A Computational Fluid Me hani s solution to the

Jean-David Benamou

Description

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 Rus 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

Sujets

• Benamou
• Problem
• Monge?Ampère equation

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

0
A

Computational
78153
Fluid
review

one

erique,
solution

to
the
the
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Monge-Kan
(
toro
F

and
h
o
mass
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transfer
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follo
Jean-Da
T
vid
to
Benamou
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Y
:Jean-
ann
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Brenier
05,
y
t
July
in
9,

2001
℄

[19
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Monge-Kan
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grangian
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and

x
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e
transfer
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total
w
R
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\m
[e-mail]

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as
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w
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b
h
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with
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d
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english
Z
v
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ersion),
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e-mail

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univ

de
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the
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p
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[31
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map
h
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transfer
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and
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the
to
in

see,
T
the
if,
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for
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all
Boltzmann
b
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subset
Astroph
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in-
of
ws
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h
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p

o
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teresting.
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to
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studied
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as
(
to
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dx:
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signal
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transfer
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ts
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matrices,
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imp
h
L
is
man
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elds
refered
Proba-
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the
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equation".
the
The
toro
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is
problem",
the

of
in
and
nding
er-Planc

℄
h
tmospheric
a

map
mo
M
Cullen
,
based
giv
arian
en
L

℄
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[26
and
media

equations
T

,

has
for
b
view
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t
solv
resp
ed
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y
℄
Moser
a
[27
oin
℄
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and
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e
Moser
t
[17
t
℄
whic
with
b
t
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w
applications,
o
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image
t
vision

men
e
in
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ds
tum
that
2

optimal
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and
oth
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lead
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to
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eectiv

e
problem
n
Tw
umerical
exp
algorithms
ts
(see
are
[7
in
℄
The
for
Monge
applications
problem
in
onds
Chemistry).
p
Clearly
1
the
has

een
problem
b
is
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underdetermined
v
and
℄
it
[19
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and
natural
relationship
to
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the
=
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(2)
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ortance
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in
p
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as
h
bilit
(or
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W

asserstein)
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[3
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(where
een
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and



T
to
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equation
y
maxw
:

d
the
p
okk
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[36
0
[22
;
A

Sciences
T
the
)
of
p
semi-geostrophic
=
del
inf
y
M
and
Z
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
gradien
:
o
j
with
denotes
ect
the
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2
norm
toro
in
h
R
[28
d
℄
and
rom
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more
inm
p
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view,
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en

among

all
a
map
aluable
M
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transp
information
orting


w
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to
densit

functions,
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.
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Whenev
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in
the
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inm
of
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ed
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y
and
some
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map
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assimilation
,
meteorology
w
o
e
y
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quan
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that

M
is