Introduction.- Analysis.- Probability Theory and Stochastic Processes.- Ordinary Differential Equations.- Markov Chains.- Stochastic Differential Equations.- Partial Differential Equations.- Invariant Manifolds for ODEs.- Averaging for Markov Chains.- Averaging for ODEs and SDEs.- Homogenization for ODEs and SDEs.- Homogenization for Elliptic PDEs.- Homogenization for Parabolic PDEs.- Averaging for Linear Transport and Parabolic PDEs.- Invariant Manifolds for ODEs: The Convergence Theorem.- Averaging for Markov Chains: The Convergence Theorem.- Averaging for SDEs: The Convergence Theorem.- Homogenization for SDEs: The Convergence Theorem.- Homogenization for Elliptic PDEs: The Convergence Theorem.- Homogenization for Parabolic PDEs: The Convergence Theorem.- Averaging for Linear Transport and Parabolic PDEs: The Convergence Theorem.
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.
The author of this book is one of the leading world experts on multiscale methods, a hot area in applied mathematics. The book is meant to be an introduction, aimed primarily towards graduate students. Part I of the book and Part III of the book are necessarily terse and present the wide range of applications of the ideas, and illustrate their unity. The presentation of the material here is particularly suited to the pedagogical goal of communicating the subject area to the wide range of mathematicians, scientists and engineers who are currently engaged in the use of these tools to tackle the enormous range of applications that require them. Extensions and generalizations of the results presented in these notes, as well as references to the literature, are given in the Discussion and Bibliography section, at the end of each chapter. With the exception of Chapter 1, all chapters are supplemented with exercises.