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### Open Access Journals

IPI

The first order optimality system of a total variation
regularization based variational model with $L^2$-data-fitting in
image denoising ($L^2$-TV problem) can be expressed as an elliptic
variational inequality of the second kind. For a finite element
discretization of the variational inequality problem, an a
posteriori error residual based error estimator is derived and its
reliability and (partial) efficiency are established. The results
are applied to solve the $L^2$-TV problem by means of the adaptive
finite element method. The adaptive mesh refinement relies on the
newly derived a posteriori error estimator and on an additional heuristic providing a
local variance estimator to cope with noisy data. The numerical
solution of the discrete problem on each level of refinement is
obtained by a superlinearly convergent algorithm based on
Fenchel-duality and inexact semismooth Newton techniques and which
is stable with respect to noise in the data. Numerical results
justifying the advantage of adaptive finite elements solutions are
presented.

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