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Grigorios A. Pavliotis, Andrew M. StuartAn Introduction to Multiscale MethodsThe aim of these notes is to describe, in a uni?ed fashion, a set of methods for the simplification of a wide variety of problems which all share the common feature of possessing multiple scales. The mathematical methods which we study are often referred to as the methods of averaging and of homogenization. The methods ap- ply to partial differential equations (PDE), stochastic differential equations (SDE), ordinary differential equations (ODE) and Markov chains. The unifying principle underlying the collection of techniques described here is the approximation of sin- gularly perturbed linear equations. The unity of the subject is most clearly visible in the application of perturbation expansions to the approximation of these singu- lar perturbation problems. A significant portion of the notes is devoted to such perturbation expansions. In this context we use the term Result to describe the conclusions of a formal perturbation argument. This enables us to derive important approximation results without the burden of rigorous proof which can sometimes obfuscate the main ideas. However, we will also study a variety of tools from analysis and probability, used to place the approximations derived on a rigorous footing. The resulting theorems are proved using a range of methods, tailored to different settings. There is less unity to this part of the subject. As a consequence considerable background is required to absorb the entire theoretical side of the sub- ject, and we devote a significant fraction of the book to this background material.Bibliographical note: |