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Section: New Results

Optimal transport

  • Participant: Bruno Lévy

Optimal transport:

Optimal Transport is not only a fundamental problem with a rich structure, but also a new computational tool, with many possible applications. To name but a few, applications of Optimal Transport comprise image registration, reflector and refractor design, histogram interpolation, artificial intelligence. In astrophysics, it is used by Early Universe Reconstruction, a difficult inverse problem that reconstructs the time evolution of the universe from the observed current state. It can be also used in meteorology, to simulate certain phenomena (semi-geostrophic currents). It is also the main component of solvers for certain equations, based on a variational formulation that leads to a gradient flow. All these applications and future developments depend on a single component: an efficient solver for the Monge-Ampère equation. We developed a new algorithm that overcome by several order of magnitude the speed of the classical "auction algorithm" (that solves in O(nlog(n)) a discrete version of the problem). The semi-discrete version of the problem that we study can be solved by extremizing a smooth objective function, thus a significantly faster speed is obtained as compared to the previous combinatorial algorithm. This year we improved our Quasi-Newton solver and replaced it with a Full-Newton solver, that gains one additional order of magnitude in speed, and we can solve semi-discrete problems with 1 million Dirac masses in a matter of minutes. We also experimented with applications of this solver to fluid simulation. Last winter (December 2015) Wenping Wang visited Nancy, and we discussed several ideas on Optimal Transport. We proposed together this year (2016) a new method to sample a surface with a power diagram [31]. The positions of the samples are optimized by a criterion similar to centroidal Voronoi tessellations, and the associated weights are used to control the areas of the power cells with prescribed values. We give the expressions of the derivatives of the combined objective function, and propose a quasi-Newton algorithm to optimize it. We describe several applications of the algorithm.

Figure 3. Semi-discrete optimal transport between a shape and a sphere, computed by our algorithm
IMG/sphere_tangled_morph.png