Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization

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 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.