The Mollification Method and the Numerical Solution of Ill-Posed Problems (Diego A. Murio)

The mollification method and the numerical solution of ill-posed problems / by Diego A. Murio. p. cm. PREFACE During the last 20 years, the subject of ill-posed problems has expanded from a collection of individual techniques to a highly developed and rich branch of applied mathematics. This textbook essentially builds, from basic mathematical concepts, the understanding of the most important aspects of the numerical treatment of the applied inverse theory. The subject has grown—and continues to grow—at such a fast pace that it is impossible to offer a complete treatment in an introductory textbook and all that can be done is to discuss a few important and interesting topics. Inevitably, in making the selection, I have been influenced by my own interests which, on the other hand, allowed me the pleasure to write about the particular problems with which I am most familiar. This book is intended to be a self-contained presentation of practical computational methods which have been extensively and successfully applied to a wide range of ill-posed problems. The nature of the subject demands the application of special mathematical techniques—rarely seen in typical science courses and strange to normal engineering curricula—with which it is initially difficult to relate the steps of a calculation with the more classical concepts of stability and accuracy. This book is intended to solve the problem by giving an account of the theory that builds from the phenomena to be explained, keeping everything in as elementary a level as possible, making it useful to a wide circle of readers. The primary goal of this book is to provide an introduction to a number of essential ideas and techniques for the study of inverse problems that are ill posed. There is a clear emphasis on the mollification method and its multiple applications when implemented as a space marching algorithm. As such, this book is intended to be an outline of the numerical results obtained with the xi xii PREFACE mollification method and a manual of various other methods which are also used in arriving at some of these results. Although the presentation concentrates mostly on problems with origins in mechanical engineering, many of the ideas and methods can be easily applied to a broad class of situations. This book—an outgrowth of classes that the author has taught at the University of Cincinnati for several years in the Seminar of Applied Mathe-matics—is organized around a series …