Iteratively solving linear inverse problems under general convex constraints

Abstract. We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algorithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively moderate assumptions on the constraint we can always prove weak convergence of the iterative scheme. In certain cases, i.e. for special families of convex constraints, weak convergence implies norm convergence. The presented approach covers a wide range of problems, e.g. Besov – or BV–restoration for which we present also numerical experiments in the context of image processing. 1. Scope of the Problem In a wide range of practical applications one has to solve the problem that the features of interest cannot be observed directly, but have to be interfered from other quantities. Very often a linear model describing the relationship between the features and the measured quantities works quite nicely, i.e. in such situations the task to solve is Tf = h. Typically, the observed data are not exactly equal to the image h = Tf, but rather a distortion of h, i.e. g = h + e = Tf + e. To find an estimate for f from an observation g, one might minimiz