Frames , Sparsity and Nonlinear Inverse Problems ∗

This work is concerned with nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to several preassigned bases or frames. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by one–homogeneous (typically weighted `p, 1 ≤ p ≤ 2) penalties on the coefficients (or isometrically transformed coefficients) of such multi–frame expansions. The computation of the solution amounts in this setting to a system of Landweber–fixed–point iterations with thresholding applied in each fixed–point iteration step.