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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.

IPI

Based on the Fenchel pre-dual of the total variation model, a nonlinear multigrid algorithm for image denoising is proposed. Due to the structure of the differential operator involved in the Euler-Lagrange equations of the dual models, line Gauss-Seidel-semismooth-Newton step is utilized as the smoother, which provides rather good smoothing rates. The paper ends with a report on numerical results and a comparison with a very recent nonlinear multigrid solver based on Chambolle's iteration [6].

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