Various iterative methods are available for the approximate solution of non-smooth minimization problems. For a popular non-smooth minimization problem arising in image processing, we discuss the suitable application of three prototypical methods and their stability.
The methods are compared experimentally with a focus on choice of stopping criteria, influence of rough initial data, step sizes as well as mesh sizes. An overview of existing algorithms is given.
The primal-dual gap is a natural upper bound for the energy error
and, for uniformly convex minimization problems, also for the error in the energy norm.
This feature can be used to construct reliable primal-dual gap error estimators for which
the constant in the reliability estimate equals one for the energy error and equals the
uniform convexity constant for the error in the energy norm. In particular, it defines
a reliable upper bound for any functions that are feasible for the primal and the
associated dual problem. The abstract a posteriori error estimate based on the
primal-dual gap is provided in this article, and the abstract theory is applied to the
nonlinear Laplace problem and the Rudin-Osher-Fatemi image denoising problem.
The discretization of the primal and dual problems with conforming, low-order
finite element spaces is addressed. The primal-dual gap error estimator is used
to define an adaptive finite element scheme and numerical experiments are presented,
which illustrate the accurate, local mesh refinement in a neighborhood of the singularities,
the reliability of the primal-dual gap error estimator and the moderate
overestimation of the error.
The discretization of a bilaterally constrained total variation minimization problem with conforming low order finite elements is analyzed and three iterative schemes are proposed which differ in the treatment of the nondifferentiable terms. Unconditional stability and convergence of the algorithms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are shown.
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