series: Inverse and Ill-Posed Problems Series

Published by De Gruyter

Inverse and Ill-Posed Problems Series

Edited by: Sergey I. Kabanikhin

ISSN: 1381-4524

Overview
Volumes

The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.

Author / Editor information

Sergey I. Kabanikhin, Sobolev Institute of Mathematics, Novosibirsk, Russia; Kazakh-British Technical University, Almaty, Kazakhstan.

book: Optimal Methods for Ill-Posed Problems

Book 2018

Optimal Methods for Ill-Posed Problems

With Applications to Heat Conduction

Vitalii P. Tanana, Anna I. Sidikova

Volume 62 in this series

The book covers fundamentals of the theory of optimal methods for solving ill-posed problems, as well as ways to obtain accurate and accurate-by-order error estimates for these methods. The methods described in the current book are used to solve a number of inverse problems in mathematical physics.

Contents
Modulus of continuity of the inverse operator and methods for solving ill-posed problems
Lavrent’ev methods for constructing approximate solutions of linear operator equations of the first kind
Tikhonov regularization method
Projection-regularization method
Inverse heat exchange problems

DOI:: https://doi.org/10.1515/9783110577211
ISBN:: 9783110577211
ISBN:: 9783110575835
ISBN:: 9783110575736
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Regularization Algorithms for Ill-Posed Problems

Book 2018

Regularization Algorithms for Ill-Posed Problems

Anatoly B. Bakushinsky, Mikhail M. Kokurin, Mikhail Yu. Kokurin

Volume 61 in this series

This specialized and authoritative book contains an overview of modern approaches to constructing approximations to solutions of ill-posed operator equations, both linear and nonlinear. These approximation schemes form a basis for implementable numerical algorithms for the stable solution of operator equations arising in contemporary mathematical modeling, and in particular when solving inverse problems of mathematical physics. The book presents in detail stable solution methods for ill-posed problems using the methodology of iterative regularization of classical iterative schemes and the techniques of finite dimensional and finite difference approximations of the problems under study. Special attention is paid to ill-posed Cauchy problems for linear operator differential equations and to ill-posed variational inequalities and optimization problems. The readers are expected to have basic knowledge in functional analysis and differential equations. The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems, and also to advanced students in these fields.

Contents
Introduction
Regularization Methods For Linear Equations
Finite Difference Methods
Iterative Regularization Methods
Finite-Dimensional Iterative Processes
Variational Inequalities and Optimization Problems

DOI:: https://doi.org/10.1515/9783110557350
ISBN:: 9783110557350
ISBN:: 9783110556384
ISBN:: 9783110556308
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Generalized Inverse Operators

Book 2016

Generalized Inverse Operators

And Fredholm Boundary-Value Problems

Alexander Andreevych Boichuk, Anatolii M. Samoilenko, Peter V. Malyshev

Volume 59 in this series

The book is devoted to the foundations of the theory of boundary-value problems for various classes of systems of differential-operator equations whose linear part is represented by Fredholm operators of the general form. A common point of view on numerous classes of problems that were traditionally studied independently of each other enables us to study, in a natural way, the theory of these problems, to supplement and improve the existing results, and in certain cases, study some of these problems for the first time.
With the help of the technique of generalized inverse operators, the Vishik– Lyusternik method, and iterative methods, we perform a detailed investigation of the problems of existence, bifurcations, and branching of the solutions of linear and nonlinear boundary-value problems for various classes of differential-operator systems and propose new procedures for their construction.
For more than 11 years that have passed since the appearance of the first edition of the monograph, numerous new publications of the authors in this direction have appeared. In this connection, it became necessary to make some additions and corrections to the previous extensively cited edition, which is still of signifi cant interest for the researchers.
For researchers, teachers, post-graduate students, and students of physical and mathematical departments of universities.

Contents:
Preliminary Information
Generalized Inverse Operators in Banach Spaces
Pseudoinverse Operators in Hilbert Spaces
Boundary-Value Problems for Operator Equations
Boundary-Value Problems for Systems of Ordinary Differential Equations
Impulsive Boundary-Value Problems for Systems of Ordinary Differential Equations
Solutions of Differential and Difference Systems Bounded on the Entire Real Axis

DOI:: https://doi.org/10.1515/9783110378443
ISBN:: 9783110378443
ISBN:: 9783110387353
ISBN:: 9783110378399
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Regularization Theory for Ill-posed Problems

Book 2013

Regularization Theory for Ill-posed Problems

Selected Topics

Shuai Lu, Sergei V. Pereverzev

Volume 58 in this series

This monograph is a valuable contribution to the highly topical and extremly productive field of regularisation methods for inverse and ill-posed problems. The author is an internationally outstanding and accepted mathematician in this field. In his book he offers a well-balanced mixture of basic and innovative aspects. He demonstrates new, differentiated viewpoints, and important examples for applications. The book demontrates the current developments in the field of regularization theory, such as multiparameter regularization and regularization in learning theory.

The book is written for graduate and PhD students and researchers in mathematics, natural sciences, engeneering, and medicine.

DOI:: https://doi.org/10.1515/9783110286496
ISBN:: 978-3-11-028646-5
ISBN:: 978-3-11-028649-6
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Computational Methods for Applied Inverse Problems

Book 2012

Computational Methods for Applied Inverse Problems

Yanfei Wang, Anatoly G. Yagola, Changchun Yang, Higher Education Press, Y. Bai, G. Bao, J. J. Cao, H. Cheng, Y. H. Dai, C. Z. Dong, G. S. Dulikravich, I. N. Egorov, C. L. Fu, L. J. Gelius, L. L. Hao, X. H. Huang, S. I. Kabanikhin, I. V. Kochikov, G. M. Kuramshina, G. S. Li, P. J. Li, Z. H. Li, J. J. Liu, X. D. Liu, Y. J. Ma, M. T. Nair, V. L. Panteleev, L. M. Pinheiro, M. A. Shishlenin, H. B. Song, Y. Song, A. V. Stepanova, T. Sun, Y. F. Wang, Z. H. Xiang, T. Y. Xiao, H. L. Xu, A. G. Yagola, C. C. Yang, H. Yang, B. Zang, H. Zhang, L. V. Zotov

Volume 56 in this series

Nowadays inverse problems and applications in science and engineering represent an extremely active research field. The subjects are related to mathematics, physics, geophysics, geochemistry, oceanography, geography and remote sensing, astronomy, biomedicine, and other areas of applications.

This monograph reports recent advances of inversion theory and recent developments with practical applications in frontiers of sciences, especially inverse design and novel computational methods for inverse problems. The practical applications include inverse scattering, chemistry, molecular spectra data processing, quantitative remote sensing inversion, seismic imaging, oceanography, and astronomical imaging.

The book serves as a reference book and readers who do research in applied mathematics, engineering, geophysics, biomedicine, image processing, remote sensing, and environmental science will benefit from the contents since the book incorporates a background of using statistical and non-statistical methods, e.g., regularization and optimization techniques for solving practical inverse problems.

DOI:: https://doi.org/10.1515/9783110259056
ISBN:: 978-3-11-025904-9
ISBN:: 978-3-11-025905-6
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Inverse and Ill-posed Problems

Book 2011

Inverse and Ill-posed Problems

Theory and Applications

Sergey I. Kabanikhin

Volume 55 in this series

The theory of ill-posed problems originated in an unusual way. As a rule, a new concept is a subject in which its creator takes a keen interest. The concept of ill-posed problems was introduced by Hadamard with the comment that these problems are physically meaningless and not worthy of the attention of serious researchers. Despite Hadamard's pessimistic forecasts, however, his unloved "child" has turned into a powerful theory whose results are used in many fields of pure and applied mathematics. What is the secret of its success? The answer is clear. Ill-posed problems occur everywhere and it is unreasonable to ignore them.

Unlike ill-posed problems, inverse problems have no strict mathematical definition. In general, they can be described as the task of recovering a part of the data of a corresponding direct (well-posed) problem from information about its solution. Inverse problems were first encountered in practice and are mostly ill-posed. The urgent need for their solution, especially in geological exploration and medical diagnostics, has given powerful impetus to the development of the theory of ill-posed problems. Nowadays, the terms "inverse problem" and "ill-posed problem" are inextricably linked to each other.

Inverse and ill-posed problems are currently attracting great interest. A vast literature is devoted to these problems, making it necessary to systematize the accumulated material. This book is the first small step in that direction. We propose a classification of inverse problems according to the type of equation, unknowns and additional information. We consider specific problems from a single position and indicate relationships between them. The problems relate to different areas of mathematics, such as linear algebra, theory of integral equations, integral geometry, spectral theory and mathematical physics. We give examples of applied problems that can be studied using the techniques we describe.

This book was conceived as a textbook on the foundations of the theory of inverse and ill-posed problems for university students. The author's intention was to explain this complex material in the most accessible way possible. The monograph is aimed primarily at those who are just beginning to get to grips with inverse and ill-posed problems but we hope that it will be useful to anyone who is interested in the subject.

DOI:: https://doi.org/10.1515/9783110224016
ISBN:: 978-3-11-022400-9
ISBN:: 978-3-11-022401-6
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Iterative Methods for Ill-Posed Problems

Book 2010

Iterative Methods for Ill-Posed Problems

An Introduction

Anatoly B. Bakushinsky, Mihail Yu. Kokurin, Alexandra Smirnova

Volume 54 in this series

Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions.

Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

DOI:: https://doi.org/10.1515/9783110250657
ISBN:: 978-3-11-025064-0
ISBN:: 978-3-11-025065-7
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Operators and Iterative Processes of Fejér Type

Book 2009

Operators and Iterative Processes of Fejér Type

Theory and Applications

Vladimir V Vasin, Ivan I. Eremin

Volume 53 in this series

This book, written by two experts in the field, deals with classes of iterative methods for the approximate solution of fixed points equations for operators satisfying a special contractivity condition, the Fejér property. The book is elementary, self-contained and uses methods from functional analysis, with a special focus on the construction of iterative schemes. Applications to parallelization, randomization and linear programming are also considered.

DOI:: https://doi.org/10.1515/9783110218190
ISBN:: 978-3-11-021818-3
ISBN:: 978-3-11-021819-0
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Numerical Methods for Solving Inverse Problems of Mathematical Physics

Book 2007

Numerical Methods for Solving Inverse Problems of Mathematical Physics

A. A. Samarskii, Petr N. Vabishchevich

Volume 52 in this series

The main classes of inverse problems for equations of mathematical physics and their numerical solution methods are considered in this book which is intended for graduate students and experts in applied mathematics, computational mathematics, and mathematical modelling.

DOI:: https://doi.org/10.1515/9783110205794
ISBN:: 978-3-11-019666-5
ISBN:: 978-3-11-020579-4
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Operator Theory and Ill-Posed Problems

Book 2006

Operator Theory and Ill-Posed Problems

Mikhail M. (Jn.) Lavrent'ev, Lev Ja. Savel'ev

Volume 50 in this series

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.

DOI:: https://doi.org/10.1515/9783110960723
ISBN:: 978-90-676-4448-8
ISBN:: 978-3-11-096072-3
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Well-posed, Ill-posed, and Intermediate Problems with Applications

Book 2005

Well-posed, Ill-posed, and Intermediate Problems with Applications

Petrov Yuri P., Valery S. Sizikov

Volume 49 in this series

This book deals with one of the key problems in applied mathematics, namely the investigation into and providing for solution stability in solving equations with due allowance for inaccuracies in set initial data, parameters and coefficients of a mathematical model for an object under study, instrumental function, initial conditions, etc., and also with allowance for miscalculations, including roundoff errors.

Until recently, all problems in mathematics, physics and engineering were divided into two classes: well-posed problems and ill-posed problems. The authors introduce a third class of problems: intermediate ones, which are problems that change their property of being well- or ill-posed on equivalent transformations of governing equations, and also problems that display the property of being either well- or ill-posed depending on the type of the functional space used.

The book is divided into two parts: Part one deals with general properties of all three classes of mathematical, physical and engineering problems with approaches to solve them; Part two deals with several stable models for solving inverse ill-posed problems, illustrated with numerical examples.

DOI:: https://doi.org/10.1515/9783110195309
ISBN:: 978-90-676-4432-7
ISBN:: 978-3-11-019530-9
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

Book 2004

Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

Sergey I. Kabanikhin, Abdigany D. Satybaev, Maxim A. Shishlenin

Volume 48 in this series

The authors consider dynamic types of inverse problems in which the additional information is given by the trace of the direct problem on a (usually time-like) surface of the domain. They discuss theoretical and numerical background of the finite-difference scheme inversion, the linearization method, the method of Gel'fand-Levitan-Krein, the boundary control method, and the projection method and prove theorems of convergence, conditional stability, and other properties of the mentioned methods.

DOI:: https://doi.org/10.1515/9783110960716
ISBN:: 978-90-676-4416-7
ISBN:: 978-3-11-096071-6
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Characterisation of Bio-Particles from Light Scattering

Book 2004

Characterisation of Bio-Particles from Light Scattering

Valeri P. Maltsev, Konstantin A. Semyanov

Volume 47 in this series

The primary aim of this monograph is to provide a systematic state-of-the-art summary of the light scattering of bioparticles, including a brief consideration of analytical and numerical methods for computing electromagnetic scattering by single particles, a detailed discussion of the instrumental approach used in measurement of light scattering, an analysis of the methods used in solution of the inverse light scattering problem, and an introduction of the results dealing with practical analysis of biosamples.

Considering the widespread need for this information in optics, remote sensing, engineering, medicine, and biology, the book is useful to many graduate students, scientists, and engineers working on various aspects of electromagnetic scattering and its applications.

DOI:: https://doi.org/10.1515/9783110915556
ISBN:: 978-90-676-4413-6
ISBN:: 978-3-11-091555-6
Subject:: Mathematics
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Biophysics
Publisher:: De Gruyter

book: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications

Book 2012

Carleman Estimates for Coefficient Inverse Problems and Numerical Applications

Michael V. Klibanov, Alexander A. Timonov

Volume 46 in this series

In this monograph, the main subject of the author's considerations is coefficient inverse problems. Arising in many areas of natural sciences and technology, such problems consist of determining the variable coefficients of a certain differential operator defined in a domain from boundary measurements of a solution or its functionals. Although the authors pay strong attention to the rigorous justification of known results, they place the primary emphasis on new concepts and developments.

DOI:: https://doi.org/10.1515/9783110915549
ISBN:: 978-90-676-4405-1
ISBN:: 978-3-11-091554-9
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Counterexamples in Optimal Control Theory

Book 2003

Counterexamples in Optimal Control Theory

Semen Ya. Serovaiskii

Volume 45 in this series

This monograph deals with cases where optimal control either does not exist or is not unique, cases where optimality conditions are insufficient of degenerate, or where extremum problems in the sense of Tikhonov and Hadamard are ill-posed, and other situations. A formal application of classical optimisation methods in such cases either leads to wrong results or has no effect. The detailed analysis of these examples should provide a better understanding of the modern theory of optimal control and the practical difficulties of solving extremum problems.

DOI:: https://doi.org/10.1515/9783110915532
ISBN:: 978-90-676-4400-6
ISBN:: 978-3-11-091553-2
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Inverse Problems of Mathematical Physics

Book 2003

Inverse Problems of Mathematical Physics

Mikhail M. (Jn.) Lavrent'ev, Alexander V. Avdeev, Viatcheslav I. Priimenko

Volume 44 in this series

This monograph deals with the theory of inverse problems of mathematical physics and applications of such problems. Besides it considers applications and numerical methods of solving the problems under study. Descriptions of particular numerical experiments are also included.

DOI:: https://doi.org/10.1515/9783110915525
ISBN:: 978-90-676-4396-2
ISBN:: 978-3-11-091552-5
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Ill-Posed Boundary-Value Problems

Book 2003

Ill-Posed Boundary-Value Problems

Serikkali E. Temirbolat

Volume 43 in this series

This monograph extends well-known facts to new classes of problems and works out novel approaches to the solution of these problems. It is devoted to the questions of ill-posed boundary-value problems for systems of various types of the first-order differential equations with constant coefficients and the methods for their solution.

DOI:: https://doi.org/10.1515/9783110915518
ISBN:: 978-90-676-4395-5
ISBN:: 978-3-11-091551-8
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Linear Sobolev Type Equations and Degenerate Semigroups of Operators

Book 2003

Linear Sobolev Type Equations and Degenerate Semigroups of Operators

Georgy A. Sviridyuk, Vladimir E. Fedorov

Volume 42 in this series

Focusing on the mathematics, and providing only a minimum of explicatory comment, this volume contains six chapters covering auxiliary material, relatively p-radial operators, relatively p-sectorial operators, relatively σ-bounded operators, Cauchy problems for inhomogenous Sobolev-type equations, bounded solutions to Sobolev-type equations, and optimal control.

DOI:: https://doi.org/10.1515/9783110915501
ISBN:: 978-90-676-4383-2
ISBN:: 978-3-11-091550-1
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Analysis
Publisher:: De Gruyter

book: Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis

Book 2003

Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis

Proceedings of the International Conference, Samarkand, Uzbekistan

Mikhail M. Lavrent'ev, Sergey I. Kabanikhin, Akbar H. Begmatov, Tukhtamurad D. Dzhuraev, Saburou Saitoh, Masahiro Yamamoto, S. Abdinazarov, B. K. Amonov, Z. R. Ashurova, Akram H. Begmatov, Abdullah Çavus, Uno Hämarik, A. R. Hashimov, Kazizat T. Iskakov, Djavvat Khadjiev, Akram Khaidarov, Bakhodir M. Kholboev, S. S. Kobilov, Mikhail M. (Jn.) Lavrent'ev, Zarif H. Ochilov, Nusrat Rajabov, D. Shodiev, Ulrich Tautenhahn, I. Yarmukhamedov, Sharov Yarmukhamedov, Yunusali Y. Zhuraev

Volume 41 in this series

These proceedings of the international Conference "Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis", held at the Samarkand State University, Uzbekistan in September 2000 bring together fundamental research articles in the major areas of the numerated fields of analysis and mathematical physics.

The book covers the following topics:

theory of ill-posed problems
inverse problems for differential equations
boundary value problems for equations of mixed type
integral geometry
mathematical modelling and numerical methods in natural sciences

DOI:: https://doi.org/10.1515/9783110936520
ISBN:: 978-3-11-036417-0
ISBN:: 978-3-11-093652-0
Subject:: Mathematics
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations

Book 2003

Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations

Alexander G. Megrabov

Volume 40 in this series

Inverse problems are an important and rapidly developing direction in mathematics, mathematical physics, differential equations, and various applied technologies (geophysics, optic, tomography, remote sensing, radar-location, etc.).

In this monograph direct and inverse problems for partial differential equations are considered. The type of equations focused are hyperbolic, elliptic, and mixed (elliptic-hyperbolic). The direct problems arise as generalizations of problems of scattering plane elastic or acoustic waves from inhomogeneous layer (or from half-space). The inverse problems are those of determination of medium parameters by giving the forms of incident and reflected waves or the vibrations of certain points of the medium. The method of research of all inverse problems is spectral-analytical, consisting in reducing the considered inverse problems to the known inverse problems for the Sturm-Liouville equation or the string equation.

Besides the book considers discrete inverse problems. In these problems an arbitrary set of point sources (emissive sources, oscillators, point masses) is determined.

DOI:: https://doi.org/10.1515/9783110944983
ISBN:: 978-90-676-4379-5
ISBN:: 978-3-11-094498-3
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Dynamical Inverse Problems of Distributed Systems

Book 2002

Dynamical Inverse Problems of Distributed Systems

Vyacheslav I. Maksimov

Volume 37 in this series

This monograph deals with problems of dynamical reconstruction of unknown variable characteristics (distributed or boundary disturbances, coefficients of operator etc.) for various classes of systems with distributed parameters (parabolic and hyperbolic equations, evolutionary variational inequalities etc.).

DOI:: https://doi.org/10.1515/9783110944839
ISBN:: 978-3-11-036413-2
ISBN:: 978-3-11-094483-9
Subject:: Mathematics
Subject:: Analysis
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Theory of Linear Ill-Posed Problems and its Applications

Book 2002

Theory of Linear Ill-Posed Problems and its Applications

Valentin K. Ivanov, Vladimir V. Vasin, Vitalii P. Tanana

Volume 36 in this series

This monograph is a revised and extended version of the Russian edition from 1978. It includes the general theory of linear ill-posed problems concerning e. g. the structure of sets of uniform regularization, the theory of error estimation, and the optimality method. As a distinguishing feature the book considers ill-posed problems not only in Hilbert but also in Banach spaces.

It is natural that since the appearance of the first edition considerable progress has been made in the theory of inverse and ill-posed problems as wall as in ist applications. To reflect these accomplishments the authors included additional material e. g. comments to each chapter and a list of monographs with annotations.

DOI:: https://doi.org/10.1515/9783110944822
ISBN:: 978-90-676-4367-2
ISBN:: 978-3-11-094482-2
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation

Book 2002

Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation

Mukarram A. Atakhodzhaev

Volume 35 in this series

Internal boundary value problems deals with the problem of determining the solution of an equation if data are given on two manifolds. One manifold is the domain boundary and the other manifold is situated inside the domain.

This monograph studies three essentially ill-posed internal boundary value problems for the biharmonic equation and the Cauchy problem for the abstract biharmonic equation, both qualitatively and quantitatively. In addition, some variants of these problems and the Cauchy problem, as well as the m-dimensional case, are considered. The author introduces some new notions, such as the notion of complete solvability.

DOI:: https://doi.org/10.1515/9783110944815
ISBN:: 978-3-11-036414-9
ISBN:: 978-3-11-094481-5
Subject:: Mathematics
Subject:: Analysis
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Investigation Methods for Inverse Problems

Book 2002

Investigation Methods for Inverse Problems

Vladimir G. Romanov

Volume 34 in this series

This monograph deals with some inverse problems of mathematical physics. It introduces new methods for studying inverse problems and gives obtained results, which are related to the conditional well posedness of the problems. The main focus lies on time-domain inverse problems for hyperbolic equations and the kinetic transport equation.

DOI:: https://doi.org/10.1515/9783110943849
ISBN:: 978-3-11-036419-4
ISBN:: 978-3-11-094384-9
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Operator Theory

Book 2002

Operator Theory

Nonclassical Problems

Sergei G. Pyatkov

Volume 33 in this series

This monograph describes mathematical methods applicable to studying nonclassical problems of mathematical physics. The emphasis of the book is on applications of the interpolar theory of Banach spaces to the theory of linear operators to be expotentially dichotomous, to some continuity properties of linear operators in Hilbert scales, to the Riesz basis property of eigenelements and associated elements of linear pencils and the correspondending elliptic problems with indefinite weight functions, and to studying nonclassical boundary value problems for first order operator-differential equations.

DOI:: https://doi.org/10.1515/9783110900163
ISBN:: 978-90-676-4363-4
ISBN:: 978-3-11-090016-3
Subject:: Mathematics
Subject:: Analysis
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Inverse Problems for Partial Differential Equations

Book 2002

Inverse Problems for Partial Differential Equations

Yurii Ya. Belov

Volume 32 in this series

This monograph is devoted to identification problems of coefficients in equations of mathematical physics. It invesitgates the existence and uniqueness of the solutions for identification coefficient problems in parabolic and hyperbolic equations and equation systems of composite type. The problems are studied with the Cauchy data and equations in which the Fourier transform with respect to the chosen variable is supposed to occur.

Differential properties of the solutions for the original direct problems and their behavior under great values of time are studied on the basis of solution properties for direct problems. The identification problems with one or two unknown coefficients are also investigated. For initial boundary value conditions linear and nonlinear parabolic equations are studied.

DOI:: https://doi.org/10.1515/9783110944631
ISBN:: 978-90-676-4358-0
ISBN:: 978-3-11-094463-1
Subject:: Mathematics
Subject:: Analysis
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Method of Spectral Mappings in the Inverse Problem Theory

Book 2002

Method of Spectral Mappings in the Inverse Problem Theory

Vacheslav A. Yurko

Volume 31 in this series

Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology and other branches of natural science.

This monograph is devoted to inverse problems of spectral analysis for ordinary differential equations. Its aim ist to present the main results on inverse spectral problems using the so-called method of spectral mappings, which is one of the main tools in inverse spectral theory.

The book consists of three chapters: In Chapter 1 the method of spectral mappings is presented in the simplest version for the Sturm-Liouville operator. In Chapter 2 the inverse problem of recovering higher-order differential operators of the form, on the half-line and on a finite interval, is considered. In Chapter 3 inverse spectral problems for differential operators with nonlinear dependence on the spectral parameter are studied.

DOI:: https://doi.org/10.1515/9783110940961
ISBN:: 978-90-676-4355-9
ISBN:: 978-3-11-094096-1
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Integral Geometry and Inverse Problems for Kinetic Equations

Book 2001

Integral Geometry and Inverse Problems for Kinetic Equations

Anvar Kh. Amirov

Volume 28 in this series

In this monograph a method for proving the solvability of integral geometry problems and inverse problems for kinetic equations is presented. The application of this method has led to interesting problems of the Dirichlet type for third order differential equations, the solvability of which appears to depend on the geometry of the domain for which the problem is stated. Another considered subject is the problem of integral geometry on paraboloids, in particular the uniqueness of solutions to the Goursat problem for a differential inequality, which implies new theorems on the uniqueness of solutions to this problem for a class of quasilinear hyperbolic equations. A class of multidimensional inverse problems associated with problems of integral geometry and the inverse problem for the quantum kinetic equations are also included.

DOI:: https://doi.org/10.1515/9783110940947
ISBN:: 978-3-11-035469-0
ISBN:: 978-3-11-094094-7
Subject:: Mathematics
Subject:: Analysis
Subject:: Geometry and Topology
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Computer Modelling in Tomography and Ill-Posed Problems

Book 2001

Computer Modelling in Tomography and Ill-Posed Problems

Mikhail M. Lavrent'ev, Sergei M. Zerkal, Oleg E. Trofimov

Volume 27 in this series

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

DOI:: https://doi.org/10.1515/9783110940930
ISBN:: 978-3-11-036412-5
ISBN:: 978-3-11-094093-0
Subject:: Medicine
Subject:: Clinical Medicine
Subject:: Radiology
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: An Introduction to Identification Problems via Functional Analysis

Book 2001

An Introduction to Identification Problems via Functional Analysis

Alfredo Lorenzi

Volume 26 in this series

this monograph is based on two courses in computational mathematics and operative research, which were given by the author in recent years to doctorate and PhD students. The text focuses on an aspect of the theory of inverse problems, which is usually referred to as identification of parameters (numbers, vectors, matrices, functions) appearing in differential – or integrodifferential – equations. The parameters of such equations are either quite unknown or partially unknown, however knowledge about these is usually essential as they describe the intrinsic properties of the material or substance under consideration.

DOI:: https://doi.org/10.1515/9783110940923
ISBN:: 978-3-11-036411-8
ISBN:: 978-3-11-094092-3
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Coefficient Inverse Problems for Parabolic Type Equations and Their Application

Book 2001

Coefficient Inverse Problems for Parabolic Type Equations and Their Application

P. G. Danilaev

Volume 25 in this series

As a rule, many practical problems are studied in a situation when the input data are incomplete. For example, this is the case for a parabolic partial differential equation describing the non-stationary physical process of heat and mass transfer if it contains the unknown thermal conductivity coefficient. Such situations arising in physical problems motivated the appearance of the present work.

In this monograph the author considers numerical solutions of the quasi-inversion problems, to which the solution of the original coefficient inverse problems are reduced.

Underground fluid dynamics is taken as a field of practical use of coefficient inverse problems. The significance of these problems for this application domain consists in the possibility to determine the physical fields of parameters that characterize the filtration properties of porous media (oil strata). This provides the possibility of predicting the conditions of oil-field development and the effects of the exploitation.

The research carried out by the author showed that the quasi-inversion method can be applied also for solution of "interior coefficient inverse problems" by reducing them to the problem of continuation of a solution to a parabolic equation. This reduction is based on the results of the proofs of the uniqueness theorems for solutions of the corresponding coefficient inverse problems.

DOI:: https://doi.org/10.1515/9783110940916
ISBN:: 978-3-11-036401-9
ISBN:: 978-3-11-094091-6
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Inverse Problems for Kinetic and Other Evolution Equations

Book 2001

Inverse Problems for Kinetic and Other Evolution Equations

Yu. E. Anikonov

Volume 24 in this series

This monograph deals with methods of studying multidimensional inverse problems for kinetic and other evolution equations, in particular transfer equations. The methods used are applied to concrete inverse problems, especially multidimensional inverse problems applicable in linear and nonlinear statements.

A significant part of the book contains formulas and relations for solving inverse problems, including formulas for the solution and coefficients of kinetic equations, differential-difference equations, nonlinear evolution equations, and second order equations.

DOI:: https://doi.org/10.1515/9783110940909
ISBN:: 978-3-11-036397-5
ISBN:: 978-3-11-094090-9
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Inverse Problems of Wave Processes

Book 2001

Inverse Problems of Wave Processes

A. S. Blagoveshchenskii

Volume 23 in this series

This monograph covers dynamical inverse problems, that is problems whose data are the values of wave fields. It deals with the problem of determination of one or more coefficients of a hyperbolic equation or a system of hyperbolic equations. The desired coefficients are functions of point. Most attention is given to the case where the required functions depend only on one coordinate.

The first chapter of the book deals mainly with methods of solution of one-dimensional inverse problems. The second chapter focuses on scalar inverse problems of wave propagation in a layered medium. In the final chapter inverse problems for elasticity equations in stratified media and acoustic equations for moving media are given.

DOI:: https://doi.org/10.1515/9783110940893
ISBN:: 978-3-11-036398-2
ISBN:: 978-3-11-094089-3
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Uniqueness Problems for Degenerating Equations and Nonclassical Problems

Book 2001

Uniqueness Problems for Degenerating Equations and Nonclassical Problems

S. P. Shishatskii, A. Asanov, E. R. Atamanov

Volume 22 in this series

DOI:: https://doi.org/10.1515/9783110920321
ISBN:: 978-3-11-036416-3
ISBN:: 978-3-11-092032-1
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Uniqueness Questions in Reconstruction of Multidimensional Objects from Tomography-Type Projection Data

Book 2000

Uniqueness Questions in Reconstruction of Multidimensional Objects from Tomography-Type Projection Data

V. P. Golubyatnikov

Volume 21 in this series

DOI:: https://doi.org/10.1515/9783110920314
ISBN:: 978-3-11-036415-6
ISBN:: 978-3-11-092031-4
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Monte Carlo Method for Solving Inverse Problems of Radiation Transfer

Book 1999

Monte Carlo Method for Solving Inverse Problems of Radiation Transfer

V. S. Antyufeev

Volume 20 in this series

DOI:: https://doi.org/10.1515/9783110920307
ISBN:: 978-3-11-036399-9
ISBN:: 978-3-11-092030-7
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Identification Problems of Wave Phenomena

Book 1999

Identification Problems of Wave Phenomena

Theory and Numerics

A. Lorenzi, S. I. Kabanikhin

Volume 18 in this series

DOI:: https://doi.org/10.1515/9783110943290
ISBN:: 978-3-11-035498-0
ISBN:: 978-3-11-094329-0
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Inverse Problems of Electromagnetic Geophysical Fields

Book 1999

Inverse Problems of Electromagnetic Geophysical Fields

P. S. Martyshko

Volume 17 in this series

DOI:: https://doi.org/10.1515/9783110943283
ISBN:: 978-3-11-035492-8
ISBN:: 978-3-11-094328-3
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Composite Type Equations and Inverse Problems

Book 1999

Composite Type Equations and Inverse Problems

A. I. Kozhanov

Volume 16 in this series

DOI:: https://doi.org/10.1515/9783110943276
ISBN:: 978-3-11-035490-4
ISBN:: 978-3-11-094327-6
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Inverse Problems of Vibrational Spectroscopy

Book 1999

Inverse Problems of Vibrational Spectroscopy

A. G. Yagola, I. V. Kochikov, G. M. Kuramshina

Volume 15 in this series

DOI:: https://doi.org/10.1515/9783110943269
ISBN:: 978-3-11-036396-8
ISBN:: 978-3-11-094326-9
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Elements of the Theory of Inverse Problems

Book 1999

Elements of the Theory of Inverse Problems

A. M. Denisov

Volume 14 in this series

DOI:: https://doi.org/10.1515/9783110943252
ISBN:: 978-3-11-035489-8
ISBN:: 978-3-11-094325-2
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Volterra Equations and Inverse Problems

Book 1999

Volterra Equations and Inverse Problems

A. L. Bughgeim

Volume 13 in this series

DOI:: https://doi.org/10.1515/9783110943245
ISBN:: 978-3-11-035488-1
ISBN:: 978-3-11-094324-5
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Regularization, Uniqueness and Existence of Solutions of Volterra Equations of the First Kind

Book 1998

Regularization, Uniqueness and Existence of Solutions of Volterra Equations of the First Kind

A. Asanov

Volume 11 in this series

DOI:: https://doi.org/10.1515/9783110943238
ISBN:: 978-90-676-4287-3
ISBN:: 978-3-11-094323-8
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Methods for Solution of Nonlinear Operator Equations

Book 1997

Methods for Solution of Nonlinear Operator Equations

V. P. Tanana

Volume 10 in this series

DOI:: https://doi.org/10.1515/9783110920291
ISBN:: 978-90-676-4272-9
ISBN:: 978-3-11-092029-1
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Inverse and Ill-Posed Sources Problems

Book 1997

Inverse and Ill-Posed Sources Problems

Yu. E. Anikonov, B. A. Bubnov, G. N. Erokhin

Volume 9 in this series

DOI:: https://doi.org/10.1515/9783110969412
ISBN:: 978-90-676-4273-6
ISBN:: 978-3-11-096941-2
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Methods for Solving Operator Equations

Book 1997

Methods for Solving Operator Equations

V. P. Tanana

Volume 8 in this series

DOI:: https://doi.org/10.1515/9783110900156
ISBN:: 978-90-676-4237-8
ISBN:: 978-3-11-090015-6
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Nonclassical and Inverse Problems for Pseudoparabolic Equations

Book 1997

Nonclassical and Inverse Problems for Pseudoparabolic Equations

A. Asanov, E. R. Atamanov

Volume 7 in this series

DOI:: https://doi.org/10.1515/9783110900149
ISBN:: 978-3-11-035535-2
ISBN:: 978-3-11-090014-9
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Formulas in Inverse and Ill-Posed Problems

Book 1997

Formulas in Inverse and Ill-Posed Problems

Yu. E. Anikonov

Volume 6 in this series

DOI:: https://doi.org/10.1515/9783110900132
ISBN:: 978-3-11-035472-0
ISBN:: 978-3-11-090013-2
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Inverse Logarithmic Potential Problem

Book 1996

Inverse Logarithmic Potential Problem

V. G. Cherednichenko

Volume 5 in this series

DOI:: https://doi.org/10.1515/9783110900125
ISBN:: 978-3-11-035536-9
ISBN:: 978-3-11-090012-5
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Multidimensional Inverse and Ill-Posed Problems for Differential Equations

Book Open Access 1995

Multidimensional Inverse and Ill-Posed Problems for Differential Equations

Yu. E. Anikonov

Volume 4 in this series

More Cite Download PDF

Inverse problems are usually nonlinear and are separated into one-dimensional and multidimensional problems, depending on whether the sought function (or functions) is a function of one variable or of many. Multidimensionality of inverse problems has particular value at present, because practice shows that many investigating processes are described by an equation, of which the co-efficient essentially depends on many variables.

This monograph is devoted to statements of multidimensional inverse problems, in particular to methods of their investigation. Questions of the uniqueness of solution, solvability and stability are studied. Methods to construct a solution are given and, in certain cases, inversion formulas are given as well. Concrete applications of the theory developed here are also given. Where possible, the author has stopped to consider the method of investigation of the problems, thereby sometimes losing generality and quantity of the problems, which can be examined by such a method.

The book should be of interet to researchers in the field of applied mathematics, geophysics and mathematical biology.

DOI:: https://doi.org/10.1515/9783110271478
ISBN:: 978-3-11-034666-4
ISBN:: 978-3-11-027147-8
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Ill-Posed Problems with A Priori Information

Book 1995

Ill-Posed Problems with A Priori Information

V. V. Vasin, A. L. Ageev

Volume 3 in this series

DOI:: https://doi.org/10.1515/9783110900118
ISBN:: 978-90-676-4191-3
ISBN:: 978-3-11-090011-8
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Inverse Problems for Maxwell's Equations

Book 1994

Inverse Problems for Maxwell's Equations

V. G. Romanov, S. I. Kabanikhin

Volume 2 in this series

DOI:: https://doi.org/10.1515/9783110900101
ISBN:: 978-3-11-035499-7
ISBN:: 978-3-11-090010-1
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Electromagnetism, Optics and Photonics
Publisher:: De Gruyter

book: Integral Geometry of Tensor Fields

Book 1994

Integral Geometry of Tensor Fields

V. A. Sharafutdinov

Volume 1 in this series

DOI:: https://doi.org/10.1515/9783110900095
ISBN:: 9789067641654
ISBN:: 9783110900095
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Inverse Problems and Carleman Estimates

Book 2021

Inverse Problems and Carleman Estimates

Global Uniqueness, Global Convergence and Experimental Data

Michael V. Klibanov, Jingzhi Li

Volume 63 in this series

This book summarizes the main analytical and numerical results of Carleman estimates. In the analytical part, Carleman estimates for three main types of Partial Differential Equations (PDEs) are derived. In the numerical part, first numerical methods are proposed to solve ill-posed Cauchy problems for both linear and quasilinear PDEs. Next, various versions of the convexification method are developed for a number of Coefficient Inverse Problems.

DOI:: https://doi.org/10.1515/9783110745481
ISBN:: 9783110745481
ISBN:: 9783110745412
ISBN:: 9783110745559
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Nonclassical Linear Volterra Equations of the First Kind

Book 2003

Nonclassical Linear Volterra Equations of the First Kind

Anatoly S. Apartsyn

Volume 39 in this series

This monograph deals with linear integra Volterra equations of the first kind with variable upper and lower limits of integration. Volterra operators of this type are the basic operators for integral models of dynamic systems.

DOI:: https://doi.org/10.1515/9783110944976
ISBN:: 978-90-676-4375-7
ISBN:: 978-3-11-094497-6
ISBN:: 9783110619935
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

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