Convergence Rates for Inverse Problems with Impulsive Noise

We study inverse problems F(f) = g with perturbed right hand side g^{obs} corrupted by so-called impulsive noise, i.e. noise which is concentrated on a small subset of the domain of definition of g. It is well known that Tikhonov-type regularization with an L^1 data fidelity term yields significantly more accurate results than Tikhonov regularization with classical L^2 data fidelity terms for this type of noise. The purpose of this paper is to provide a convergence analysis explaining this remarkable difference in accuracy. Our error estimates significantly improve previous error estimates for Tikhonov regularization with L^1-fidelity term in the case of impulsive noise. We present numerical results which are in good agreement with the predictions of our analysis

A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space

We consider the task of computing an approximate minimizer of the sum of a smooth and non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are studied. Furthermore, the results are applied to the minimization of Tikhonov-functionals associated with linear inverse problems and semi-norm penalization in Banach spaces. With the help of Bregman-Taylor-distance estimates, rates of convergence for the forward-backward splitting procedure are obtained. Examples which demonstrate the applicability are given, in particular, a generalization of the iterative soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as well as total-variation based image restoration in higher dimensions are presented

Convergence Rates for Exponentially Ill-Posed Inverse Problems with Impulsive Noise

This paper is concerned with exponentially ill-posed operator equations with additive impulsive noise on the right hand side, i.e. the noise is large on a small part of the domain and small or zero outside. It is well known that Tikhonov regularization with an $L^1$ data fidelity term outperforms Tikhonov regularization with an $L^2$ fidelity term in this case. This effect has recently been explained and quantified for the case of finitely smoothing operators. Here we extend this analysis to the case of infinitely smoothing forward operators under standard Sobolev smoothness assumptions on the solution, i.e. exponentially ill-posed inverse problems. It turns out that high order polynomial rates of convergence in the size of the support of large noise can be achieved rather than the poor logarithmic convergence rates typical for exponentially ill-posed problems. The main tools of our analysis are Banach spaces of analytic functions and interpolation-type inequalities for such spaces. We discuss two examples, the (periodic) backwards heat equation and an inverse problem in gradiometry.Comment: to appear in SIAM J. Numer. Ana