A Newton Algorithm for Semidiscrete Optimal Transport with Storage Fees

A Newton algorithm for semi-discrete optimal transport

Many problems in geometric optics or convex geometry can be recast as optimal transport problems and a popular way to solve these problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We introduce a damped Newton’s algorithm for this type of problems, which is experimentally efficient, and we establish its ...

The Semidiscrete Filtered Backprojection Algorithm Is Optimal for Tomographic Inversion

Minimal Geodesics Along Volume-Preserving Maps, Through Semidiscrete Optimal Transport

We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L metric, which as observed by Arnold [Arn66] solve the Euler equations of inviscid incompressible uids. The method relies on the generalized polar decomposition of Brenier [Bre91], numerically implemented through semi-discrete optimal transport. It is robust enough to e...

A sparse algorithm for dense optimal transport

Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport we provide a framework to verify global optimality of a discrete transport plan locally. This allows construction of a new sparse algorithm to solve large dense problems by considering a sequenc...

Newton-Product Integration for a Stefan Problem with Kinetics

Stefan problem with kinetics is reduced to a system of nonlinear Volterra integral equations of second kind and Newton's method is applied to linearize it. Product integration solution of the linear form is found and sufficient conditions for convergence of the numerical method are given. An example is provided to illustrated the applicability of the method.