Making use of a partial order in solving inverse problems

Abstract

In many applications, the concepts of inequality and comparison play an essential role, and the nature of the objects under consideration is better described by means of partial order relations. To reflect this nature, the conventional problem statements in normed spaces have to be modified. There is a need to enrich the structure of the functional spaces employed. In this paper, we consider inverse problems in Banach lattices—functional spaces endowed with both norm and partial order. In this new problem statement, we are able to construct a linearly constrained set that contains all admissible approximate solutions given the approximate data, the approximation errors and prior restrictions that are available. We do not suppose the problem to be either well posed or ill posed since we believe that the concepts and techniques we describe here can be useful in both cases. The range of applications includes medical imaging (CT, PET, SPECT, photo- and thermoacoustics), geophysics (e.g., inverse gravimetry problems), engineering design and other inverse problems.