Multi-Frames in Thresholding Iterations for Nonlinear Operator Equations with Mixed Sparsity Constraints

This paper is concerened 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. 1 Scope of the problem We consider the computation of an approximation to a solution of a nonlinear operator equation T (x) = y , (1.1) where T : X → Y is an operator between Hilbert spaces X, Y . In case of having only noisy data y with ‖y − y‖ ≤ δ available, there might be the problem of ill-posedness (in the sense of a discontinuous dependency of the solution on the data). Thus problem (1.1) has to be stabilized by regularization methods. In recent years, many of the well known methods for linear inverse problems have been generalized to nonlinear operator equations. But so far all the proposed schemes for nonlinear problems incorporate at most quadratic regularization whereas in many applications the solution is assumed to have sparse expansion which immediately leads to the involvement of nonquadratic penalties, e.g. `p norms with p < 2. In linear lore, this problem is still solved, see [2]. In nonlinear inverse problems there is an approach, see [5], which solves nonlinear operator equations with sparsity constraints. However, recent developments indicate that (higly) redundant systems, such as frames or systems of frames may yield a gain in this context (optimal representation/decomposition of the solution to be reconstructed). When dealing with dictionaries of frame systems, there exist certain methods, e.g. such as basis ∗The author is with the Department of Mathematics, University of Bremen, Germany. G. T. was partially supported by Deutsche Forschungsgemeinschaft Grants TE 354/1-2, TE 354/3-1.