Abstract

Many problems in mathematical imaging can be phrased as convex minimization problems or convex-concave saddle point problems. In both cases, the respective optimality system is an inclusion with a (maximally) monotone operator. To solve these inclusions, several splitting methods have been proposed which rely on the idea that the monotone operator can be split up into simpler parts for which, e.g. the resolvent can be applied easily.

In this talk we present a fairly general splitting method which works for inclusions in which the operator can be split such that one operator is co-coercive and for the other a certain preconditioned resolvent is easily applicable. We arrive at an inertial forward backward splitting method for which we prove weak convergence under fairly general assumptions. It is show that the methods covers several existing methods such as Polyak’s heavy ball method, Nesterov’s accelerated gradient descent, the forward-backward splitting method and Beck and Teboulle’s FISTA . We illustrate the applicability and performance on numerous problems such as the Rudin-Osher-Fatemi denoising and deconvolution or the Osher-Sole-Vese denoising.

This is joint work with Thomas Pock (TU Graz).