Volume 39, pp. 476-507, 2012.

On the minimization of a Tikhonov functional with a non-convex sparsity constraint

Ronny Ramlau and Clemens A. Zarzer

Abstract

In this paper we present a numerical algorithm for the optimization of a Tikhonov functional with an $\ell_p$-sparsity constraints and $p<1$. Recently, it was proven that the minimization of this functional provides a regularization method. We show that the idea used to obtain these theoretical results can also be utilized in a numerical approach. In particular, we exploit the technique of transforming the Tikhonov functional to a more viable one. In this regard, we consider a surrogate functional approach and show that this technique can be applied straightforwardly. It is proven that at least a critical point of the transformed functional is obtained, which directly translates to the original functional. For a special case, it is shown that a gradient based algorithm can be used to reconstruct the global minimizer of the transformed and the original functional, respectively. Moreover, we apply the developed method to a deconvolution problem and a parameter identification problem in the field of physical chemistry, and we provide numerical evidence for the theoretical results and the desired sparsity promoting features of this method.

Full Text (PDF) [889 KB]

Key words

sparsity, surrogate functional, inverse problem, regularization

AMS subject classifications

65L09, 65J22, 65J20, 47J06, 94A12

Links to the cited ETNA articles

[37]Vol. 30 (2008), pp. 54-74 Ronny Ramlau: Regularization properties of Tikhonov regularization with sparsity constraints
[38]Vol. 37 (2010), pp. 87-104 Ronny Ramlau and Elena Resmerita: Convergence rates for regularization with sparsity constraints

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