William Rundell – författare

Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.In the opening chapter, Päivärinta collects the mathematical tools needed in the subsequent chapters and gives references for further study. Colton's chapter focuses on electromagnetic scattering problems. As an application he considers the problem of detecting and monitoring leukemia. Rundell's chapter deals with inverse spectral problems. He describes several exact and algorithmic methods for reconstructing an unknown function from the spectral data. Chadan provides an introduction to quantum mechanical inverse scattering problems. As an application he explains the celebrated method of Gardner, Greene, Kruskal, and Miura for solving nonlinear evolution equations such as the Korteweg_de Vries equation. Each chapter provides full references for further study.

As the title of the book indicates, this is primarily a book on partial differential equations (PDEs) with two definite slants: toward inverse problems and to the inclusion of fractional derivatives. The standard paradigm, or direct problem, is to take a PDE, including all coefficients and initial/boundary conditions, and to determine the solution. The inverse problem reverses this approach asking what information about coefficients of the model can be obtained from partial information on the solution. Answering this question requires knowledge of the underlying physical model, including the exact dependence on material parameters.The last feature of the approach taken by the authors is the inclusion of fractional derivatives. This is driven by direct physical applications: a fractional derivative model often allows greater adherence to physical observations than the traditional integer order case.The book also has an extensive historical section and the material that can be called ""fractional calculus"" and ordinary differential equations with fractional derivatives. This part is accessible to advanced undergraduates with basic knowledge on real and complex analysis. At the other end of the spectrum, lie nonlinear fractional PDEs that require a standard graduate level course on PDEs.

14 contributions present mathematical models for different imaging techniques in medicine and nondestructive testing. The underlying mathematical models are presented in a way that also newcomers in the field have a chance to understand the relation between the special applications and the mathematics needed for successfully treating these problems. The reader gets an insight into a modern field of scientific computing with applications formerly not presented in such form, leading from the basics to actual research activities.

Inverse problems are concerned with determining causes for observed or desired effects. Problems of this type appear in many application fields both in science and in engineering. The mathematical modelling of inverse problems usually leads to ill-posed problems, i.e., problems where solutions need not exist, need not be unique or may depend discontinuously on the data. For this reason, numerical methods for solving inverse problems are especially difficult, special methods have to be developed which are known under the term "regularization methods". This volume contains twelve survey papers about solution methods for inverse and ill-posed problems and about their application to specific types of inverse problems, e.g., in scattering theory, in tomography and medical applications, in geophysics and in image processing. The papers have been written by leading experts in the field and provide an up-to-date account of solution methods for inverse problems.

Inverse problems are concerned with determining causes for observed or desired effects. Problems of this type appear in many application fields both in science and in engineering. The mathematical modelling of inverse problems usually leads to ill-posed problems, i.e., problems where solutions need not exist, need not be unique or may depend discontinuously on the data. For this reason, numerical methods for solving inverse problems are especially difficult, special methods have to be developed which are known under the term "regularization methods". This volume contains twelve survey papers about solution methods for inverse and ill-posed problems and about their application to specific types of inverse problems, e.g., in scattering theory, in tomography and medical applications, in geophysics and in image processing. The papers have been written by leading experts in the field and provide an up-to-date account of solution methods for inverse problems.