Computer Modelling In Tomography And Ill Posed Problems
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Computer Modelling in Tomography and Ill-Posed Problems
Author: Mikhail M. Lavrent'ev
language: en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date: 2014-07-24
Comparatively weakly researched untraditional tomography problems are solved because of new achievements in calculation mathematics and the theory of ill-posed problems, the regularization process of solving ill-posed problems, and the increase of stability. Experiments show possibilities and applicability of algorithms of processing tomography data. This monograph is devoted to considering these problems in connection with series of ill-posed problems in tomography settings arising from practice.The book includes chapters to the following themes: Mathematical basis of the method of computerized tomography Cone-beam tomography reconstruction Inverse kinematic problem in the tomographic setting
Computer Modelling in Tomography and Ill Posed Problems
The last decades of the 20th century were marked by the appearance of a new field of mathematics: computerized tomography. Its theory forms the basis for the solution of many applied problems. The methods of computerized tomography make it possible study the interior structure of a body by examining the characteristics of radiation passing through the object under study (transmission tomography). Depending on the type of radiation used, X-ray, optical, seismic, and some other kinds of tomography can be distinguished. Comparatively weakly researched, untraditional tomography problems are being solved because of new achievements in calculation mathematics and the theory of ill-posed problems (3D cone-beam tomography, geo-tomography). Experiments show possibilities and applicability of algorithms of processing tomography data. This monograph is devoted to considering these problems in connection with series of ill-posed problems in tomography settings, arising from practice. The basic themes of the book are: mathematical basis of the method of computerized tomography; algorithms for 3D cone-beam tomography; and inverse kinematics problems in tomographic settings (geo-tomography). This volume in the Inverse and Ill-Posed Problems Series will be of interest to researchers, graduates and post-graduates in X-ray, optical, seismic, as well as some other kinds of tomography in both academia and industry.
Operator Theory and Ill-Posed Problems
Author: Mikhail M. Lavrent'ev
language: en
Publisher: Walter de Gruyter
Release Date: 2011-12-22
This book consists of three major parts. The first two parts deal with general mathematical concepts and certain areas of operator theory. The third part is devoted to ill-posed problems. It can be read independently of the first two parts and presents a good example of applying the methods of calculus and functional analysis. The first part "Basic Concepts" briefly introduces the language of set theory and concepts of abstract, linear and multilinear algebra. Also introduced are the language of topology and fundamental concepts of calculus: the limit, the differential, and the integral. A special section is devoted to analysis on manifolds. The second part "Operators" describes the most important function spaces and operator classes for both linear and nonlinear operators. Different kinds of generalized functions and their transformations are considered. Elements of the theory of linear operators are presented. Spectral theory is given a special focus. The third part "Ill-Posed Problems" is devoted to problems of mathematical physics, integral and operator equations, evolution equations and problems of integral geometry. It also deals with problems of analytic continuation. Detailed coverage of the subjects and numerous examples and exercises make it possible to use the book as a textbook on some areas of calculus and functional analysis. It can also be used as a reference textbook because of the extensive scope and detailed references with comments.