Iteratively solving linear inverse problems under general convex constraints

Abstract

We consider linear inverse problems where the solution is assumed to fulfillsome generalhomogeneous convex constraint. We develop an algorithm thatamounts to a projected Landweber iteration and that provides and iterativeapproach to the solution of this inverse problem. For relativelymoderate assumptions on the constraint we can always prove weakconvergence of the iterative scheme. In certain cases, i.e. for specialfamilies of convex constraints, weak convergence implies normconvergence. The presented approach covers a wide range of problems,e.g. Besov-- or BV--restoration for which we present also numericalexperiments in the context of image processing.