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 welland 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 -- 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 fromlinear 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.