The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations

We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Ampère type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e...

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Vydáno v:	Numerische Mathematik Ročník 145; číslo 4; s. 771 - 836
Hlavní autor:	Berman, Robert J.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2020 Springer Nature B.V
Témata:	Algorithms boundary-value problem Complexity convergence Matematik Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematical sciences Mathematics Mathematics and Statistics Monte Carlo simulation neumann Numerical Analysis Numerical and Computational Physics numerical-solution Parabolic differential equations Partial differential equations polar factorization regularity schemes Simulation Theoretical Toruses 35J60 65N99 35K55 90C08
ISSN:	0029-599X, 0945-3245, 0945-3245
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Popis
Shrnutí:	We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Ampère type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0029-599X 0945-3245 0945-3245
DOI:	10.1007/s00211-020-01127-x