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An Introduction to the Mathematical Theory of Inverse Problems

2011-03-24 Mathematics
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Release : 2011-03-24
Genre : Mathematics
Kind : eBook
Book Rating : 747/5 ( reviews)

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Book Synopsis An Introduction to the Mathematical Theory of Inverse Problems by : Andreas Kirsch

Download or read book An Introduction to the Mathematical Theory of Inverse Problems written by Andreas Kirsch. This book was released on 2011-03-24. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the reader to the area of inverse problems. The study of inverse problems is of vital interest to many areas of science and technology such as geophysical exploration, system identification, nondestructive testing and ultrasonic tomography. The aim of this book is twofold: in the first part, the reader is exposed to the basic notions and difficulties encountered with ill-posed problems. Basic properties of regularization methods for linear ill-posed problems are studied by means of several simple analytical and numerical examples. The second part of the book presents two special nonlinear inverse problems in detail - the inverse spectral problem and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness and continuous dependence on parameters. Then some theoretical results as well as numerical procedures for the inverse problems are discussed. The choice of material and its presentation in the book are new, thus making it particularly suitable for graduate students. Basic knowledge of real analysis is assumed. In this new edition, the Factorization Method is included as one of the prominent members in this monograph. Since the Factorization Method is particularly simple for the problem of EIT and this field has attracted a lot of attention during the past decade a chapter on EIT has been added in this monograph as Chapter 5 while the chapter on inverse scattering theory is now Chapter 6.The main changes of this second edition compared to the first edition concern only Chapters 5 and 6 and the Appendix A. Chapter 5 introduces the reader to the inverse problem of electrical impedance tomography.

An Introduction to the Mathematical Theory of Inverse Problems

2012-08-14 Science
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Release : 2012-08-14
Genre : Science
Kind : eBook
Book Rating : 389/5 ( reviews)

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Book Synopsis An Introduction to the Mathematical Theory of Inverse Problems by : Andreas Kirsch

Download or read book An Introduction to the Mathematical Theory of Inverse Problems written by Andreas Kirsch. This book was released on 2012-08-14. Available in PDF, EPUB and Kindle. Book excerpt: Following Keller [119] we call two problems inverse to each other if the for mulation of each of them requires full or partial knowledge of the other. By this definition, it is obviously arbitrary which of the two problems we call the direct and which we call the inverse problem. But usually, one of the problems has been studied earlier and, perhaps, in more detail. This one is usually called the direct problem, whereas the other is the inverse problem. However, there is often another, more important difference between these two problems. Hadamard (see [91]) introduced the concept of a well-posed problem, originating from the philosophy that the mathematical model of a physical problem has to have the properties of uniqueness, existence, and stability of the solution. If one of the properties fails to hold, he called the problem ill-posed. It turns out that many interesting and important inverse in science lead to ill-posed problems, while the corresponding di problems rect problems are well-posed. Often, existence and uniqueness can be forced by enlarging or reducing the solution space (the space of "models"). For restoring stability, however, one has to change the topology of the spaces, which is in many cases impossible because of the presence of measurement errors. At first glance, it seems to be impossible to compute the solution of a problem numerically if the solution of the problem does not depend continuously on the data, i. e. , for the case of ill-posed problems.

Inverse Problems in the Mathematical Sciences

2013-12-14 Technology & Engineering
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Release : 2013-12-14
Genre : Technology & Engineering
Kind : eBook
Book Rating : 020/5 ( reviews)

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Book Synopsis Inverse Problems in the Mathematical Sciences by : Charles W. Groetsch

Download or read book Inverse Problems in the Mathematical Sciences written by Charles W. Groetsch. This book was released on 2013-12-14. Available in PDF, EPUB and Kindle. Book excerpt: Inverse problems are immensely important in modern science and technology. However, the broad mathematical issues raised by inverse problems receive scant attention in the university curriculum. This book aims to remedy this state of affairs by supplying an accessible introduction, at a modest mathematical level, to the alluring field of inverse problems. Many models of inverse problems from science and engineering are dealt with and nearly a hundred exercises, of varying difficulty, involving mathematical analysis, numerical treatment, or modelling of inverse problems, are provided. The main themes of the book are: causation problem modeled as integral equations; model identification problems, posed as coefficient determination problems in differential equations; the functional analytic framework for inverse problems; and a survey of the principal numerical methods for inverse problems. An extensive annotated bibliography furnishes leads on the history of inverse problems and a guide to the frontiers of current research.

Inverse Problems

2016-11-24 Mathematics
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Release : 2016-11-24
Genre : Mathematics
Kind : eBook
Book Rating : 846/5 ( reviews)

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Book Synopsis Inverse Problems by : Mathias Richter

Download or read book Inverse Problems written by Mathias Richter. This book was released on 2016-11-24. Available in PDF, EPUB and Kindle. Book excerpt: The overall goal of the book is to provide access to the regularized solution of inverse problems relevant in geophysics without requiring more mathematical knowledge than is taught in undergraduate math courses for scientists and engineers. From abstract analysis only the concept of functions as vectors is needed. Function spaces are introduced informally in the course of the text, when needed. Additionally, a more detailed, but still condensed introduction is given in Appendix B. A second goal is to elaborate the single steps to be taken when solving an inverse problem: discretization, regularization and practical solution of the regularized optimization problem. These steps are shown in detail for model problems from the fields of inverse gravimetry and seismic tomography. The intended audience is mathematicians, physicists and engineers having a good working knowledge of linear algebra and analysis at the upper undergraduate level.

Inverse Problem Theory and Methods for Model Parameter Estimation

2005-01-01 Mathematics
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Release : 2005-01-01
Genre : Mathematics
Kind : eBook
Book Rating : 921/5 ( reviews)

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Book Synopsis Inverse Problem Theory and Methods for Model Parameter Estimation by : Albert Tarantola

Download or read book Inverse Problem Theory and Methods for Model Parameter Estimation written by Albert Tarantola. This book was released on 2005-01-01. Available in PDF, EPUB and Kindle. Book excerpt: While the prediction of observations is a forward problem, the use of actual observations to infer the properties of a model is an inverse problem. Inverse problems are difficult because they may not have a unique solution. The description of uncertainties plays a central role in the theory, which is based on probability theory. This book proposes a general approach that is valid for linear as well as for nonlinear problems. The philosophy is essentially probabilistic and allows the reader to understand the basic difficulties appearing in the resolution of inverse problems. The book attempts to explain how a method of acquisition of information can be applied to actual real-world problems, and many of the arguments are heuristic.

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