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

✍ Scribed by Petrov Yuri P.; Valery S. Sizikov

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 244
• Series: Inverse and Ill-Posed Problems Series; 49
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

PART I THREE CLASSES OF PROBLEMS IN MATHEMATICS, PHYSICS, AND ENGINEERING
Chapter 1. Simplest ill-posed problems
1.1. Statement of the problem. Examples
1.2. Definitions
1.3. Examples and approaches to solving ill-posed problems
1.4. Ill-posed problems of synthesis for optimum control systems
1.5. Ill-posed problems on finding eigenvalues for systems of linear homogeneous equations
1.6. Solution of systems of differential equations. Do solutions always depend on parameters continuously?
1.7. Conclusions
Chapter 2. Problems intermediate between weiland ill-posed problems35
2.1. The third class of problems in mathematics, physics and engineering, and its significance
2.2. Transformations equivalent in the classical sense
2.3. Discovered paradoxes
2.4. Transformations equivalent in the widened sense
2.5. Problems intermediate between well- and ill-posed problems
2.6. Applications to control systems and some other objects described by differential equations
2.7. Applications to practical computations
2.8. Conclusions from Chapters 1 and 2
Chapter 3. Change of sensitivity to measurement errors under integral transformations used in modeling of ships and marine control systems
3.1. Application of integral transformations to practical problems
3.2. Properties of correlation functions
3.3. Properties of spectra
3.4. Correctness of integral transformations
3.5. Problems low sensitive to errors in the spectrum
3.6. Differentiation of distorted functions
3.7. Prognostication
Bibliography to Part I
PART II STABLE METHODS FOR SOLVING INVERSE PROBLEMS
Chapter 4. Regular methods for solving ill-posed problems
4.1. Elements of functional analysis
4.2. Some facts from linear algebra
4.3. Basic types of equations and transformations
4.4. Well- and ill-posedness according to Hadamard
4.5. Classical methods for solving Fredholm integral equations of the first kind
4.6. Gauss least-squares method and Moore-Penrose inverse-matrix method
4.7. Tikhonov regularization method
4.8. Solution-on-the-compact method
Chapter 5. Inverse problems in image reconstruction and tomography
5.1. Reconstruction of blurred images
5.2. Reconstruction of defocused images
5.3. X-ray tomography problems
5.4. Magnetic-field synthesis in an NMR tomograph
Bibliography to Part II