ILL-Posed Problems and a priori information

Abstract

This chapter describes the method of using a priori information for constructing regularizing algorithms and error estimation while solving ill-posed problems. The chapter considers the examples of compactness of a set of solutions and source-wise representation of a solution with a compact operator as typical illustrations of a priori information. The first and the main rule that should be used before solving a practical ill-posed problem is to study all physical or technical details concerning an unknown solution. It means that it is necessary to include all a priori information (a priori constraints) in a statement of a mathematical problem before it is solved. This rule, which is good for all kinds of mathematical problems, is extremely important for ill-posed problems. An ill-posed problem has very unpleasant features. A stable method of its solution (regularizing algorithm), if it exists, does not guarantee the possibility of error estimation or comparison of convergence rates. But in some cases there can be an error of the approximate solution if additional information about the structure of a set to which the exact solution belongs is known.