The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations
Journal article, 2020

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.


Robert Berman

Chalmers, Mathematical Sciences, Algebra and geometry

Numerische Mathematik

0029-599X (ISSN) 0945-3245 (eISSN)

Vol. 145 4 771-836

Subject Categories

Computational Mathematics

Control Engineering

Mathematical Analysis



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9/3/2020 1