgomot alb spațial și multiplicativ. Prima parte se concentrează pe ecuații diferențiale ordinare cu întârziere, în timp ce a doua parte introduce metodele Wiener chaos și colocația stocastică pentru ecuații de advecție-difuzie-reacție. Un element distinctiv îl reprezintă Partea a III-a, unde sunt abordate ecuațiile eliptice neliniare și metodele de reducere a modelului bazate pe calculul Wick-Malliavin.

Considerăm acest titlu o alternativă tehnică riguroasă la An Introduction to Computational Stochastic PDEs de Gabriel J. Lord pentru cursurile de analiză numerică avansată. În timp ce lucrarea lui Lord pune accent pe metodele Monte Carlo și elemente finite, volumul de față, publicat în seria Applied Mathematical Sciences, aduce avantajul unei analize detaliate a expansiunii polinomiale chaos și a metodelor de colocație aplicate legilor de conservare stocastice. Experiența de lectură este susținută de un echilibru între demonstrații matematice și aplicații practice, autorii oferind fișiere Matlab pentru a ilustra convergența și stabilitatea algoritmilor prezentați.

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Specificații

ISBN-13: 9783319861814
ISBN-10: 3319861816
Pagini: 412
Ilustrații: XV, 394 p. 36 illus., 34 illus. in color.
Dimensiuni: 155 x 235 x 23 mm
Greutate: 0.62 kg
Ediția:Softcover reprint of the original 1st edition 2017
Editura: Springer
Colecția Applied Mathematical Sciences
Seria Applied Mathematical Sciences

Locul publicării:Cham, Switzerland

De ce să citești această carte

Recomandăm această lucrare ca o resursă fundamentală pentru cei care doresc să stăpânească metodele numerice de ultimă oră în domeniul ecuațiilor stocastice. Cititorul câștigă acces la tehnici avansate de discretizare și aproximare, validate prin exerciții practice și suport computațional. Este un instrument esențial pentru tranziția de la teoria pură a calculului stocastic la implementarea unor simulări numerice robuste în fizică sau inginerie.

Descriere scurtă

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.

Cuprins

Preface.- Prologue.- Brownian Motion and Stochastic Calculus.- Numerical Methods for Stochastic Differential Equations.- Part I Stochastic Ordinary Differential Equations.- Numerical Schemes for SDEs with Time Delay Using the Wong-Zakai Approximation.- Balanced Numerical Schemes for SDEs with non-Lipschitz Coefficients.- Part II Temporal White Noise.- Wiener Chaos Methods for Linear Stochastic Advection-Diffusion-Reaction Equations.- Stochastic Collocation Methods for Differential Equations with White Noise.- Comparison Between Wiener Chaos Methods and Stochastic Collocation Methods.- Application of Collocation Method to Stochastic Conservation Laws.- Part III Spatial White Noise.- Semilinear Elliptic Equations with Additive Noise.- Multiplicative White Noise: The Wick-Malliavin Approximation.- Epilogue.- Appendices.- A. Basics of Probability.- B. Semi-analytical Methods for SPDEs.- C. Gauss Quadrature.- D. Some Useful Inequalities and Lemmas.- E. Computation of Convergence Rate.

Recenzii

“Zhang and Karniadakis’ book may be used as a textbook, but it may also be considered as a reference for the state of the art concerning the numerical solution of stochastic differential equations involving white noise/Wiener processes/ Brownian motion. … Bibliographic notes address the state of the art in the field. Appendices give the necessary background in probability, stochastic calculus, semi-analytical approximation methods for stochastics differential equation, Gauss quadrature … . “ (José Eduardo Souze de Cursi, Mathematical Reviews, September, 2018)

“It is an interesting book on numerical methods for stochastic partial differential equations with white noise through the framework of Wong-Zakai approximation. ... . It is to be noted that the authors provide a thorough review of topics both theoretical and computational exercises to justify the effectiveness of the developed methods. Further, the MATLAB files are made available to the researchers and readers to understand the state of art of numerical methods for stochastic partial differential equations.” (Prabhat Kumar Mahanti, zbMATH 1380.65021, 2018)

Caracteristici

Includes both theoretical and computational exercises, allowing for use with mixed-level classes Provides Matlab codes for examples The first book to emphasizes the Wong-Zakai approximation Offers an approach to stochastic modeling other than the common Monte Carlo methods Includes supplementary material: sn.pub/extras Includes supplementary material: sn.pub/extras