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Parallel Radiation Transport Algorithms
and
Associated Architectural Requirements
Jim Morel, Randal Baker, and James Warsa
Computer and Computational Sciences Division
Los Alamos National Laboratory
[email protected], [email protected], [email protected]
Presentation at the Conference on High Speed Computing, Gleneden Beach, OR, April 19-22, 2004 Slide 1/27
Ove
rvie
w
Intr
od
uct
ion
Th
eTr
ansp
ort
Eq
uat
ion
So
urc
eIt
erat
ion
Co
nver
gen
ceo
fS
ou
rce
Iter
atio
n
Co
nver
gen
ceA
ccel
erat
ion
Kry
lov
Met
ho
ds
and
Pre
con
dit
ion
ers
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e2/
27
Ove
rvie
w
Rec
tan
gu
lar-
Mes
hP
aral
lelS
wee
pA
lgo
rith
ms
Rec
tan
gu
lar-
Mes
hS
wee
pL
ayo
uts
Par
alle
lSw
eep
so
nU
nst
ruct
ure
dM
esh
es
Par
alle
lSw
eep
so
nS
tru
ctu
red
AM
RM
esh
es
Do
mai
nD
eco
mp
osi
tio
nS
olu
tio
nA
lgo
rith
ms
Co
mp
ute
rA
rch
itec
ture
Req
uir
emen
ts
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e3/
27
Intr
od
uct
ion
•Tr
ansp
ortc
alcu
latio
nsar
eof
fund
amen
tali
mpo
rtan
ceto
AS
CI.
•Tr
ansp
ortc
alcu
latio
nsre
quire
far
mor
em
emor
yan
dC
PU
-tim
eth
anan
y
othe
rty
peof
phys
ics
calc
ulat
ion
inan
AS
CIc
ode.
•T
here
are
thre
ere
leva
ntty
pes
oftr
ansp
ortc
alcu
latio
ns:
neut
ron,
ther
mal
radi
atio
n,an
dch
arge
d-pa
rtic
le.
•T
henu
mer
ical
met
hods
for
neut
ron
tran
spor
tcal
cula
tions
are
very
mat
ure
and
gene
rally
adeq
uate
.
•T
henu
mer
ical
met
hods
for
char
ged-
part
icle
tran
spor
tand
ther
mal
radi
atio
ntr
ansp
orta
refa
rle
ssm
atur
ean
dth
eir
adeq
uacy
for
mul
tidim
ensi
onal
calc
ulat
ions
isno
twel
lest
ablis
hed.
•Fr
oma
num
eric
alm
etho
dsst
andp
oint
,the
rmal
radi
atio
ntr
ansp
orti
s
sign
ifica
ntly
mor
ede
man
ding
than
char
ged-
part
icle
tran
spor
t,an
d
char
ged-
part
icle
tran
spor
tis
sign
ifica
ntly
mor
ede
man
ding
than
neut
ron
tran
spor
t.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e4/
27
Intr
od
uct
ion
•F
orth
em
ostp
art,
the
sam
eba
sic
para
llels
olut
ion
tech
niqu
esca
nbe
used
for
allt
hree
type
sof
calc
ulat
ions
.
•P
aral
lels
olut
ion
tech
niqu
esar
equ
item
atur
efo
rre
ctan
gula
rsp
atia
l
mes
hes,
butt
hey
rem
ain
are
sear
chto
pic
for
stru
ctur
edA
MR
mes
hes
and
fully
unst
ruct
ured
mes
hes.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e5/
27
Th
eTr
ansp
ort
Eq
uat
ion
•T
hefo
llow
ing
neut
ron
tran
spor
tequ
atio
nis
repr
esen
tativ
eof
allt
hree
type
sof
tran
spor
tequ
atio
ns:
1 v
∂ψ ∂t
+−→ Ω
·−→ ∇ψ
+σ
tψ=
∫ ∞ 0
∫ 4π
σs(E
′ →E
,−→ Ω′ ·−→ Ω
)ψ(−→ Ω
′ ,E′ )
dΩ′ d
E′ +
Q.
•W
eus
eop
erat
orno
tatio
nto
expr
ess
this
equa
tion
asfo
llow
s:
Lψ
=S
ψ+
Q.
•T
hepr
imar
yun
know
nis
the
angu
lar
flux,
ψ,w
hich
isse
ven-
dim
ensi
onal
:
ψ=
ψ(t
,−→ r,−→ Ω
,E) .
•t
istim
e,−→ r
isth
epa
rtic
lepo
sitio
n,−→ Ω
isth
epa
rtic
ledi
rect
ion,
and
Eis
the
part
icle
ener
gy.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e6/
27
Th
eTr
ansp
ort
Eq
uat
ion
•T
his
equa
tion
isju
sta
stat
emen
tofp
artic
leco
nser
vatio
nin
adi
ffere
ntia
l
phas
e-sp
ace
volu
me,
dP
=dV
dΩ
dE
.
•T
hedi
ffere
ntia
lpha
sesp
ace
volu
me
cons
ists
ofth
edi
ffere
ntia
lspa
tial
volu
me
abou
t−→ r
,the
diffe
rent
ials
olid
angl
eab
out−→ Ω
,and
the
diffe
rent
iale
nerg
yin
terv
alab
outE
.
•T
heeq
uatio
nst
ates
that
the
rate
ofch
ange
ofth
enu
mbe
rof
part
icle
sin
the
diffe
rent
ialv
olum
edP
iseq
ualt
oth
epa
rtic
leso
urce
rate
sm
inus
the
part
icle
sink
rate
s.
•S
inks
fordP
incl
ude
•st
ream
ing
outo
fthe
diffe
rent
ials
patia
lvol
ume
abou
t−→ r
.
•be
ing
eith
erab
sorb
edor
scat
tere
dou
toft
hedi
ffere
ntia
lsol
idan
gle
abou
t−→ Ωan
dth
edi
ffere
ntia
lene
rgy
inte
rval
abou
tthe
ener
gyE
.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e7/
27
Th
eTr
ansp
ort
Eq
uat
ion
•S
ourc
esin
clud
e
•st
ream
ing
into
the
diffe
rent
ials
patia
lvol
ume
abou
t−→ r
.
•be
ing
scat
tere
din
toth
edi
ffere
ntia
lsol
idan
gle
abou
t−→ Ω
and
the
diffe
rent
iale
nerg
yin
terv
alab
outt
heen
ergy
E.
•B
eing
expl
icitl
ycr
eate
dw
ithin
dP
.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e8/
27
So
urc
eIt
erat
ion
•T
hest
anda
rdte
chni
que
for
solv
ing
the
tran
spor
tequ
atio
nis
the
sour
ce
itera
tion
tech
niqu
e.
•S
ince
the
sour
cete
rmco
ntai
nsal
lthe
angl
e-en
ergy
coup
ling,
itis
itera
tivel
yla
gged
:
ψ�+
1=
L−1
Sψ
�+
L−1
Q.
•O
nre
ctan
gula
rm
eshe
s,th
eop
erat
orL
repr
esen
tsa
bloc
k
low
er-t
riang
ular
mat
rixw
ithea
chbl
ock
corr
espo
ndin
gto
the
unkn
owns
in
asi
ngle
cell.
•S
uch
mat
rices
are
very
easy
toin
vert
ona
scal
arm
achi
ne.
•In
part
icul
ar,o
nese
quen
tially
solv
esfo
rth
eun
know
nsin
each
mes
hce
ll,
mov
ing
acro
ssth
em
esh
inth
edi
rect
ion
ofpa
rtic
leflo
w.
•B
ecau
seof
this
mov
emen
t,th
ein
vers
ion
proc
ess
isca
lled
asw
eep.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e9/
27
So
urc
eIt
erat
ion
•T
hest
anda
rdte
chni
que
for
solv
ing
the
tran
spor
tequ
atio
nis
the
sour
ce
itera
tion
tech
niqu
e.
•S
ince
the
sour
cete
rmco
ntai
nsal
lthe
angl
e-en
ergy
coup
ling,
itis
itera
tivel
yla
gged
:
ψ�+
1=
L−1
Sψ
�+
L−1
Q.
•O
nre
ctan
gula
rm
eshe
s,th
eop
erat
orL
repr
esen
tsa
bloc
k
low
er-t
riang
ular
mat
rixw
ithea
chbl
ock
corr
espo
ndin
gto
the
unkn
owns
in
asi
ngle
cell.
•S
uch
mat
rices
are
very
easy
toin
vert
ona
scal
arm
achi
ne.
•In
part
icul
ar,o
nese
quen
tially
solv
esfo
rth
eun
know
nsin
each
mes
hce
ll,
mov
ing
acro
ssth
em
esh
inth
edi
rect
ion
ofpa
rtic
leflo
w.
•B
ecau
seof
this
mov
emen
t,th
ein
vers
ion
proc
ess
isca
lled
asw
eep.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e10
/27
Co
nver
gen
ceo
fS
ou
rce
Iter
atio
n
•A
ssum
ing
aze
roin
itial
gues
sfo
rth
esc
atte
ring
sour
ce,e
ach
sour
ce
itera
tion
adds
the
cont
ribut
ion
toth
eso
lutio
nfr
oman
othe
rge
nera
tion
of
scat
terin
g.
•In
othe
rw
ords
,the
first
sour
ceite
ratio
nyi
elds
the
solu
tion
due
topa
rtic
les
that
have
noty
etsc
atte
red.
The
seco
ndso
urce
itera
tion
adds
the
cont
ribut
ion
toth
eso
lutio
nfr
ompa
rtic
les
that
have
scat
tere
don
ce.
The
third
sour
ceite
ratio
nad
dsth
eco
ntrib
utio
nto
the
solu
tion
from
part
icle
s
that
have
scat
tere
dtw
ice,
and
soon
.
•T
hus
the
sour
ceite
ratio
npr
oces
sco
nver
ges
rapi
dly
inpr
oble
ms
inw
hich
part
icle
son
lysc
atte
ra
few
times
onav
erag
ebe
fore
eith
erbe
ing
abso
rbed
orst
ream
ing
outo
fthe
syst
em.
•A
rbitr
arily
slow
conv
erge
nce
rate
sca
nbe
obta
ined
indi
ffusi
vepr
oble
ms,
i.e.,
prob
lem
sw
ithlit
tleab
sorp
tion
that
appe
arve
ryth
ick
toth
epa
rtic
les.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e11
/27
Co
nver
gen
ceA
ccel
erat
ion
•In
diffu
sive
prob
lem
s,th
eso
urce
itera
tion
proc
ess
mus
tbe
acce
lera
ted.
Thi
sis
abso
lute
lycr
itica
lin
ther
mal
radi
atio
ntr
ansp
ortc
alcu
latio
ns.
•In
gene
ral,
the
resu
lting
solu
tion
tech
niqe
sca
nbe
thou
ghto
fas
atw
o-gr
id
met
hod,
with
adi
ffusi
oneq
uatio
npl
ayin
gth
ero
leof
the
“coa
rse-
grid
”
tran
spor
tope
rato
r.
•T
hedi
ffusi
oneq
uatio
nm
ustb
edi
scre
tized
inac
cord
ance
with
the
disc
retiz
atio
nof
the
tran
spor
tequ
atio
nto
avoi
din
stab
ilitie
s.
•B
ecau
seth
etr
ansp
orte
quat
ion
isge
nera
llydi
scre
tized
usin
g
disc
ontin
uous
finite
-ele
men
tmet
hods
,thi
sle
ads
tono
n-st
anda
rddi
ffusi
on
disc
retiz
atio
nsth
atar
eve
rydi
fficu
ltto
solv
e.
•F
urth
erm
ore,
the
acce
lera
tion
tech
niqu
eca
nbe
com
ein
crea
sing
ly
inef
fect
ive
with
seve
redi
scon
tinui
ties
inm
ater
ialp
rope
rtie
s.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e12
/27
Kry
lov
Met
ho
ds
and
Pre
con
dit
ion
ers
•A
nal
tern
ativ
eto
the
two-
grid
appr
oach
isto
use
aK
rylo
vm
etho
dto
solv
e
the
tran
spor
tequ
atio
n,an
dto
reca
stth
edi
ffusi
on-b
ased
acce
lera
tion
tech
niqu
eas
adi
ffusi
on-b
ased
prec
ondi
tione
r.
•T
his
appr
oach
rela
xes
the
cons
iste
ncy
requ
irem
ents
onth
edi
ffusi
on
disc
retiz
atio
n,an
dis
easi
lyim
plem
ente
din
exis
ting
code
s.
•H
ighl
yef
fect
ive
prec
ondi
tioni
ngis
achi
eved
via
ast
anda
rddi
ffusi
on
disc
retiz
atio
nra
ther
than
adi
scon
tinuo
usdi
ffusi
ondi
scre
tizat
ion,
and
the
mat
eria
ldis
cont
inut
itypr
oble
mis
elim
inat
ed.
•T
hepr
econ
ditio
ned
Kry
lov
appr
oach
isha
ving
are
volu
tiona
ryim
pact
upon
solu
tion
tech
niqu
esfo
rth
etr
ansp
orte
quat
ion.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e13
/27
Kry
lov
Met
ho
ds
and
Pre
con
dit
ion
ers
•T
heeq
uatio
nw
hich
the
Kry
lov
met
hod
solv
esis
not
Lψ
=S
ψ+
Q,
butr
athe
r
(I−
L−1
S)ψ
=L−1
Q.
•T
hela
tter
equa
tion
has
far
bette
rsp
ectr
alpr
oper
ties,
butr
equi
res
asw
eep
toev
alua
teits
actio
n.
•T
hus
swee
psan
ddi
ffusi
onso
lves
are
requ
ired
with
both
trad
ition
alan
d
adva
nced
tran
spor
tsol
utio
nte
chni
ques
.
•O
urpa
ralle
lmet
hods
for
solv
ing
diffu
sion
equa
tions
are
take
nfr
omth
e
appl
ied
mat
hem
atic
sco
mm
unity
.
•T
hus
we
dono
tdis
cuss
them
furt
her.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e14
/27
Rec
tan
gu
lar-
Mes
hP
aral
lelS
wee
pA
lgo
rith
ms
•O
na
N-c
ell1
-Dm
esh,
the
swee
pst
arts
atth
ebo
unda
ryce
llfo
rw
hich
the
part
icle
dire
ctio
nis
inci
dent
.
•S
olvi
ngfo
rth
eun
know
nsin
the
first
cell
prov
ides
the
inci
dent
solu
tion
valu
esre
quire
dto
solv
efo
rth
eun
know
nsin
the
seco
ndce
ll,an
dso
on,
asill
ustr
ated
inF
ig.1
.
Fig
ure
1:1-
DS
wee
p
12
34
•T
heso
lutio
npr
oces
sis
inhe
rent
lyse
quen
tiala
ndpa
ralle
lism
isno
t
poss
ible
.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e15
/27
Rec
tan
gu
lar-
Mes
hP
aral
lelS
wee
pA
lgo
rith
ms
•O
na
N×
N2-
Dm
esh,
the
swee
pst
arts
atth
eco
rner
mes
hce
llfo
r
whi
chth
epa
rtic
ledi
rect
ion
isin
cide
nton
two
cell
face
s.
•S
olvi
ngfo
rth
eun
know
nsin
the
corn
erm
esh
cell
prov
ides
the
inci
dent
solu
tion
valu
esre
quire
dto
solv
efo
rth
eun
know
nsin
the
next
two
cells
,
and
soon
,as
illus
trat
edin
Fig
.2.
Fig
ure
2:2-
DS
wee
p
12
2
3
3
3
4
4
4
4
5
5
56
67
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e16
/27
Rec
tan
gu
lar-
Mes
hP
aral
lelS
wee
pA
lgo
rith
ms
•2N
−1
sequ
entia
lsol
utio
nst
eps
are
requ
ired
with
N2
cells
,so
one
can
sim
ulta
neou
sly
solv
efo
rO
(N/2)
unkn
owns
per
step
onth
eav
erag
e.
•T
hus
para
lleliz
atio
nis
poss
ible
.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e17
/27
Rec
tan
gu
lar-
Mes
hP
aral
lelS
wee
pA
lgo
rith
ms
•O
na
N×
N×
N3-
Dm
esh,
the
swee
pst
arts
atth
eco
rner
mes
hce
llfo
r
whi
chth
epa
rtic
ledi
rect
ion
isin
cide
nton
thre
ece
llfa
ces.
•S
olvi
ngfo
rth
eun
know
nsin
the
corn
erm
esh
cell
prov
ides
the
inci
dent
solu
tion
valu
esre
quire
dto
solv
efo
rth
eun
know
nsin
the
next
thre
ece
lls,
and
soon
,as
illus
trat
edin
Fig
.3.
Fig
ure
3:3-
DS
wee
p
21
22
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e18
/27
Rec
tan
gu
lar-
Mes
hP
aral
lelS
wee
pA
lgo
rith
ms
•3N
−2
sequ
entia
lsol
utio
nst
eps
are
requ
ired
with
N3
cells
,so
one
can
sim
ulta
neou
sly
solv
efo
rO
(N2/3
)un
know
nspe
rst
epon
the
aver
age.
•T
hus
para
lleliz
atio
nis
poss
ible
.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e19
/27
Rec
tan
gu
lar-
Mes
hS
wee
pL
ayo
uts
•T
heda
tala
yout
for
a2-
Dtr
ansp
ortc
alcu
latio
nis
1-D
and
the
data
layo
ut
for
a3-
Dca
lcul
atio
nis
2-D
.
•F
orin
stan
ce,c
onsi
der
a4×
4m
esh.
•T
hece
llda
tafo
ral
ldire
ctio
nsan
den
ergi
esw
ould
bem
appe
dto
four
proc
esso
rsas
follo
ws: Tabl
e1:
Layo
utfo
r4×
4m
esh.
P1
P2
P3
P4
45
67
34
56
23
45
12
34
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e20
/27
Rec
tan
gu
lar-
Mes
hS
wee
pL
ayo
uts
•T
his
isa
non-
stan
dard
layo
ut.
•T
here
isa
load
bala
ncin
gpr
oble
mth
atis
addr
esse
dby
star
ting
the
solu
tion
proc
ess
for
addi
tiona
ldire
ctio
nsbe
fore
any
one
dire
ctio
nfin
ishe
s.
•O
nre
ctan
gula
r3-
Dm
eshe
s,pa
ralle
leffi
cien
cies
onth
eor
der
of50
-80
perc
ent(
with
wea
ksc
alin
g)ha
vebe
enac
hiev
edw
ithth
ousa
nds
of
proc
esso
rs.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e21
/27
Par
alle
lSw
eep
so
nU
nst
ruct
ure
dM
esh
es
•M
atte
rsbe
com
em
uch
mor
eco
mpl
icat
edon
unst
ruct
ured
mes
hes.
•A
nor
derin
gof
the
unkn
owns
that
lead
sto
abl
ock
low
er-t
riang
ular
stru
ctur
efo
rth
esw
eep
equa
tions
alw
ays
exis
tson
“rea
sona
ble”
2-D
unst
ruct
ured
mes
hes,
butt
his
isno
tnec
essa
rily
the
case
for
3-D
mes
hes.
•A
bloc
klo
wer
-tria
ngul
aror
derin
ges
sent
ially
alw
ays
exis
tsfo
r“r
easo
nabl
e”
tetr
ahed
ralm
eshe
s,bu
tita
lmos
tnev
erex
ists
for
dist
orte
dhe
xahe
dral
mes
hes.
•E
ven
whe
na
bloc
klo
wer
-tria
ngul
aror
derin
gex
ists
,the
num
ber
ofst
eps
asso
ciat
edw
itha
mes
hof
Nce
llsva
ries
alon
gw
ithth
enu
mbe
rof
cells
in
each
step
.
•O
ptim
alpa
rtiti
onin
gis
anN
P-c
ompl
ete
prob
lem
.
•P
aral
lelu
nstr
uctu
red-
mes
hsw
eep
algo
rithm
sar
est
illa
rese
arch
topi
c.
•N
onet
hele
ss,v
ery
good
para
llele
ffici
enci
esha
vebe
enob
tain
edw
ith
thou
sand
sof
proc
esso
rson
tetr
ahed
ral-m
eshe
s.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e22
/27
Par
alle
lSw
eep
so
nS
tru
ctu
red
AM
RM
esh
es
•T
hedi
fficu
ltyof
defin
ing
anef
ficie
ntsw
eep
algo
rithm
for
ast
ruct
ured
AM
R
mes
hca
nbe
sim
ilar
toth
atfo
rei
ther
unst
ruct
ured
orre
ctan
gula
rm
eshe
s.
•T
hedi
fficu
ltyof
defin
ing
anef
ficie
ntsw
eep
algo
rithm
for
ast
ruct
ured
AM
R
mes
hus
ually
incr
ease
sw
ithth
efle
xibi
lity
ofth
em
esh
gene
ratio
npr
oces
s.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e23
/27
Do
mai
nD
eco
mp
osi
tio
nS
olu
tio
nA
lgo
rith
ms
•P
aral
lelt
rans
port
solu
tion
algo
rithm
sba
sed
upon
stan
dard
spat
iald
omai
n
deco
mpo
sitio
nar
ebe
ing
inve
stig
ated
.
•In
effe
ct,s
ourc
eite
ratio
nsar
epe
rfor
med
with
inea
chsu
b-do
mai
nan
dth
e
solu
tion
valu
eson
the
sub-
dom
ain
boun
darie
sar
eite
rate
dup
on.
•T
his
appr
oach
appe
ars
toha
veco
nsid
erab
lepr
omis
eas
long
asea
ch
spat
ials
ub-d
omai
nap
pear
sth
ick
toth
epa
rtic
les.
•W
hen
the
sub-
dom
ains
beco
me
thin
toth
epa
rtic
les,
the
algo
rithm
s
beco
me
inef
ficie
ntdu
eto
slow
conv
erge
nce
ofth
esu
b-do
mai
nbo
unda
ry
unkn
owns
.
•T
hese
appr
oach
esca
nbe
com
bine
dw
ithK
rylo
vm
etho
ds.
•W
hile
Kry
lov
met
hods
can
impr
ove
effic
ienc
yin
the
thic
kca
se,t
hey
are
inef
fect
ive
for
the
thin
case
.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e24
/27
Co
mp
ute
rA
rch
itec
ture
Req
uir
emen
ts
•A
light
wei
ghtO
Sth
atm
inim
izes
the
num
ber
ofin
terr
upts
gene
rate
d.
•T
hesw
eep
algo
rithm
istig
htly
coup
led
inth
ese
nse
that
the
equa
tions
for
the
dow
nwin
dce
llsca
nnot
beso
lved
until
thos
efo
rth
eup
win
dce
lls
have
been
solv
ed.
•If
ado
wnw
ind
proc
esso
rha
sno
tfini
shed
it’s
com
puta
tion-
com
mun
icat
ion
cycl
edu
eto
anO
Sin
terr
upto
fsom
eso
rt,
then
ever
ythi
ngha
ltsun
tilit
isfr
eed
up.
•A
high
perf
orm
ance
mem
ory
sub-
syst
em.
•V
ery
few
float
ing
poin
tope
ratio
nsar
epe
rfor
med
per
mem
ory
load
,so
data
quic
kly
mov
esth
roug
hth
eca
che.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e25
/27
Co
mp
ute
rA
rch
itec
ture
Req
uir
emen
ts
•A
low
late
ncy,
high
band
wid
thin
terc
onne
ctto
min
imiz
eco
mm
unic
atio
n
cost
s.N
ote
that
this
also
requ
ires
ahi
ghpe
rfor
man
ceM
PIl
ibra
rysi
nce
on
mod
ern
hard
war
em
osto
fthe
late
ncy
isdu
eto
softw
are,
noth
ardw
are.
•D
ata
isco
mm
unic
ated
betw
een
proc
esso
rsaf
ter
each
sequ
entia
lste
p
inth
esw
eep.
•T
here
lativ
eam
ount
ofda
taco
mm
unic
ated
isla
rge
enou
ghto
requ
ire
sign
ifica
ntba
ndw
idth
,but
smal
leno
ugh
tobe
sens
itive
tola
tenc
y.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e26
/27
Co
mp
ute
rA
rch
itec
ture
Req
uir
emen
ts
•T
hese
requ
irem
ents
are
best
met
bya
syst
emco
mpo
sed
ofa
few
thou
sand
high
perf
orm
ance
proc
esso
rsw
itha
larg
eam
ount
ofm
emor
y
per
proc
esso
r.
•M
inim
izin
gth
enu
mbe
rof
proc
esso
rsw
hile
mee
ting
the
requ
irem
ents
for
spee
dan
dm
emor
ym
akes
the
desi
gnan
dim
plem
enta
tion
ofa
high
perf
orm
ance
inte
rcon
nect
easi
er.
•A
lthou
ghm
oder
nco
des
have
been
exte
nsiv
ely
para
lleliz
ed,A
mda
hl’s
Law
still
com
esin
toef
fect
asth
enu
mbe
rof
proc
esso
rsin
crea
ses.
•P
roce
ssor
relia
bilit
ybe
com
espr
oble
mat
icw
ithm
ore
than
afe
wth
ousa
nd
proc
esso
rs.
•E
xist
ing
AS
CIs
oftw
are
isde
sign
edfo
rR
ISC
-bas
edC
PU
’s.
Tran
sitio
ning
tove
ctor
-typ
eC
PU
’sco
uld
resu
ltin
sign
ifica
ntre
desi
gnco
sts.
Pre
sent
atio
nat
the
Con
fere
nce
onH
igh
Spe
edC
ompu
ting,
Gle
nede
nB
each
,OR
,Apr
il19
-22,
2004
Slid
e27
/27
