The term "ill-posed problem" appeared due to Jacques Hadamard who formulated criteria for well-posed problems and gave the first example of the problem which is not well-posed, namely, the Cauchy problem for Laplace equation. All problems which violate one or more of Hadamard's well-posedness criteria are now called ill-posed problems. For 30 years mathematicians believed that ill-posed problems made no practical or physical sense, and there was no need to try to solve them. However, it has subsequently been shown that ill-posed inverse problems are ubiquitous throughout science and engineering.

Unlike the ill-posed problems, inverse problems have no strict mathematical definition. In general, they can be described as problems of recovering a part of data of a corresponding direct (well-posed) problem from additional information about its solution. The inverse problems have arisen in practice. The urgent need in their solution, especially in geological exploration and medical diagnostics, gave a powerful impetus to the development of the theory of ill-posed problems. Nowadays, the terms "inverse problem" and "ill-posed problem" are inextricably related to each other.

The inverse and ill-posed problems are currently of 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 ill-posed problems in different areas of mathematics (such as linear algebra, theory of integral equations, integral geometry, spectral theory, and mathematical physics) from a single position and indicate relationships between them. We give examples of applied problems which can be studied using the described techniques.

We aimed to present such a complex subject as theoretical aspects of ill-posed and inverse problems in the most accessible form. This monograph is based on the Russian version "Inverse and Ill-Posed Problems" which has been published twice in Russia. It is officially recommended by the Ministry of Education and Science of the Russian Federation as a textbook for university students. Intending the book to those who are just beginning to get familiar with inverse and ill-posed problems, we hope that it will be useful to anyone who is interested in this subject.

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.