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Adaptive Mesh Refinement Methods and Parallel Computing
Rich Hornung
hornung@llnl.govwww.llnl.gov/CASC/SAMRAI
The Conference on High-Speed Computing Gleneden Beach, OR
April 19-22, 2004
Center for Applied Scientific ComputingLawrence Livermore National Laboratory
UCRL-PRES-203583
This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National
Laboratory under contract No. W-7405-Eng-48.
Pres
enta
tion
Out
line
�O
verv
iew
of a
dapt
ive
mes
h re
finem
ent (
AM
R) a
ppro
ache
s—
Gen
eral
issu
es—
Uns
truc
ture
d A
MR
—C
ell-b
y-ce
ll st
ruct
ured
AM
R—
Patc
h-ba
sed
stru
ctur
ed A
MR
—H
ybrid
AM
R m
etho
ds�
SAM
RA
I alg
orith
m d
evel
opm
ent t
o im
prov
e pa
ralle
l sca
labi
lity
�C
oncl
udin
g re
mar
ks
Ada
ptiv
e M
esh
Ref
inem
ent (
AM
R) t
arge
ts
“mul
ti-re
solu
tion”
pro
blem
s
�M
any
scie
nce
& e
ngin
eerin
g pr
oble
ms
exhi
bit s
olut
ions
with
larg
egr
adie
nts
sepa
rate
d by
rela
tivel
y la
rge
smoo
th re
gion
s�
Num
eric
al s
olut
ion
of P
DEs
invo
lves
a d
iscr
ete
dom
ain
(i.e.
, grid
) an
d al
gebr
aic
appr
oxim
atio
n of
equ
atio
ns—
fine
grid
s re
quire
d to
reso
lve
loca
l fea
ture
s—
coar
ser g
rids
suffi
ce in
regi
ons
whe
re s
olut
ion
is s
moo
th�
Grid
spa
cing
det
erm
ines
acc
urac
y an
d co
st o
f com
puta
tion
—fin
e gr
id e
very
whe
re m
ay b
e in
effic
ient
—lo
catio
n an
d re
solu
tion
of fi
ne g
rid n
ot a
lway
s kn
own
a pr
iori
�A
MR
dyn
amic
ally
focu
ses
com
puta
tiona
l effo
rt
—ad
just
grid
reso
lutio
n to
reso
lve
loca
l fea
ture
s w
ithou
t in
curr
ing
cost
of g
loba
lly fi
ne g
rid—
com
pute
d so
lutio
n sh
ould
hav
e gl
obal
ly “
unifo
rm”
accu
racy
AM
R is
incr
easi
ngly
impo
rtan
t for
larg
e-sc
ale
prob
lem
s, p
rese
nts
chal
leng
es
�A
MR
has
dem
onst
rate
d its
use
fuln
ess
in m
any
appl
icat
ion
area
s—
shoc
ks, f
luid
/mat
eria
l int
erfa
ces
—pl
asm
a ph
ysic
s—
radi
atio
n di
ffusi
on, t
rans
port
—
mag
neto
hydr
odyn
amic
s, s
pace
wea
ther
—co
mbu
stio
n —
solid
s, fr
actu
re, s
hear
ban
ds—
astr
ophy
sics
, cos
mol
ogy
—el
ectr
omag
netic
s, s
emic
ondu
ctor
s—
com
plex
geo
met
ry in
eng
inee
ring
prob
lem
s—
man
y ot
hers
…
�A
dapt
ive
met
hods
use
eith
er u
nstr
uctu
red
or s
truc
ture
d m
eshe
s —
All
star
t with
coa
rse
base
grid
and
add
loca
l ref
inem
ent a
s ne
eded
—M
athe
mat
ical
cha
lleng
es: n
umer
ical
app
roxi
mat
ion,
erro
r est
imat
ion
—C
ompu
tatio
nal c
halle
nges
: opt
imiz
e dy
nam
ic lo
ad b
alan
ce, d
ata
(re)d
istri
butio
n, c
ompl
ex c
omm
unic
atio
n (o
verh
eads
can
not b
e am
ortiz
ed)
—D
etai
ls a
re s
peci
fic to
cho
ice
of A
MR
met
hodo
logy
Uns
truc
ture
d ad
aptiv
e m
esh
refin
emen
t�
Typi
cally
use
d w
ith F
E, F
V m
etho
ds—
good
for c
ompl
exge
omet
ries
—m
athe
mat
ical
ly ri
goro
us e
rror
est
. (FE
M)
�K
ey p
aral
lel A
MR
issu
es—
trad
eoff
betw
een
dyna
mic
mes
h/da
ta
mig
ratio
n &
load
bal
ance
–m
esh/
data
repa
rtiti
oned
ent
irely
or
part
ially
dur
ing
(de)
refin
emen
t–
com
mun
icat
ion
hard
to fa
ctor
into
lo
ad b
alan
cere
finem
ent
�A
MR
mes
h &
std
. uns
truct
ured
mes
h si
mila
r—
AM
R m
esh
typi
cally
a “
sing
le-le
vel”
mes
h—
ofte
n re
lies
on m
esh
“sm
ooth
ing”
to
ensu
re m
esh/
elem
ent q
ualit
y (e
ltan
gle
size
, elt
aspe
ct ra
tio, e
tc.)
—m
esh
part
ition
ing
algo
rithm
s (g
raph
-ba
sed
or g
eom
etry
-bas
ed) u
sed
to a
ssig
n m
esh
poin
ts to
pro
cess
ors
J. L
ou, C
. Nor
ton,
T. C
wik
(Pyr
amid
) NA
SA/J
PL/C
alte
ch
Cel
l-by-
cell
(str
uctu
red)
mes
h re
finem
ent
�U
sed
with
FE
, FV
, or F
D m
etho
ds—
stru
ctur
ed m
ulti-
leve
l sol
vers
—fa
cilit
ates
loca
l tim
este
ppin
g
…
�A
MR
mes
h or
gani
zed
as a
hie
rarc
hy—
hier
arch
y of
nes
ted
leve
ls—
typi
cally
use
s sm
all,
sam
e-si
zed
bloc
ks
—ty
pica
lly u
ses
octr
ee(q
uadt
ree
in 2
d) to
m
aint
ain
pare
nt-c
hild
cel
l rel
atio
nshi
ps
�K
ey p
aral
lel A
MR
issu
es—
dist
ribut
ion
of tr
ee s
truc
ture
� ���po
tent
ially
hi
gh s
tora
ge, b
ookk
eepi
ng o
verh
ead
—tr
adeo
ff be
twee
n lo
ad b
alan
ce &
in
terp
roce
ssor
data
com
mun
icat
ion � ���
split
pa
rent
s &
chi
ldre
n ac
ross
pro
cs?
—tr
adeo
ff be
twee
n bl
ock
size
and
co
mm
unic
atio
n co
mpl
exity
/cos
t
Spac
e fil
ling
curv
es a
re a
use
ful t
ool t
o fa
cilit
ate
data
loca
lity
& lo
ad b
alan
ce�
Spa
ce fi
lling
cur
ve is
a m
appi
ng:
�G
oal i
s to
opt
imiz
e lo
calit
y in
a g
loba
l sen
se—
num
eric
al a
lgor
ithm
loca
lity �
cell
inde
x lo
calit
y �
data
loca
lity
1N
Nd
→
Pean
o-H
ilber
t cur
ve o
rder
ing
�En
able
s dy
nam
ic lo
ad b
alan
ce &
redu
ces
inte
rpro
cess
or c
omm
in A
MR
(Bro
wne
&
Par
asha
r--D
AG
H/G
rAC
E)
—re
fined
regi
ons
“ins
erte
d” in
to c
urve
—
recu
rsiv
ely
fille
d w
/ cur
ve o
f sam
e st
ruct
ure
Patc
h-ba
sed
stru
ctur
ed m
esh
refin
emen
t�
Use
d w
ith F
D, F
V m
etho
ds—
stru
ctur
ed m
ulti-
leve
l sol
vers
—
faci
litat
es lo
cal t
imes
tepp
ing,
lega
cy
num
eric
al k
erne
ls
�A
MR
mes
h or
gani
zed
as a
hie
rarc
hy—
hier
arch
y of
nes
ted
leve
ls—
cells
clu
ster
ed in
to “
patc
hes”
of
vary
ing
size
� ���in
crea
sed
com
p/co
mm
volu
me
ratio
—si
mpl
e m
odel
of d
ata
loca
lity
(no
indi
rect
ion)
—lo
w o
verh
ead
mes
h de
scrip
tion
�K
ey p
aral
lel A
MR
issu
es—
cell
“clu
ster
ing”
sca
labi
lity
—tr
adeo
ff be
twee
n pa
tch
size
and
co
mm
unic
atio
n co
mpl
exity
/cos
t &
load
bal
ance
—
smal
l pat
ches
mor
e fle
xibl
e fo
r LB
, bu
t com
mun
icat
ion
over
head
gre
ater
SAM
RA
I pro
ject
(LLN
L)w
ww
.llnl
.gov
/CA
SC/S
AM
RA
I
AM
R h
elps
reso
lve
com
plex
geo
met
ry
CA
RT3
D, B
erge
r (N
YU),
Afto
smis
et. a
l (N
AS
A) g
ener
ates
AM
R m
eshe
s to
reso
lve
feat
ures
aro
und
embe
dded
bou
ndar
ies
http
://pe
ople
.nas
.nas
a.go
v/%
7Eaf
tosm
is/c
art3
d/
Grif
fith,
Pes
kin
(NYU
) dev
elop
ing
elec
trica
l-m
echa
nica
l hea
rt m
odel
com
bini
ng
imm
erse
d bo
unda
ries
and
AM
R (S
AM
RA
I)
ALE
-AM
R c
ombi
nes
ALE
inte
grat
ion
with
AM
R
Mov
ing-
defo
rmin
g A
MR
grid
3D IC
F hy
dro
calc
ulat
ion
smal
l-sca
le R
Min
stab
ility
And
erso
n, E
lliott,
Pem
ber,
“An
Arb
itrar
y La
gran
gian
-Eul
eria
nM
etho
ds w
ith A
dapt
ive
Mes
h R
efin
emen
t fo
r the
Sol
utio
n of
the
Eule
r Equ
atio
ns”,
sub.
to J
. Com
p. P
hys.
, UC
RL-
JC-1
5243
5. A
lso
see
ww
w.ll
nl.g
ov/C
AS
C/p
roje
cts.
shtm
l
�A
dvan
tage
s of
ALE
(mul
tiple
mat
eria
ls, m
ovin
g in
terfa
ces)
�
Adv
anta
ges
of A
MR
(dyn
amic
add
ition
& re
mov
al o
f mes
h po
ints
)
Con
tinuu
m
repr
esen
tatio
n (E
uler
, N
avie
r-St
okes
) aw
ay
from
inte
rfac
e
fluid
Aflu
id B
DSM
C re
pres
enta
tion
at in
terf
ace
Ada
ptiv
e M
esh
and
Alg
orith
m R
efin
emen
t (A
MA
R) r
efin
es m
esh
and
num
eric
al m
odel
�A
MR
is u
sed
to re
fine
cont
inuu
m
calc
ulat
ion
�A
lgor
ithm
sw
itche
s to
dis
cret
e at
omis
tic m
etho
d to
incl
ude
phys
ics
abse
nt in
con
tinuu
m m
odel
Wije
sing
he, H
ornu
ng, G
arci
a, H
adjic
onst
antin
ou, “
“Thr
ee-d
imen
sion
al H
ybrid
Con
tinuu
m-A
tom
istic
Si
mul
atio
ns fo
r Mul
tisca
leH
ydro
dyna
mic
s”, P
roc.
200
3 AS
ME
Int’l
Mec
h. E
ng. C
ong.
and
R&D
Exp
o ,
Was
hing
ton,
D.C
., N
ov. 1
5-21
, 200
3. U
CR
L-JC
-151
323.
Als
o se
e w
ww
.llnl
.gov
/CAS
C/S
AMR
AI.
Part
icle
s re
solv
e m
olec
ular
-sca
le
dyna
mic
s of
mix
ing
regi
on in
ada
ptiv
e sh
ear l
ayer
cal
cula
tion
Pres
enta
tion
Out
line
�O
verv
iew
of a
dapt
ive
mes
h re
finem
ent (
AM
R) a
ppro
ache
s—
Gen
eral
issu
es—
Uns
truc
ture
d A
MR
—C
ell-b
y-ce
ll st
ruct
ured
AM
R—
Patc
h-ba
sed
stru
ctur
ed A
MR
—H
ybrid
AM
R m
etho
ds�
SAM
RA
I alg
orith
m d
evel
opm
ent t
o im
prov
e pa
ralle
l sca
labi
lity
�C
oncl
udin
g re
mar
ks
Wis
sink
, A.,
D. H
ysom
, and
R. H
ornu
ng, "
Enh
anci
ng S
cala
bilit
y of
Par
alle
l Stru
ctur
ed A
MR
C
alcu
latio
ns,"
Proc
eedi
ngs
of th
e 17
th A
nnua
l Ass
ocia
tion
of C
ompu
ting
Mac
hine
ry In
tern
atio
nal
Con
fere
nce
on S
uper
com
putin
g,S
an F
ranc
isco
, Cal
iforn
ia, J
une
23-2
6, 2
003,
pp.
336
-347
Box
cal
culu
s is
fund
amen
tal t
o pa
tch-
base
d st
ruct
ured
AM
R
�S
eria
l num
eric
al ro
utin
es o
pera
te o
n pa
tche
s af
ter b
ound
ary
data
exch
ange
exch
ange
pat
ch b
ound
ary
data
fora
llpa
tche
s ii
n le
vel X
call
do_s
ome_
wor
k(X
(i))
end
fora
llse
rial c
ode
“par
alle
l” lo
op
para
llel d
ata
com
mun
icat
ion
1) B
ox re
gion
s ge
nera
ted
by
“clu
ster
ing”
tagg
ed c
ells
3) P
atch
es m
appe
d to
pro
cess
ors
Pro
cA
Pro
cB
Pro
cC
Pro
cD
Pro
cE
01 2 3
4
2) B
oxes
“ch
oppe
d” to
form
pat
ch re
gion
s0 1 32 4
�P
atch
es a
re d
istri
bute
d am
ong
proc
esso
rs o
ne le
vel a
t a ti
me
�C
ost o
f cre
atin
g se
nd/re
ceiv
e se
ts c
an b
e am
ortiz
ed o
ver m
ultip
le
com
mun
icat
ion
cycl
es
�D
ata
from
var
ious
sou
rces
pac
ked
into
sin
gle
mes
sage
str
eam
—su
ppor
ts a
rbitr
ary,
var
iabl
e-le
ngth
dat
a ty
pes
—tw
ose
nds
per p
roce
ssor
pai
r(lo
w la
tenc
y)
Com
mun
icat
ion
sche
dule
s cr
eate
dat
a de
pend
enci
es fr
om g
loba
l mes
h in
form
atio
n
Send
Set
Rec
eive
Set
mes
sage
buf
fer
MPI
sen
dC
ell D
ata
(dou
ble)
Part
icle
s
packStream(...);
Our
exp
erie
nce:
dat
a co
mm
unic
atio
n ef
ficie
nt;
adap
tive
mes
hing
ope
ratio
ns m
ay s
cale
poo
rly
Non
-sca
led
4 Le
vel S
edov
Pro
blem
100
1000
1000
0
1000
00
3264
128
256
512
Proc
esso
rs
Wallclock Time
Scal
ed3
Leve
l Adv
ectin
g Fr
ont P
robl
em
0
500
1000
1500
2000
2500
3000
3264
128
256
512
1024
Proc
esso
rs
Wallclock Time
Ada
ptiv
e m
eshi
ng (t
ag, c
lust
er, l
oad
bal,
crea
te s
ched
, mov
e da
ta)
Num
eric
al o
pera
tions
and
dat
a co
mm
unic
atio
nTo
tal
4 Le
vel S
edov
Prob
lem
--N
on-s
cale
d � ���
sam
e-si
ze p
robl
em o
n al
l pro
c co
unts
3 Le
vel A
dvec
tion
Prob
lem
--Sc
aled
� ���sa
me
wor
kloa
d on
eac
h pr
oc
Com
mun
icat
ion
sche
dule
con
stru
ctio
n
�D
ata
depe
nden
cies
bet
wee
n pa
tche
s de
term
ined
by
iden
tifyi
ng b
ox in
ters
ectio
ns
�O
rigin
al im
plem
enta
tion
com
pare
d ea
ch
dest
inat
ion
patc
h w
ith a
ll po
tent
ial s
ourc
e pa
tche
s (n
ot ju
st n
eigh
bors
)
�C
ompl
exity
O(N
2 ),
N =
num
ber o
f pat
ches
�N
gro
ws
prop
ortio
nally
with
pro
blem
siz
e
Com
mun
icat
ion
sche
dule
co
nstr
uctio
n co
sts
grow
dr
amat
ical
ly w
ith p
robl
em s
ize
Rec
ursi
ve B
inar
y B
ox T
ree
(RB
BT)
ef
ficie
ntly
des
crib
es s
patia
l rel
atio
nshi
ps
Con
stru
ctR
BB
TNod
e(bo
xlis
tB,d
irect
ion
d)
1.C
ompu
te a
bou
ndin
g bo
x ar
ound
B
2.Bi
sect
bou
ndin
g bo
x al
ong
dax
is -
form
L,R
par
ts
3.Fo
rm 3
new
box
lists
:�
BL
= bo
xes
entir
ely
cont
aine
d in
L p
art
�B
R=
boxe
s en
tirel
y co
ntai
ned
in R
par
t�
Bno
de=
boxe
s ly
ing
on in
terfa
ce
4.Fo
rm c
hild
nod
es b
y re
curs
ing
BL,B
Rw
ith s
ame
d. F
orm
“s
econ
dary
” tre
e by
recu
rsin
gB
node
with
new
dire
ctio
n d.
�Bu
ild a
tree
con
tain
ing
spat
ial r
elat
ions
hips
bet
wee
n bo
xes,
bas
ed o
n th
eir c
orne
r ind
ices
�bo
xlis
tis
a lis
t of b
oxes
; dis
a d
irect
ion
x, y
, or z
.
Use
RB
BT
in c
omm
unic
atio
n sc
hedu
le
cons
truc
tion
�Fa
st d
eter
min
atio
n of
whi
ch b
oxes
inte
rsec
t (lim
its b
ox
com
paris
ons
to n
eigh
bors
)
�Ex
ampl
e:Is
box
B in
the
neig
borh
ood
of b
ox A
?—
“Wal
k th
e tre
e” –
is B
insi
de A
’s b
ound
ing
box?
If s
o,
recu
rse
to c
hild
nod
e.
—La
st n
ode
in th
e re
curs
ion
prov
ides
a s
mal
l sub
set o
f bo
xes
that
may
inte
rsec
t A
—Th
en w
e ap
ply
naïv
e O
(M2 )
com
paris
on a
lgor
ithm
, M
is n
umbe
r of n
eigh
bors
of A
�C
ompl
exity
ana
lysi
s:
—S
etup
: O(N
log(
N)),
don
e on
ce fo
r eac
h le
vel &
reus
ed—
Que
ry (f
or e
ach
box)
: O(lo
g(N
))—
Run
time
com
plex
ity: O
(N lo
g(N
)) –
varie
s w
ith b
ox
layo
ut, b
ut v
alid
for n
early
all
case
s en
coun
tere
d
A
B
Non
-sca
led
Sedo
vpr
oble
m –
sam
e pr
oble
m s
ize
on a
ll pr
oces
sor c
ount
s
3D S
edov
shoc
k -E
uler
hyd
rody
nam
ics �
Tota
l wor
k in
crea
ses
durin
g si
mul
atio
n
�P
er-p
roce
ssor
wor
kloa
d de
crea
ses
as
num
ber o
f pro
cess
ors
incr
ease
d
Prob
lem
Siz
e on
Eac
h Le
vel
012345678910 0.E+
001.
E-03
2.E-
033.
E-03
4.E-
03Si
mul
atio
n Ti
me
Number of Gridcells (in Millions)
Leve
l 3Le
vel 2
Leve
l 1Le
vel 0
�4
leve
ls
�R
efin
e ra
tio =
4 b
etw
een
leve
ls
Scal
ing
of n
on-s
cale
dSe
dov
prob
lem
Non
-sca
led
4 le
vel S
edov
Prob
lem
ASC
I IB
M B
lue
Paci
fic
100
1000
1000
0
1000
00
3264
128
256
512
Proc
esso
rs
Wallclock Time
Idea
lTo
tal
Tim
e Ad
vanc
eAd
aptiv
e G
riddi
ngO
ther
100
1000
1000
0
1000
00
3264
128
256
512
Proc
esso
rs
Wallclock Time
Orig
inal
With
new
alg
orith
ms
Scal
edlin
ear a
dvec
tion
prob
lem
–pr
oble
m s
ize
incr
ease
d w
ith p
roc
coun
t
�P
er-p
roce
ssor
wor
kloa
d re
mai
ns
roug
hly
cons
tant
as
num
ber o
f pr
oces
sors
is in
crea
sed
�C
ompu
tatio
nally
sim
ple,
che
ap
num
eric
al k
erne
ls �
high
rela
tive
cost
of a
dapt
ivity
, com
mun
icat
ion
Num
ber o
f Pat
ches
on
Fine
st L
evel
0510152025
00.
10.
20.
30.
40.
50.
6
Sim
ulat
ion
Tim
e
Number of Patches(in Thousands)
1024
pro
c
512
proc
256
proc
128
proc
64 p
roc
32 p
roc
3D s
inus
oida
l fro
nt –
linea
r adv
ectio
n �
3 le
vels
�
Ref
ine
ratio
= 4
bet
wee
n le
vels
Scal
ing
of s
cale
d lin
ear a
dvec
tion
prob
lem
Scal
ed
3 le
vel l
inea
r adv
ectio
n pr
oble
mLL
NL
Linu
x M
CR
Clu
ster
0
500
1000
1500
2000
2500
3000
3264
128
256
512
1024
Proc
esso
rs
Wallclock Time
0
500
1000
1500
2000
2500
3000
3264
128
256
512
1024
Proc
esso
rs
Wallclock Time
Idea
lTo
tal
Tim
e A
dvan
ceA
dapt
ive
Gri
ddin
gO
ther
Orig
inal
With
new
alg
orith
ms
We
are
deve
lopi
ng n
ew a
lgor
ithm
s to
fu
rthe
r im
prov
e sc
alin
g
�M
ost “
AM
R o
verh
ead”
cur
rent
ly s
cale
s as
O(N
) for
sto
rage
and
O
(Nlo
gN) f
or o
pera
tion
com
plex
ity, N
= n
umbe
r of p
atch
es�
New
app
roac
hes
nece
ssar
y fo
r O(1
0K-1
00K
) pro
cess
ors
(Blu
e G
ene/
L)—
Asy
nchr
onou
s po
int c
lust
erin
g al
gorit
hm p
aral
leliz
es s
tand
ard
“ser
ial”
algo
rithm
(Ber
ger-R
igou
tsos
)–
Cur
rent
impl
emen
tatio
n re
duce
s co
llect
ive
com
mun
icat
ion,
but
stil
l se
rializ
es o
pera
tions
on
inde
pend
ent s
ubdo
mai
ns–
New
impl
emen
tatio
n ov
erla
ps c
omm
unic
atio
n w
ith c
ompu
tatio
n to
re
duce
pro
cess
or w
ait t
ime
—“D
istri
bute
dbo
x gr
aph”
app
roac
h re
duce
sAM
R m
esh
desc
riptio
nto
incl
ude
only
loca
l pat
ch a
nd n
eigh
borp
atch
info
rmat
ion
–R
ecas
t all
glob
al d
omai
n op
erat
ions
in te
rms
of lo
cal a
nd n
eare
st
neig
hbor
oper
atio
ns–
Use
pre
viou
sly
disc
arde
d in
cide
ntal
dat
a to
repl
ace
glob
ally
com
pute
d da
ta (i
.e.,
recy
cle
grap
h in
form
atio
n at
regr
idst
eps)
Pres
enta
tion
Out
line
�O
verv
iew
of a
dapt
ive
mes
h re
finem
ent (
AM
R) a
ppro
ache
s—
Gen
eral
issu
es—
Uns
truc
ture
d A
MR
—C
ell-b
y-ce
ll st
ruct
ured
AM
R—
Patc
h-ba
sed
stru
ctur
ed A
MR
—H
ybrid
AM
R m
etho
ds�
SAM
RA
I alg
orith
m d
evel
opm
ent t
o im
prov
e pa
ralle
l sca
labi
lity
�C
oncl
udin
g re
mar
ks
Con
clud
ing
rem
arks
�A
MR
is a
n im
porta
nt te
chno
logy
for l
arge
-sca
le s
cien
ce &
eng
inee
ring
prob
lem
s th
at e
xhib
it be
havi
or re
quiri
ng fi
ne lo
cal g
rids
to re
solv
e�
AM
R d
ynam
ical
ly fo
cuse
s co
mpu
tatio
nal e
ffort
to a
void
cos
t of g
loba
lly fi
ne g
rid�
Ther
e ar
e a
varie
ty o
f AM
R a
ppro
ache
s ai
med
at d
iffer
ent p
robl
ems,
sol
n. a
lgs
�B
alan
ce o
f wor
k as
sign
ed to
pro
cess
ors
and
cost
of c
ompl
ex in
terp
roce
ssor
com
mun
icat
ion
are
key
scal
abilit
y is
sues
—lo
ad b
alan
ce, d
ata
(re)d
istri
butio
n, c
ompl
ex d
ata
com
mun
icat
ion
patte
rns
–tra
deof
fs a
mon
g th
ese
are
dyna
mic
–ov
erhe
ads
that
can
not b
e am
ortiz
ed—
AM
R s
how
s go
odsc
alin
g to
O(1
K) p
roce
ssor
s—
New
dev
elop
men
ts s
how
pro
mis
e to
sca
le to
O(1
0K+)
pro
cess
ors �
redu
ce s
tora
ge a
nd o
pera
tion
over
head
s—
Com
bini
ng te
chni
ques
from
diff
eren
t AM
R a
ppro
ache
s is
ben
efic
ial
�M
ultip
hysi
cspr
oble
ms
that
com
bine
alg
orith
ms
with
diff
eren
t per
form
ance
ch
arac
teris
tics
is a
maj
or c
halle
nge
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