Contemporary Mathematics

Volume 353, 2004

Numerical Analysis of a Multi-Phasic Mass Transport

Problem

Jean-David Benamou, Yann Brenier, and Kevin Guittet

ABSTRACT. We propose a multi-phasic extension of the

£

2-Monge Kantorovitch

transport problem and adapt the numerical method introduced in

[1]

for its

resolution.

1.

Introduction

We extend the time-continuous reformulation and the numerical method intro-

duced in

[1]

in a multi-phasic setting. Preliminary reading of this first paper is

recommended as the present work is in large part similar to the simplest mono-

phasic case. The range of applications of the classical Monge-Kantorovitch mass

transfer problem evolves rapidly

[9, 10, 7, 6, 12, 23, 17, 24, 13, 22, 14, 8, 16, 18,

19, 20, 21]

and certainly opens new potential generalizations for the multi-phasic

problem.

Let us first recall the mono-phasic "classical" formulation of the Monge-Kantoro-

vitch problem (MKP):

Given two bounded, non-negative measurable functions p0 and pr with same

mass in the torus D

=

JR_d /7ld, find an application M which realizes the transport

from

p0

to PT in the following sense : For all Borel set

A, M

satisfies

(1.1)

{ Po(x)dx

=

J

Pr(x)dx,

JM-l(A) A

and achieves the minimal cost

(1.2)

This minimal cost defines the so-called Wasserstein metric between the densities

p0

and

py.

In order to compute this optimal map, a time-continuous reformulation

of the problem was introduced in

[1] :

1991 Mathematics Subject Classification. Primary 90B99, 65K10 ; Secondary 35J60, 78A05.

Key words and phrases. Monge-Kantorovitch Problem, Multi-Phasic fluid model.

©

2004 American Mathematical Society

http://dx.doi.org/10.1090/conm/353/06428