Unnormalized optimal transport

Wilfrid Gangbo UCLA Wuchen Li UCLA Stanley Osher UCLA Michael Puthawala Rice

Numerical Analysis and Scientific Computing mathscidoc:2004.25002

Journal of Computational Physics, 2019.12
We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier. Our extension allows optimal transfer of unnormalized and unequal masses. We obtain a one-parameter family of simple modifications of the formulation in [4]. This leads us to a new Monge-Ampere type equation and a new Kantorovich duality formula. These can be solved efficiently by, for example, the Chambolle-Pock primal-dual algorithm. This solution to the extended mass transfer problem gives us a simple metric for computing the distance between two unnormalized densities. The L1 version of this metric was shown in [23] (which is a precursor of our work here) to have desirable properties.
Optimal transport; Unnormalized density space; Unnormalized Monge-Ampére equation
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  title={Unnormalized optimal transport},
  author={Wilfrid Gangbo, Wuchen Li, Stanley Osher, and Michael Puthawala},
  booktitle={Journal of Computational Physics},
Wilfrid Gangbo, Wuchen Li, Stanley Osher, and Michael Puthawala. Unnormalized optimal transport. 2019. In Journal of Computational Physics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427014406145588648.
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