Inverse and Ill-posed Problems: Theory and Applications

✍ Scribed by Sergey I. Kabanikhin

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 475
• Series: Inverse and Ill-Posed Problems Series; 55
• Category: Library

✦ Synopsis

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.

• Covers a topic that is abundant in all areas of science and technology
• Includes methods for incomplete input data
• Enables readers to tackle real-world problems

✦ Table of Contents

Preface
Denotations
1 Basic concepts and examples
1.1 On the definition of inverse and ill-posed problems
1.2 Examples of inverse and ill-posed problems
2 Ill-posed problems
2.1 Well-posed and ill-posed problems
2.2 On stability in different spaces
2.3 Quasi-solution. The Ivanov theorems
2.4 The Lavrentiev method
2.5 The Tikhonov regularization method
2.6 Gradient methods
2.7 An estimate of the convergence rate with respect to the objective functional
2.8 Conditional stability estimate and strong convergence of gradient methods applied to ill-posed problems
2.9 The pseudoinverse and the singular value decomposition of an operator
3 Ill-posed problems of linear algebra
3.1 Generalization of the concept of a solution. Pseudo-solutions
3.2 Regularization method
3.3 Criteria for choosing the regularization parameter
3.4 Iterative regularization algorithms
3.5 Singular value decomposition
3.6 The singular value decomposition algorithm and the Godunov method
3.7 The square root method
3.8 Exercises
4 Integral equations
4.1 Fredholm integral equations of the first kind
4.2 Regularization of linear Volterra integral equations of the first kind
4.3 Volterra operator equations with boundedly Lipschitz-continuous kernel
4.4 Local well-posedness and uniqueness on the whole
4.5 Well-posedness in a neighborhood of the exact solution
4.6 Regularization of nonlinear operator equations of the first kind
5 Integral geometry
5.1 The Radon problem
5.2 Reconstructing a function from its spherical means
5.3 Determining a function of a single variable from the values of its integrals. The problem of moments
5.4 Inverse kinematic problem of seismology
6 Inverse spectral and scattering problems
6.1 Direct Sturm-Liouville problem on a finite interval
6.2 Inverse Sturm-Liouville problems on a finite interval
6.3 The Gelfand-Levitan method on a finite interval
6.4 Inverse scattering problems
6.5 Inverse scattering problems in the time domain
7 Linear problems for hyperbolic equations
7.1 Reconstruction of a function from its spherical means
7.2 The Cauchy problem for a hyperbolic equation with data on a time-like surface
7.3 The inverse thermoacoustic problem
7.4 Linearized multidimensional inverse problem for the wave equation
8 Linear problems for parabolic equations
8.1 On the formulation of inverse problems for parabolic equations and their relationship with the corresponding inverse problems for hyperbolic equations
8.2 Inverse problem of heat conduction with reverse time (retrospective inverse problem)
8.3 Inverse boundary-value problems and extension problems
8.4 Interior problems and problems of determining sources
9 Linear problems for elliptic equations
9.1 The uniqueness theorem and a conditional stability estimate on a plane
9.2 Formulation of the initial boundary value problem for the Laplace equation in the form of an inverse problem. Reduction to an operator equation
9.3 Analysis of the direct initial boundary value problem for the Laplace equation
9.4 The extension problem for an equation with self-adjoint elliptic operator
10 Inverse coefficient problems for hyperbolic equations
10.1 Inverse problems for the equation utt = uxx – q(x)u + F(x,t)
10.2 Inverse problems of acoustics
10.3 Inverse problems of electrodynamics
10.4 Local solvability of multidimensional inverse problems
10.5 Method of the Neumann to Dirichlet maps in the half-space
10.6 An approach to inverse problems of acoustics using geodesic lines
10.7 Two-dimensional analog of the Gelfand-Levitan-Krein equation
11 Inverse coefficient problems for parabolic and elliptic equations
11.1 Formulation of inverse coefficient problems for parabolic equations. Association with those for hyperbolic equations
11.2 Reducing to spectral inverse problems
11.3 Uniqueness theorems
11.4 An overdetermined inverse coefficient problem for the elliptic equation. Uniqueness theorem
11.5 An inverse problem in a semi-infinite cylinder
Appendix A
A.1 Spaces
A.2 Operators
A.3 Dual space and adjoint operator
A.4 Elements of differential calculus in Banach spaces
A.5 Functional spaces
A. 6 Equations of mathematical physics
Appendix B
B.1 Supplementary exercises and control questions
B.2 Supplementary references
Epilogue
Bibliography
Index