A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints

In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one–homogeneous (typically weighted `p) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For p < 2, the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber–fixed–point iteration with a projection applied in each fixed–point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse SPECT (Single Photon Emission Computerized Tomography) problem.