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

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.

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 well and ill-posed problems 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 102 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