Multiscale Methods for Fredholm Integral Equations

The recent appearance of wavelets as a new computational tool in applied mathematics has given a new impetus to the field of numerical analysis of Fredholm integral equations. This book gives an account of the state of the art in the study of fast multiscale methods for solving these equations based on wavelets. The authors begin by introducing essential concepts and describing conventional numerical methods. They then develop fast algorithms and apply these to solving linear, nonlinear Fredholm integral equations of the second kind, ill-posed integral equations of the first kind and eigen-problems of compact integral operators. Theorems of functional analysis used throughout the book are summarised in the appendix. The book is an essential reference for practitioners wishing to use the new techniques. It may also be used as a text, with the first five chapters forming the basis of a one-semester course for advanced undergraduates or beginning graduates.Preface; Introduction; 1. A review on the Fredholm approach; 2. Fredholm equations and projection theory; 3. Conventional numerical methods; 4. Multiscale basis functions; 5. Multiscale Galerkin methods; 6. Multiscale Petrov-Galerkin methods; 7. Multiscale collocation methods; 8. Numerical integrations and error control; 9. Fast solvers for discrete systems; 10. Multiscale methods for nonlinear integral equations; 11. Multiscale methods for ill-posed integral equations; 12. Eigen-problems of weakly singular integral operators; Appendix. Basic results from functional analysis; References; Symbols; Index.