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arXiv · 2205.12172

Data driven gradient flows

Jan-F. Pietschmann·Matthias Schlottbom
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Abstract

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 specialise 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 independent of their respective properties (such as finite speed of propagation or blow-up). We then focus on the numerical implementation of our approach using an 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 detailed numerical examples.

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Jan-F. Pietschmann, Matthias Schlottbom. 2022-05-24. Data driven gradient flows. https://doi.org/10.1553/etna_vol57s193

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