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 -sparsity
constraints and . 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.
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Key words
sparsity, surrogate functional, inverse problem, regularization
AMS subject classifications
65L09, 65J22, 65J20, 47J06, 94A12
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