Multiscale Methods: Averaging and Homogenization

Grigorios A. Pavliotis, Andrew M. Stuart (auth.)9780387738284, 0387738282, 0387738290

This introduction to multiscale methods gives readers a broad overview of the many uses and applications of the methods. The book begins by setting the theoretical foundations of the subject area, and moves on to develop a unified approach to the simplification of a wide range of problems which possess multiple scales, via perturbation expansions; differential equations and stochastic processes are studied in one unified framework. The book concludes with an overview of a range of theoretical tools used to justify the simplified models derived via the perturbation expansions.

The presentation of the material is particularly suited to the range of mathematicians, scientists and engineers who want to exploit multiscale methods in applications. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable readers to build their 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. All of the twenty-one chapters are supplemented with exercises.

Grigorios Pavliotis is a Lecturer of Mathematics at Imperial College London.

Andrew Stuart is a Professor of Mathematics at Warwick University.

Table of contents :
Front Matter….Pages I-XVIII
Front Matter….Pages 1-1
Introduction….Pages 1-10
Front Matter….Pages 11-11
Analysis….Pages 13-35
Probability Theory and Stochastic Processes….Pages 37-57
Ordinary Differential Equations….Pages 59-72
Markov Chains….Pages 73-84
Stochastic Differential Equations….Pages 85-101
Partial Differential Equations….Pages 103-124
Front Matter….Pages 125-125
Invariant Manifolds for ODEs….Pages 127-135
Averaging for Markov Chains….Pages 137-143
Averaging for ODEs and SDEs….Pages 145-156
Homogenization for ODEs and SDEs….Pages 157-182
Homogenization for Elliptic PDEs….Pages 183-202
Homogenization for Parabolic PDEs….Pages 203-226
Averaging for Linear Transport and Parabolic PDEs….Pages 227-236
Front Matter….Pages 237-237
Invariant Manifolds for ODEs: The Convergence Theorem….Pages 239-244
Averaging for Markov Chains: The Convergence Theorem….Pages 245-248
Averaging for SDEs: The Convergence Theorem….Pages 249-253
Homogenization for SDEs: The Convergence Theorem….Pages 255-262
Homogenization for Elliptic PDEs: The Convergence Theorem….Pages 263-272
Homogenization for Elliptic PDEs: The Convergence Theorem….Pages 273-278
Front Matter….Pages 237-237
Averaging for Linear Transport and Parabolic PDEs: The Convergence Theorem….Pages 279-285
Back Matter….Pages 287-307