Explicit a posteriori error representation for variational problems and application to TV-minimization
Year: 2023
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Bartels, Sören, Kaltenbach, Alex
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4383987677