Ball Stabilization in Inverse Problems

Abstract

We take as starting point a continuous, bijective map M from a Banach space C to a Banach space S ¹) The inverse problem starts from an initial data function D ≡ D(s), which is a point in S, and seeks, by using a properly defined generalized inverse map M _Q ⁻¹ to obtain as solution a well-defined function ϕ 22 ≡(t) belonging to C. Continuity of M does not ensure that the inverse map M ⁻¹ will be continuous (see Fig. I below); more generally the stabilization of the generalized (quasisolution) inverse map M _Q ⁻¹ is a central element in solving the inverse problem. The need to introduce quasisolutions arises because D, although it belongs to S, may not be in M(C).