Everyone solves inverse and ill-posed problem every minute. Humans solve them, as a rule, quickly and efficiently (if they are in good health and a clear mind). Let's consider, for example, visual perception. It is known that our vision fixes only a finite number of points around us. Then how are we seeing everything? Brain (in this situation - a powerful personal computer) uses seen points (interpolates and extrapolates) to recover all that the eye did not percept. It is clear that to recover true picture (in the general case - the volume and color) using only several points can only be the case when it is already more or less familiar with (most of the subjects and images we have seen, and sometimes touched by hands). That is, despite the strong ill-posedness (non-uniqueness and instability of the solution) of the problem (to restore the observed object and all that surrounds it by several points), the brain solves this problem pretty quickly. Why? It uses a wealth of experience (a priori information). And anyway, if we want to understand something complex enough to solve the problem, the probability of error which is large enough, we tend to arrive at an unstable (ill-posed) problem.
Inverse and ill-posed problems are united by one important feature - the instability of solutions to small errors in measurement data. In most interesting cases, inverse problems are ill-posed and such problems, as a rule, can be formulated as inverse with respect to certain direct (well posed) problems. But since inverse and ill-posed problems are historically formulated and studied quite often independently and in parallel, nowadays both terms are used in the scientific literature.
Summarizing, we can say that experts in the inverse and ill-posed problems
are involved in the study of properties and methods for the regularization of unstable problems. In other words, mathematicians are trying to create and explore sustainable methods of approximation of unstable maps. From the viewpoint of linear algebra this is a search of approximate methods for finding the normal pseudosolutions of systems of linear algebraic equations with rectangular, singular or poorly conditioned matrices. In functional analysis, a prime example of ill-posed problem is an operator equation Aq = f, where A is compact (completely continuous). Recently published studies interpret some problems of mathematical statistics as the inverse problem of probability theory. From the viewpoint of information theory, experts in the inverse and ill-posed problems investigate the properties of maps of compacts with a high epsilon-entropy in the table with a small epsilon-entropy.