Making use of a partial order in solving inverse problems: II.

Abstract

Mathematical formulations of applied inverse problems often involve operator equations in normed functional spaces. In many cases, these spaces can, in addition, be endowed with a partial order relation, which turns them into Banach lattices. The fact that two tools, such as a partial order relation and a monotone (with respect to this partial order) norm, are available to the researchers, gives them a clear advantage of having more freedom in the problem formulations. For instance, errors in the approximate data are sometimes easier to describe in terms of pointwise bounds. Inverse problems in partially ordered normed spaces (Banach lattices) have been studied before in the case when a compact set containing the unknown exact solution is available a priori. It turned out that under this assumption it is possible, even in the ill-posed case, to compute ‘pointwise’ bounds for the unknown exact solution (or rather, bounds by means of the appropriate partial order), thus providing an error estimate by means of the partial order. Also, a useful property of this approach was that one was able to quantify the uncertainty in the operator by means of linear inequalities that were included in the corresponding optimization problems as (linear) constraints and made the computations easier. However, the compactness assumption might sometimes be too demanding in practice. This paper aims at revealing the possibilities and advantages of using partial order in solving inverse problems in the case when no compact set of prior restrictions is available, concentrating on linear inverse (possibly ill-posed) problems.