Termine Angewandte Mathematik Münster

The family of Zolotarev distances between two probabilities naturally extends the Wasserstein-1 metric to higher orders: one bounds the Lipschitz constant of the relevant derivatives of the potential. So far, however, the optimal transport perspective has been available only for the first order. In my talk I will demonstrate how a PDE motivation revolving around optimal elastic structures has led us to a new (OT) framework for the second-order Zolotarev distance. The new duality theory paves a way to efficient computational method, including a variant of the famous Sinkhorn algorithm. The talk is partially based on a joint work with Guy Bouchitté (Université de Toulon).

Many linear and nonlinear partial differential equations (PDEs) arise from the minimization of an underlying energy functional. Examples are ubiquitous and arise, for instance, in the study of non-Newtonian fluids, minimal surfaces, or nonlinear mechanics. Whereas classical numerical methods and their analysis mostly focus entirely on the solution of the associated PDE/Euler-Lagrange equations, recent contributions have started to take the energetic structure into account. This is particularly relevant for engineering applications where the energetic behavior of minimizers can be more relevant than the solution itself. In addition, analysis in terms of energy often naturally fits the structure of the problem, which can be exploited to derive error bounds with explicit constants. We will introduce the method of flux-equilibration which yields highly effective a posteriori error bounds that can subsequently be used to steer adaptive mesh-refinement. After a general introduction to the concept, we highlight recent advances and generalizations to the nonlinear setting and present numerical results highlighting the efficiency of the method.

TBA