Data-driven gradient flows

  • We present a framework enabling variational data assimilation for gradient flows in general metric spaces, based on the minimizing movement (or Jordan–Kinderlehrer–Otto) approximation scheme. After discussing stability properties in the most general case, we specialize to the space of probability measures endowed with the Wasserstein distance. This setting covers many non-linear partial differential equations (PDEs), such as the porous-medium equation or general drift–diffusion–aggregation equations, which can be treated by our methods independently of their respective properties (such as finite speed of propagation or blow-up). We then focus on the numerical implementation using a primal–dual algorithm. The strength of our approach lies in the fact that, by simply changing the driving functional, a wide range of PDEs can be treated without the need to adopt the numerical scheme. We conclude by presenting several numerical examples.

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Author:Jan-Frederik PietschmannORCiDGND, Matthias Schlottbom
Frontdoor URL
Parent Title (English):ETNA - Electronic Transactions on Numerical Analysis
Publisher:Verlag der Österreichischen Akademie der Wissenschaften
Place of publication:Wien
Year of first Publication:2022
Release Date:2023/02/20
Tag:Analysis; Applied Mathematics
First Page:193
Last Page:215
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Inverse Probleme