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One feasible way to minimize a non-smooth functional is to replace it by some smoothed version, which leads to a surrogate minimization problem that is easily treated by standard means. A prominent example of such a problem is given by a Tikhonov-type functional incorporating a sparsity-enforcing penalty term. It has received enormous attention in recent years, yet its efficient minimization remains challenging. In this paper we consider general Tikhonov-type functionals and show, under mild conditions, the stability of their minimizer with respect to the replacement of the penalty term with an appropriate approximation. In particular, we consider the case of separable penalty terms. Finally, we apply the proposed strategy to the inverse medium problem and demonstrate numerical results that indicate the efficiency of the approach.

This paper presents Tikhonov- and iterated soft-shrinkage regularization methods for nonlinear inverse medium scattering problems. Motivated by recent sparsity-promoting reconstruction schemes for inverse problems, we assume that the contrast of the medium is supported within a small subdomain of a known search domain and minimize Tikhonov functionals with sparsity-promoting penalty terms based on Lp-norms. Analytically, this is based on scattering theory for the Helmholtz equation with the refractive index in Lp, 1 < p < ∞, and on crucial continuity and compactness properties of the contrast-to-measurement operator. Algorithmically, we use an iterated soft-shrinkage scheme combined with the differentiability of the forward operator in Lp to approximate the minimizer of the Tikhonov functional. The feasibility of this approach together with the quality of the obtained reconstructions is demonstrated via numerical examples.

We consider the Tikhonov functional incorporating a lp-norm as a penalty term. The minimizer of this functional is investigated in regard to its sensitivity with respect to the parameter p. To quantify the sensitivity of the minimizer, we determine the derivative with respect to p. The underlying idea is based on the implicit function theorem. We show that the minimizer of the Tikhonov functional is differentiable with respect to p for any 1 < p < 2. For p = 1, the derivative exists only under additional assumptions.

We investigate a method of accelerated Landweber type for the iterative regularization of nonlinear ill-posed operator equations in Banach spaces. Based on an auxiliary algorithm with a simplified choice of the step-size parameter we present a convergence and stability analysis of the algorithm under consideration. We will close our discussion with the presentation of a numerical example.