scholarly journals Second-Order Models for Optimal Transport and Cubic Splines on the Wasserstein Space

2019 ◽  
Vol 19 (5) ◽  
pp. 1113-1143 ◽  
Author(s):  
Jean-David Benamou ◽  
Thomas O. Gallouët ◽  
François-Xavier Vialard
Keyword(s):  
Optimal Transport ◽  
Cubic Splines ◽  
Second Order ◽  
Wasserstein Space

USING CUBIC SPLINES TO SOLVE SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS: A TEACHING APPLICATION

2018 ◽  
Author(s):  
Fernando Giménez Palomares ◽  
José Fernando Giménez Luján ◽  
Juan Antonio Monsoriu Serrá ◽  
Pedro Fernández de Córdoba Castellá
Keyword(s):  
Differential Equations ◽  
Cubic Splines ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linear Differential ◽  
Teaching Application

scholarly journals A GEOMETRIC STUDY OF WASSERSTEIN SPACES: HADAMARD SPACES

2012 ◽  
Vol 04 (04) ◽  
pp. 515-542 ◽  
Author(s):  
JÉRÔME BERTRAND ◽  
BENOÎT KLOECKNER
Keyword(s):  
Large Scale ◽  
Euclidean Plane ◽  
Optimal Transport ◽  
Probability Measures ◽  
Hadamard Space ◽  
Geodesic Boundary ◽  
Wasserstein Space ◽  
Scale Properties ◽  
Positively Curved ◽  
Geometric Study

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space [Formula: see text]. In this paper we investigate the geometry of [Formula: see text] when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, [Formula: see text] is not non-positively curved, our results show that [Formula: see text] have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for [Formula: see text] that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in [Formula: see text].

scholarly journals An Optimal Transport View of Schrödinger's Equation

2012 ◽  
Vol 55 (4) ◽  
pp. 858-869 ◽  
Author(s):  
Max-K. von Renesse
Keyword(s):  
Symplectic Structure ◽  
Optimal Transport ◽  
Riemannian Metric ◽  
Symplectic Space ◽  
Probability Measures ◽  
Extended System ◽  
Third Law ◽  
Classical Potential ◽  
Wasserstein Space ◽  
Complex Valued

AbstractWe show that the Schrödinger equation is a lift of Newton's third law of motion on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential μ → F(μ) is the sum of the total classical potential energy (V, μ) of the extended system and its Fisher information . The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures.

scholarly journals Statistical data analysis in the Wasserstein space

2020 ◽  
Vol 68 ◽  
pp. 1-19
Author(s):  
Jérémie Bigot
Keyword(s):  
Optimal Transport ◽  
Point Clouds ◽  
Data Set ◽  
Inference Problems ◽  
Research Perspectives ◽  
Random Probability ◽  
Wasserstein Space ◽  
Geometric Variation ◽  
Random Probability Measures ◽  
Wasserstein Distances

This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on the use of Wasserstein distances and tools from optimal transport to analyse such data. In particular, we highlight the benefits of using the notions of barycenter and geodesic PCA in the Wasserstein space for the purpose of learning the principal modes of geometric variation in a dataset. In this setting, we discuss existing works and we present some research perspectives related to the emerging field of statistical optimal transport.

scholarly journals Characterization of barycenters in the Wasserstein space by averaging optimal transport maps

2018 ◽  
Vol 22 ◽  
pp. 35-57 ◽  
Author(s):  
Jérémie Bigot ◽  
Thierry Klein
Keyword(s):  
Optimal Transport ◽  
Probability Measures ◽  
Reference Measure ◽  
Precise Characterization ◽  
Fixed Reference ◽  
Random Probability ◽  
Wasserstein Space ◽  
Transport Maps ◽  
Random Probability Measures

This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier [SIAM J. Math. Anal. 43 (2011) 904–924], and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.

scholarly journals Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport

2020 ◽  
Author(s):  
Matt Jacobs ◽  
Inwon Kim ◽  
Alpár R. Mészáros
Keyword(s):  
Surface Tension ◽  
Weak Solutions ◽  
Mean Curvature Flow ◽  
Optimal Transport ◽  
Heat Content ◽  
Threshold Dynamics ◽  
Muskat Problem ◽  
Equilibrium Configurations ◽  
Wasserstein Space ◽  
Energy Convergence

Abstract Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoḡlu–Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations.

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