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
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