Abstract
In recent works—both experimental and theoretical—it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge–Ampère equations. This yields H^{1}-converge rates, in terms of the corresponding spatial resolution h, of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on new quantitative stability results for optimal transport maps, shown using complex geometry.
Highlights
The theory of optimal transport [47], which was originally motivated by applications to logistics and economics, has generated a multitude of applications ranging from meteorology and cosmology to image processing and computer graphics in more recent years [44,45]
Since the complex Monge–Ampère measures of locally bounded psh functions do not charge complex analytic subvarieties, formula 3.5 on C∗n follows from formula 3.5 on XC
The general case follows from the previous case and basic qualitative stability properties for the solution of the second boundary value problem for M A on X with respect to variations of the target domain Y
Summary
The theory of optimal transport [47], which was originally motivated by applications to logistics and economics, has generated a multitude of applications ranging from meteorology and cosmology to image processing and computer graphics in more recent years [44,45] This has led to a rapidly expanding literature on numerical methods to construct optimal transport maps, using an appropriate discretization scheme. The present paper is concerned with a particular discretization scheme, known as semi-discrete optimal transport in the optimal transport literature (see [6] and references therein for other discretization schemes, based on finite differences) This approach uses computational geometry to compute a solution to the corresponding discretized Monge–Ampère equation and exhibits remarkable numerical performance, using a damped Newton iteration [34,38]. The convergence of the iteration toward the discrete solution was recently settled in [31], and one of the main aims of the present paper is to establish quantitative convergence rates of the discrete solutions, as the spatial resolution h tends to zero
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