Alex Kaltenbach (TU Berlin):
Vortragsreihe im Oberseminar: Alex Kaltenbach (TU Berlin) - A priori and a posteriori error identities for convex minimization problems based on convex duality relations
Zeit und Ort
Mittwoch, 14.8.24, 14:00-15:00, (TBA)
Zusammenfassung
A priori and a posteriori error identities for convex minimization problems based on convex duality relations\n\nA. Kaltenbach (TU Berlin)\n\nThe objective of this mini course is to develop a thorough error analysis for, in particular, non-smooth convex minimization problems on the basis of convex duality:\n\nAs a motivation example, we consider the celebrated Prager-Synge identity, the most famous example of an a posteriori error identity for the approximation of the Poisson problem. The original proof of the Prager-Synge identity resorts to Pythagoras' theorem, so that, initially, it seems like that the Prager-Synge identity cannot be generalized to non-linear or non-smooth problems. In this mini course, we will find that this is not true. More precisely, replacing Pythagoras' theorem by basic concepts from convex duality, we will find that the Prager-Synge identity can be generalized to a vast class of non-linear and non-smooth convex minimization problems.\n\nTo begin with, we recapitulate the most important concepts of convex duality theory: from basic notions from convex analysis via convex duality in the senses of Lagrange and Fenchel to the celebrated Fenchel duality theorem.\n\nThen, we apply the general Fenchel duality theory to a class of non-smooth convex minimization problems given through integral functionals and derive a generalized Prager-Synge identity, the so-called primal-dual gap identity.\n\nTo make the primal-dual gap identity practicable from a numerical point of view, using orthogonality relations between the Crouzeix-Raviart and the Raviart-Thomas elements, we transfer all convex duality relations to a discrete level.\n\nThe thus derived discrete convex duality relations, in turn, allow to derive an a posteriori error identity on a discrete level, which, eventually, turns out to be an a priori error identity.