49 research outputs found
Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re
equation with second boundary value condition, provided the target is a convex
set. This yields a fast adaptive method to numerically solve the Optimal
Transport problem between two absolutely continuous measures, the second of
which has convex support. The proposed numerical method actually captures a
specific Brenier solution which is minimal in some sense. We prove the
convergence of the method as the grid stepsize vanishes and we show with
numerical experiments that it is able to reproduce subtle properties of the
Optimal Transport problem
Multi-physics Optimal Transportation and Image Interpolation
International audienceOptimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by Benamou and Brenier \cite{BB00} where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpolation path, and it is actually not very difficult to exhibit test cases where the algorithm produces a path of images where high density regions split at the beginning before merging back at its end. However, in some applications to image interpolation this behaviour is not physically realistic. Hence, this paper aims at studying how some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation
The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Amp\ue8re 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–Amp\ue8re 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
From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit
Let X be a compact complex manifold equipped with a smooth (but not
necessarily positive) closed form theta of one-one type. By a well-known
envelope construction this data determines a canonical theta-psh function u
which is not two times differentiable, in general. We introduce a family of
regularizations of u, parametrized by a positive number beta, defined as the
smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is
shown that, as beta tends to infinity, the regularizations converge to the
envelope u in the strongest possible Holder sense. A generalization of this
result to the case of a nef and big cohomology class is also obtained. As a
consequence new PDE proofs are obtained for the regularity results for
envelopes in [14] (which, however, are weaker than the results in [14] in the
case of a non-nef big class). Applications to the regularization problem for
quasi-psh functions and geodesic rays in the closure of the space of Kahler
metrics are given. As briefly explained there is a statistical mechanical
motivation for this regularization procedure, where beta appears as the inverse
temperature. This point of view also leads to an interpretation of the
regularizations as transcendental Bergman metrics.Comment: 28 pages. Version 2: 29 pages. Improved exposition, references
updated. Version 3: 31 pages. A direct proof of the bound on the
Monge-Amp\`ere mass of the envelope for a general big class has been included
and Theorem 2.2 has been generalized to measures satisfying a
Bernstein-Markov propert