Data‐Driven Optimal Transport

The problem of optimal transport between two distributions ρ(x) and μ(y) is extended to situations where the distributions are only known through a finite number of samples {x_i} and {y_j}. A weak formulation is proposed, based on the dual of the Kantorovich formulation, with two main modifications: replacing the expected values in the objective function by their empirical means over the {x_i} and {y_j}, and restricting the dual variables u(x) and v(y) to a suitable set of test functions adapted to the local availability of sample points. A procedure is proposed and tested for the numerical solution of this problem, based on a fluidlike flow in phase space, where the sample points play the role of active Lagrangian markers. © 2016 Wiley Periodicals, Inc.     

Boundary value problems in elastostatics with singular data

Lithuanian Mathematical Journal - We consider themain boundary value problems of linear elastostatics with nonregular data. We prove existence and uniqueness results for bounded and exterior...     

A regularity result for a class of non-uniformly elliptic operators

Archiv der Mathematik - We obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations led by operators that are neither...     

The pinched Veronese is Koszul

In this paper we prove that the coordinate ring of the pinched Veronese, k[X ³,X ² Y,XY ²,Y ³,X ² Z,Y ² Z,XZ ²,YZ ²,Z ³], is Koszul. The result is obtained by combining the use of a flat deformation induced by a distinguished weight together with a generalization of the notion of Koszul filtrations.     

Global solution to a phase transition model based on a microforce balance

Our aim in this article is to prove the existence and the uniqueness of a global solution to a nonisothermal Ginzburg-Landau (Allen-Cahn) system. This system is obtained by considering, in addition to the fundamental laws of thermodynamics, a balance law for internal microforces proposed by M. Gurtin.     

Compactness and Vitali’s compactness criterion in vector-valued F-seminormed function spaces

Results of compactness for vector-valued F-seminormed function spaces and a general Vitali’s compactness criterion under the W-property are exhibited.

     

Explanation of Variability and Removal of Confounding Factors from Data through Optimal Transport

A methodology based on the theory of optimal transport is developed to attribute variability in data sets to known and unknown factors and to remove such attributable components of the variability from the data. Denoting by x the quantities of interest and by z the explanatory factors, the procedure transforms x into filtered variables y through a z‐dependent map, so that the conditional probability distributions ρ(x|z) are pushed forward into a target distribution μ(y), independent of z. Among all maps and target distributions that achieve this goal, the procedure selects the one that minimally distorts the original data: the barycenter of the ρ(x|z). Connections are found to unsupervised learning and to fundamental problems in statistics such as conditional density estimation and sampling. Particularly simple instances of the methodology are shown to be equivalent to k‐means and principal component analysis. An application is shown to a time series of ground temperature hourly data across the United States.© 2017 Wiley Periodicals, Inc.