Ill-Posed Problems and Regularization

In this chapter we consider the equation $$ Ax = y $$ (2.0) linear) continuous operator acting from a subset X of a Banach space into a subset Y of another Banach space, and x ∈ X is to be found given y. We discuss solvability of this equation when A −1 does not exist by outlining basic results of the theory created in the 1960s by Ivanov, John, Lavrent’ev, and Tikhonov. In Section 2.1 we give definitions of well- and ill-posedness, together with important illustrational examples. In Section 2.2 we describe a class of equations (2.0) that can be numerically solved in a stable way. Section 2.3 is to the variational construction of algorithms of solutions by minimizing Tikhonov stabilizing fimctionals. In Section 2.4 we show that stability estimates for equation (2.0) imply convergence rates for numerical algorithms and discuss the relation between convergence of these algorithms and the existence of a solution to (2.0). The final section, Section 2.5, describes some iterative regularization algorithms.