Fast minimization of Tikhonov functionals with sparsity constraints

In this work, we consider ill-posed linear problems, where instead of exact data we are given noisy measurements only. By ill-posedness we mean that the solution of the problem does not depend continuously on the data. Due to this ill-posedness, we have to employ regularization methods for the computation of approximate solutions. In particular, we are interested in sparse solutions. A widely used regularization method that yields sparse solutions is the minimization of a Tikhonov functional with an l 1-sparsity constraint. Instead of minimizing this functional directly, we consider a transformation to a differentiable standard form with an l 2-constraint. This transformation allows us to utilize iterative methods that use information about the derivatives of the functional. We check whether these methods yield a speed-up compared to methods that minimize the nontransformed functional. We test this on a number of numerical examples based on a tomography problem and on a localization problem in microscopy.submitted by Fabian HintererUniversität Linz, Masterarbeit, 2019(VLID)378027