Sparsity Reconstruction by the Standard Tikhonov Regularization

It is a common belief that Tikhonov scheme with ‖ · ‖L2 -penalty fails to reconstruct a sparse structure with respect to a given system {φi}. However, in this paper we present a procedure for sparsity reconstruction, which is totally based on the standard Tikhonov method. This procedure consists of two steps. At first Tikhonov scheme is used as a sieve to find the coefficients near φi, which are suspected to be non-zero. Within this step the performance of the standard Tikhonov method is controlled in some sparsity promoting space rather than in original Hilbert one. In the second step of proposed procedure the coefficients with indices selected in the previous step are estimated by means of data functional strategy. The choice of the regularization parameter is the crucial issue for both steps. We show that recently developed parameter choice rule called the balancing principle can be effectively used here. We also present the results of computational experiments giving the evidence of the reliability of our approach.