Explicit a posteriori error representation for variational problems and application to TV-minimization
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In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix-Raviart and the Raviart-Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation – initially based on continuous arguments only – practicable from a numerical point of view. In addition, we provide a generalized Marini formula for the primal solution that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin-Osher-Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.
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