Work

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

Year: 2024

Type: article

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 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.

Source: Foundations of Computational Mathematics

DOI: 10.1007/s10208-024-09676-5

Authors: Sören Bartels, Alex Kaltenbach

Institutions: University of Freiburg, Technische Universität Berlin

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funder: Technische Universität Berlin