Doctoral School Mathematics and Scientific Computing

Abstract: A nonconvex variational model is introduced which contains the l^q-"norm", q in (0,1),of the gradient of the image to be reconstructed as the regularization term together with a least-squares type data fidelity term which may depend on a possibly spatially dependent weighting parameter. Hence, the regularization term in this functional is a nonconvex compromise between the minimization of the support of the reconstruction and the classical convex total variation model. In the discrete setting, existence of a minimizer is proven, a Newton-type solution algorithm is introduced and its global as well as locally superlinear convergence is established. The potential indefiniteness (or negative definiteness) of the Hessian of the objective during the iteration is handled by a trust-region based regularization scheme. The performance of the new algorithm is studied by means of a series of numerical tests. For the associated infinite dimensional model an existence result based on the weakly lower semicontinuous envelope is established and its relation to the original problem is discussed. [hide abstract]