JKO schemes with general transport costs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929258020995072 |
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| author | Rankin, Cale Wong, Ting-Kam Leonard |
| author_facet | Rankin, Cale Wong, Ting-Kam Leonard |
| contents | We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges under suitable conditions to the corresponding Riemannian Fokker--Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker--Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_17681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | JKO schemes with general transport costs Rankin, Cale Wong, Ting-Kam Leonard Analysis of PDEs Probability 35K57 (Primary) 58J35, 82C31 (Secondary) We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges under suitable conditions to the corresponding Riemannian Fokker--Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker--Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence. |
| title | JKO schemes with general transport costs |
| topic | Analysis of PDEs Probability 35K57 (Primary) 58J35, 82C31 (Secondary) |
| url | https://arxiv.org/abs/2402.17681 |