In this book the authors present a number of examples which lead to ill-posed problems arising with the processing and interpretation of data of physical measurements. Basic postulates and some results in the general theory of ill-posed problems follow. The exposition also includes problems of analytic continuation from continua and discrete sets, analogous problems of continuation of solutions of elliptic and parabolic equations, the main ill-posed boundary value problem for partial differential equations, and results on the theory of Volterra equations of the first kind. A very broad presentation is given of modern results on the problem of uniqueness in integral geometry and on inverse problems for partial differential equations.
Author(s): V. G. Romanov, and S. P. Shishatskii M. M. Lavrentev
Series: Translations of Mathematical Monographs, Vol. 64
Publisher: American Mathematical Society
Year: 1986
Language: English
Pages: C, VI+290, B
Cover
IIl-posed Problems of Mathematical Physics and Analysis
Copyright © 1986 by the American Mathematical Society
ISBN 0-8218-4517-9
QC20.7.B6L3813 1986 530.1'55
LCCN 86-3642
Contents
Preface
Introduction
CHAPTER I Physical Formulations Leading to Ill-posed Problems
§1. Continuation of static fields
§2. Problems for the diffusion equation
§3. Continuation of fields from discrete sets
§4. Processing readings of physical instruments
§5. Inverse problems of geophysics
§6. Inverse problems of gravimetry
§7. The inverse kinematic problem of seismology
CHAPTER II Basic Concepts of the Theory of Ill-posed Problems
§1. Problems well-posed in the Tikhonov sense
§2. Regularization
§3. Linear ill-posed problems
CHAPTER III Analytic Continuation
§1. Formulations of problems and classical results
§2. Analytic continuation from continua
§3. Analytic continuation from classes of sets including discrete sets
§4. Recovery of solutions of elliptic and parabolic equations from their values on sets lying inside the domain of regularity
CHAPTER IV Boundary Value Problems for Differential Equations
§1. The noncharacteristic Cauchy problem for,a parabolic equation. The Cauchy problem for an elliptic equation
§2. A mixed problem for a parabolic equation with decreasing time
§3. Cauchy problems with data on a segment of the time axis for degenerate parabolic and pseudoparabolic equations
§4. Cauchy problems with data on a timelike surface for hyperbolic and ultrahyperbolic equations
CHAPTER V Volterra Equations
§1. Regularization of a Volterra equation of the first kind
§2. Operator Volterra equations of the first kind
CHAPTER VI Integral Geometry
§1. The problem of finding a function from its spherical means
§2. Problems of integral geometry on a family of manifolds which is invariant under a group of transformations of the space
§3. Integral geometry in special classes of functions
§4. Integral geometry "in the small"
§5. The problem of integral geometry on plane curves and energy inequalities
CHAPTER VII Multidimensional Inverse Problems for Linear Differential Equations
§1. Examples of formulations of multidimensional inverse problems. Mathematical problems connected with investigating them
§2. A general approach to investigating questions of uniqueness and stability of inverse problems
§3. Inverse problems for hyperbolic equations of second order
§4. Inverse problems for first-order hyperbolic systems
§5. Inverse problems for parabolic equations of second order
1. The connection between solutions of direct problems for equations of hyperbolic and parabolic types and inverse problem
2. The method of descent in inverse problems.
3. Inverse problems for equations of parabolic type
§6. An abstract inverse problem and questions of its being well-posed
1. Reduction to the investigation of a two-parameter family of linear equations.
2. The method of linearization in investigating the inverse problem
Bibliography*
Back Cover
