A Posteriori Error Control for Nonconvex Problems via Calibration

SIAM Journal on Numerical Analysis, vol. 64, pp. 350–369

Abstract

Abstract. In this paper, a posteriori error estimates are derived for the approximation error of minimizers of functionals on the space of functions with bounded variation with a nonconvex lower-order term. To this end, the calibration method by Alberti, Bouchitté, and Dal Maso [ Calc. Var. Partial Differential Equations, 16 (2003), pp. 299–333] allows the problem to be reformulated as a uniformly convex variational problem over characteristic functions of subgraphs in one dimension higher. A primal-dual approach is formulated where the duality of divergence and gradient properly incorporates boundary conditions for the primal variable. Based on this, a posteriori error estimates can be derived first for the relaxed problem in the [Formula: see text]-norm. A cut-out argument allows converting this into an [Formula: see text]-error estimate for the characteristic subgraph functions apart from the jump interface, whereas the area of the interfacial region is estimated separately. To apply the estimate, we consider as one possible discretization a conforming finite element space for the primal variable and a nonconforming space for the dual variable. Finally, we validate the a posteriori error estimates in numerical experiments for a prototypical nonconvex functional in one and two dimensions as well as depth estimation in stereo imaging, a classical computer vision problem.

Authors 4

1. Benjamin Berkels Aachen Aachen Institute for Advanced Study in Computational Engineering Science (AICES)

   Aachen Institute for Advanced Study in Computational Engineering Science · RWTH Aachen University

   Affiliation as printed

   Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, 52056 Aachen, Germany

2. Alexander Effland

   University of Bonn

   Affiliation as printed

   Institute for Applied Mathematics, University of Bonn, 53115 Bonn, Germany

3. Martin Rumpf

   University of Bonn

   Affiliation as printed

   Institute for Numerical Simulation and Institute for Applied Mathematics, University of Bonn, 53115 Bonn, Germany

4. Jan Verhülsdonk

   University of Bonn

   Affiliation as printed

   Institute for Applied Mathematics, University of Bonn, 53115 Bonn, Germany