Explicit a posteriori error representation for variational problems and application to TV-minimization

TL;DR

This paper develops an explicit a posteriori error representation for convex minimization problems, enabling practical numerical error estimation and adaptive algorithms, demonstrated on TV-minimization with proven convergence.

Contribution

It introduces a novel discrete a posteriori error representation for convex problems using duality and orthogonality, applicable to TV-minimization, with a new primal solution formula.

Findings

Achieved a practical error estimator for convex minimization.

Demonstrated an adaptive algorithm with linear convergence.

Validated approach on the TV-minimization model.

Abstract

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.