Abstract
It is a common belief that the Tikhonov scheme with the \({\|\cdot\|_{L_2}}\)-penalty fails to reconstruct a sparse structure with respect to a given system {ϕ i }. However, in this paper we present a procedure for the sparse recovery, which is totally based on the standard Tikhonov method. This procedure consists of two steps. At first the Tikhonov scheme is used as a sieve to find the coefficients near ϕ i , 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 the original Hilbert one. In the second step of the proposed procedure, the coefficients with indices selected in the previous step are estimated by means of the data functional strategy. The choice of the regularization parameter is a crucial issue for both steps. We show that a 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.
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Lu, S., Pereverzev, S.V. Sparse recovery by the standard Tikhonov method. Numer. Math. 112, 403–424 (2009). https://doi.org/10.1007/s00211-009-0214-x
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DOI: https://doi.org/10.1007/s00211-009-0214-x