 # A Computational Fluid Me hani s solution to the

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A Computational Fluid Me hani s solution to the Monge-Kantorovi h mass transfer problem. Jean-David Benamou Yann Brenier y July 9, 2001 Abstra t The L 2 Monge-Kantorovi h mass transfer problem [31? is reset in a Fluid Me hani s framework and numeri ally solved by an augmented La- grangian method. 1 Introdu tion The rst mass transfer problem was onsidered by Monge in 1781 in his \memoire sur la theorie des deblais et des remblais, a Civil Engineering problem where par els of materials have to be displa ed from one site to another one with minimal transportation ost. A modern treatment of this problem has been initiated by Kantorovi h in 1942 ( f. [23? for the english version), leading to the so- alled Monge-Kantorovi h problem whi h has re eived a onsiderable interest in the re ent years, with a wide range of potential appli ations and extensions. A re ent omprehensive review an be found in the new books by Ra hev and Rus hendorf [31?, the le ture notes by Evans [19? and the review paper by M Cann and Gangbo [21?. The framework of the Monge-Kantorovi h problem is as follows. Two density fun tions 0 (x) 0 and T (x) 0 of x 2 R d , that we assume to be bounded with total mass one Z

• fun tions

• ee tive

• tive proof

• problem

• monge-ampere equation

• lagrangian method

• problem has

• standard boundary

• alled alg2

• numeri al

Subjects

##### Monge?Ampère equation

Informations

Report a problem

0
A

Computational
78153
Fluid
review

one

erique,
solution

to
the
the
2
Monge-Kan
(
toro
F

and
h
o
mass
b
transfer
Gangb
problem.
follo
Jean-Da
T
vid
to
Benamou
)

Y
:Jean-
ann
aris
Brenier
05,
y
t
July
in
9,

2001

[19
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densit
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that
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ounded
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[31
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grangian
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metho
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d.
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1
functions
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and

x
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rst
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mass
e
transfer
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problem
total
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R
as
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y
)
Monge
;
in
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1781
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in
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rance,
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vid.Benamou@inria.fr

ersitaire
emoire
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th
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to
ans
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ortation
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as
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mo
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t
(
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has
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,
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w
toro
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h
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in
with
1942
mass
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d

0
for
x
the
dx
english
Z
v
d
ersion),
T
x
to
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the
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toro
de

oluceau,
h
.105
problem
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whic
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h
F
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the
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a
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p
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to
in

see,
T
the
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for
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p

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to
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mo
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as
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to
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just
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of
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y

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man
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refered
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as
and
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equation".
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is
problem",
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of
in
and
nding
er-Planc

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tmospheric
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map
mo
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Cullen
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[26
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equations
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w
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algorithms
ts
(see
are
[7
in

The
for
Monge
applications
problem
in
onds
Chemistry).
p
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1
the
has

een
problem
b
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underdetermined
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it
[19
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to
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dels).
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the
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dressed
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as
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asserstein)

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to
dened
homogeneous
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equation
y
maxw
:

d
the
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okk
(
k

[36
0
[22
;
A

Sciences
T
the
)
of
p
semi-geostrophic
=
del
inf
y
M
and
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is
j
on
M
v
(
t
x
the
)
2
x
[16
j

p
ysics

0
orous
(
equations,
x
w
)
(with
dx;
new
(4)
h
where
tro
p
b

Otto
1
dissipativ
is
PDEs
xed,
ed
j
:
o
j
with
denotes
ect
the
the

2
norm
toro
in
h
R
[28
d

and
rom
the
more
inm
p
um
t
is
view,
tak
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en

among

all
a
map
aluable
M
titativ
transp
information
orting

w
0
dieren
to
densit

functions,
T
h
.
y
Whenev
e
er
in
the
arious
inm
of
um

is
as
ac
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ed
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y
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some
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map
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assimilation
,
meteorology
w
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e
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y

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