High
-Per
form
ance
Sim
ulat
ions
of
Cohe
rent
Syn
chro
tron
Rad
iatio
n on
M
ultic
ore
GPU
and
CPU
Pla
tfor
ms
Balš
aTe
rzić
, PhD
Depa
rtm
ent o
f Phy
sics,
Old
Dom
inio
n U
nive
rsity
Cent
er fo
r Acc
eler
ator
Stu
dies
(CAS
), O
ld D
omin
ion
Uni
vers
ity
2015
IPAC
, Ric
hmon
d, 4
May
201
5
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
1
Colla
bora
tors
May
4, 2
015
2
Cent
er fo
r Acc
eler
ator
Sci
ence
(CAS
) at O
ld D
omin
ion
Uni
vers
ity (O
DU):
Prof
esso
rs:
Phys
ics:
Alex
ande
r God
unov
Com
pute
r Sci
ence
: M
oham
mad
Zub
air,
Desh
Ranj
anPh
D st
uden
t: Co
mpu
ter S
cien
ce:
Kam
esh
Arum
ugam
Early
adv
ance
s on
this
proj
ect b
enef
ited
from
my
colla
bora
tion
with
Ru
iLi (
Jeffe
rson
Lab
)
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
Out
line
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
3
•Co
here
nt S
ynch
rotr
on R
adia
tion
(CSR
)•
Phys
ical
pro
blem
•Co
mpu
tatio
nal c
halle
nges
•N
ew 2
D Pa
rtic
le-In
-Cel
l CSR
Cod
e•
Out
line
of th
e ne
w a
lgor
ithm
•Pa
ralle
l im
plem
enta
tion
CPU
/GPU
clu
ster
s•
Benc
hmar
king
aga
inst
ana
lytic
al re
sults
•St
ill to
Com
e
•Su
mm
ary
CSR:
Phy
sica
l Pro
blem
Be
am’s
self-
inte
ract
ion
due
to C
SR c
an le
ad to
a h
ost o
f adv
erse
ef
fect
s
Incr
ease
in e
nerg
y sp
read
Em
ittan
cede
grad
atio
n
Long
itudi
nal i
nsta
bilit
y (m
icro
-bun
chin
g)
Be
ing
able
to q
uant
itativ
ely
simul
ate
CSR
is th
e fir
st st
ep
tow
ard
miti
gatin
g its
adv
erse
effe
cts
It
is vi
tally
impo
rtan
t to
have
a tr
ustw
orth
y 2D
CSR
cod
e
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
4
CSR:
Com
puta
tiona
l Cha
lleng
es
•Dy
nam
ics g
over
ned
by th
e Lo
rent
z for
ce:
•:
exte
rnal
EM
fiel
ds•
: se
lf-in
tera
ctio
n (C
SR) }re
tard
ed
pote
ntia
ls( r,t)
A( r,t)
( r'
,t')
J( r'
,t')
d r'
r r'
Char
ge d
ensit
y:N
eed
to tr
ack
the
entir
ehi
stor
y of
the
bunc
hCu
rren
t den
sity:
( r,t)
f( r, v,t)
d v
J( r,t)
vf( r, v,t)
d v
reta
rded
tim
et't
r r' c
Eself
1 c A t
Bself A
d dtm
e v
e E B
v c
E Eext Eself
B Bext Bself
Beam
dist
ribut
ion
func
tion
(DF)
:f( r, v,t)
Eext , Bext
Eself, Bself
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
5
LARG
E CA
NCE
LLAT
ION
NU
MER
ICAL
NO
ISE
DUE
TO G
RADI
ENTS
ENO
RMO
US
COM
PUTA
TIO
NAL
AN
D M
EMO
RY LO
AD
ACCU
RATE
2D
INTE
GRAT
ION
CSR:
Com
puta
tiona
l Cha
lleng
es
O
ur n
ew c
ode
solv
es th
e m
ain
com
puta
tiona
l cha
lleng
es a
ssoc
iate
d w
ith th
e nu
mer
ical
sim
ulat
ion
of C
SR e
ffect
s
Enor
mou
s com
puta
tiona
l and
mem
ory
load
(s
torin
g an
d in
tegr
atio
n ov
er b
eam
’s hi
stor
y)Pa
ralle
l im
plem
enta
tion
on G
PU/C
PU p
latfo
rms
La
rge
canc
ella
tion
in th
e Lo
rent
z for
ceDe
velo
ped
high
-acc
urac
y, ad
aptiv
e m
ultid
imen
siona
l int
egra
tor f
or G
PUs
Sc
alin
g of
the
beam
self-
inte
ract
ion
Part
icle
-in-C
ell (
PIC)
cod
e•S
elf-i
nter
actio
n in
PIC
cod
es sc
ales
as g
rid re
solu
tion
squa
red
(Poi
nt-to
-poi
nt c
odes
: sca
les a
s num
ber o
f mac
ropa
rtic
less
quar
ed)
N
umer
ical
noi
seN
oise
rem
oval
usin
g w
avel
ets
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
6
Nm
acro
part
icle
sat
t=t k
syst
em a
t t=t
k+∆t
Adva
nce
part
icle
s by ∆t
Stor
e di
strib
utio
n on
Nx×
N ygr
id
Npo
int-
part
icle
sat
t=t k
Bin
part
icle
s on
N x×N y
grid
Inte
rpol
ate
to o
btai
n fo
rces
on
eac
h pa
rticl
e
Inte
grat
e ov
er g
rid h
istor
ies t
o co
mpu
te re
tard
ed p
oten
tials
and
corr
espo
ndin
g fo
rces
on th
e N x×
N ygr
id
New
Cod
e: T
he B
ig P
ictu
re
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
7
NO
N-S
TAN
DARD
FO
R PI
C CO
DES
New
Cod
e: C
ompu
ting
Reta
rded
Pot
entia
ls
•Ca
rry
out i
nteg
ratio
n ov
er h
istor
y:
•De
term
ine
limits
of i
nteg
ratio
n in
lab
fram
e:co
mpu
te R
max
and
(θm
ini , θ
max
i )
For e
ach
grid
poin
t, in
depe
nden
tly,
do th
e sa
me
inte
grat
ion
over
bea
m’s
hist
ory
Obv
ious
can
dida
te fo
rpa
ralle
l com
puta
tion
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
8
•Pa
ralle
l com
puta
tion
on G
PUs
•Id
eally
suite
d fo
r alg
orith
ms w
ith h
igh
arith
met
ic op
erat
ion/
mem
ory
acce
ss ra
tio•
Sam
e In
stru
ctio
n M
ultip
le D
ata
(SIM
D)•
Seve
ral t
ypes
of m
emor
iesw
ith v
aryi
ng a
cces
s tim
es (g
loba
l, sh
ared
, reg
ister
s)•
Use
s ext
ensio
n to
exi
stin
g pr
ogra
mm
ing
lang
uage
s to
hand
le n
ew a
rchi
tect
ure
•GP
Us h
ave
man
y sm
alle
r cor
es (~
400-
500)
des
igne
d fo
r par
alle
l exe
cutio
n•
Avoi
d br
anch
ing
and
com
mun
icatio
n be
twee
n co
mpu
tatio
nal t
hrea
ds
CPU
GPU
Para
llel C
ompu
tatio
n on
GPU
s
Mor
e sp
ace
for A
LU,
less
for c
ache
an
d flo
w co
ntro
lGP
U:
grid
bl
ocks
th
read
s
Exam
ple:
NVI
DIA
GeFo
rce
GTX
480
GPU
has
448
cor
esM
ay 4
, 201
5 C
SR S
imul
atio
ns o
n M
ultic
ore
Plat
form
s9
Para
llel C
ompu
tatio
n on
GPU
s
Com
putin
g th
e re
tard
ed p
oten
tials
requ
ires i
nteg
ratin
g ov
er
the
entir
e bu
nch
hist
ory
–ve
ry sl
ow!M
ust p
aral
leliz
e.
In
tegr
atio
n ov
er a
grid
is id
eally
suite
d fo
r GPU
s
No
need
for c
omm
unic
atio
n be
twee
n gr
idpo
ints
Sa
me
kern
elex
ecut
ed fo
r all
Ca
n re
mov
e al
l bra
nche
s fro
m th
e al
gorit
hm
W
e de
signe
d a
new
ada
ptiv
e m
ultid
imen
siona
l int
egra
tion
algo
rithm
opt
imize
d fo
r GPU
s[A
rum
ugam
, God
unov
, Ran
jan,
Terz
ić&
Zub
air2
013a
,b]
N
VIDI
A’s C
UDA
fram
ewor
k (e
xten
sion
to C
++)
Ab
out 2
ord
ers o
f mag
nitu
de sp
eedu
p ov
er a
seria
l im
plem
enta
tion
U
sefu
l bey
ond
this
proj
ect
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
10
Perf
orm
ance
Com
paris
on: C
PU V
s. G
PU
Com
paris
on: 1
CPU
vs.
1 G
PU;
8 CP
Us v
s. 4
GPU
s (on
e co
mpu
te n
ode)
1
GPU
ove
r 50
x fa
ster
than
1 C
PU
Both
line
arly
scal
e w
ith m
ultic
ores
: 4 G
PUs 2
5x fa
ster
than
8 C
PUs
Hy
brid
CPU
/GPU
impl
emen
tatio
n m
argi
nally
bet
ter t
han
GPU
s alo
ne
Exec
utio
n tim
e re
duce
sas t
he n
umbe
r of p
oint
-par
ticle
s gro
ws
M
ore
part
icle
s, le
ss n
umer
ical
noi
se, f
ewer
func
tion
eval
uatio
ns n
eede
d fo
r hig
h-ac
cura
cy in
tegr
atio
n
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
11
GPU
Clu
ster
Impl
emen
tatio
n
The
high
er th
e re
solu
tion,
the
larg
er th
e fra
ctio
n of
tim
e sp
ent
on c
ompu
ting
inte
gral
s (an
d th
eref
ore
the
spee
dup)
We
expe
ct th
e sc
alin
g at
larg
er re
solu
tions
to b
e ne
arly
line
ar
1 st
ep o
f the
sim
ulat
ion
on a
128
x128
grid
and
32
GPU
s: ~
10
s
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
12
1 2 4 8 16
32 1
2 4
8 1
6 3
2
Speedup
Num
ber o
f GPU
sGrid
Res
olut
ion
128
x 12
8
Grid
Res
olut
ion
64 x
64
N=10
2400
0
Benc
hmar
king
Aga
inst
Ana
lytic
1D
Resu
lts•
Anal
ytic
stea
dy st
ate
solu
tion
avai
labl
e fo
r a ri
gid
line
Gaus
sian
bunc
h [D
erbe
nev
& S
hilts
ev19
96, S
LAC-
Pub
7181
]
•Ex
celle
nt a
gree
men
t bet
wee
n an
alyt
ic a
nd c
ompu
ted
solu
tions
pro
vide
sapr
oof o
f con
cept
for t
he n
ew c
ode
N=5
1200
0N
x=Ny=6
4
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
13
LON
GITU
DIN
ALTR
ANSV
ERSE
-7-6-5-4-3-2-1 0 1
-4-2
0 2
4
Effective Transverse CSR Force [keV/m]
s/ s
anal
ytic
com
pute
d+ +
-500
-400
-300
-200
-100 0
100
200
-4-2
0 2
4
Effective Longitudinal CSR Force [keV/m]
s/ s
anal
ytic
com
pute
d+ +
Larg
e Ca
ncel
latio
n in
the
Lore
ntz F
orce
•Tr
aditi
onal
ly d
iffic
ult t
o tr
ack
larg
e qu
antit
ies w
hich
mos
tly c
ance
l out
:
•Hi
gh a
ccur
acy
of th
e im
plem
enta
tion
able
to tr
ack
accu
rate
ly th
ese
canc
ella
tions
ove
r 5 o
rder
s of m
agni
tude
4×10
76×
102
N=1
2800
0N
x=Ny=3
2
Effe
ctiv
e Lo
ngitu
dina
l For
ce:
ϕ−β s
Αs
s s
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
14
Effo
rts C
urre
ntly
Und
erw
ay
Co
mpa
re to
2D
sem
i-ana
lytic
al re
sults
(chi
rped
bun
ch)
[Li 2
008,
PR
STAB
11,
024
401]
Co
mpa
re to
oth
er 2
D co
des (
for i
nsta
nce
Bass
iet a
l. 20
09)
Si
mul
ate
a te
st c
hica
ne
Fu
rthe
r Afie
ld:
Va
rious
bou
ndar
y co
nditi
ons
Sh
ield
ing
U
se w
avel
ets t
o re
mov
e nu
mer
ical
noi
se (i
ncre
ase
effic
ienc
y an
d ac
cura
cy)
Ex
plor
e th
e ne
ed a
nd fe
asib
ility
of g
ener
alizi
ng th
e co
de fr
om 2
D to
3D
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
15
Sum
mar
y
Pres
ente
d th
e ne
w 2
D PI
C co
de:
Re
solv
es tr
aditi
onal
com
puta
tiona
l diff
icul
ties b
y op
timizi
ng o
ur a
lgor
ithm
on
a G
PU p
latfo
rm
Proo
f of c
once
pt: e
xcel
lent
agr
eem
ent w
ith a
naly
tical
1D
resu
lts
O
utlin
ed o
utst
andi
ng is
sues
that
will
soon
be
impl
emen
ted
Cl
osin
g in
on
our g
oal
Ac
cura
te a
nd e
ffici
ent c
ode
whi
ch fa
ithfu
lly si
mul
ates
CSR
effe
cts
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
16
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
17
Back
up S
lides
Impo
rtan
ce o
f Num
eric
al N
oise
•Si
gnal
-to-n
oise
ratio
in P
IC si
mul
atio
ns sc
ales
as N
ppc1/
2
[Ter
zić, P
ogor
elov
& B
ohn
2007
, PR
STAB
10,
034
021]
•Th
en th
e nu
mer
ical
noi
se sc
ales
as N
ppc-1
/2(N
ppc:
avg.
# o
f par
ticle
s per
cel
l)
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
18
128
x 12
8 gr
id
Less
num
eric
al n
oise
= m
ore
accu
rate
and
fast
er si
mul
atio
ns[T
erzić
, Pog
orel
ov&
Boh
n 20
07, P
R ST
AB 1
0, 0
3402
1; Te
rzić
& B
assi
2011
, PR
STAB
14,
070
701]
Exec
utio
n tim
e fo
r int
egra
lev
alua
tion
also
scal
es a
s Npp
c-1/2
W
hen
the
signa
l is k
now
n, o
ne c
an
com
pute
Sig
nal-t
o-No
ise R
atio
(SNR
):
N ppc: a
vg. #
of p
artic
les p
er c
ell
Npp
c= N/
N cells
2D su
perim
pose
d Ga
ussia
ns o
n 25
6×25
6 gr
id
Wav
elet
den
oisin
gyi
elds
a re
pres
enta
tion
whi
ch is
:
-Ap
prec
iabl
y m
ore
accu
rate
than
non
-den
oise
dre
pres
enta
tion
-Sp
arse
(if c
leve
r, w
e ca
n tr
ansla
te th
is sp
arsit
yin
to c
ompu
tatio
nal e
ffici
ency
)
Wav
elet
Den
oisi
ngan
d Co
mpr
essi
on
CO
MPA
CT:
onl
y 0.
12%
of c
oeffs
AN
ALY
TIC
AL
Npp
c=3
SNR
=2.0
2N
ppc=
205
SNR=1
6.89
WAV
ELET
TH
RES
HO
LDIN
GD
EN
OIS
ED
CO
MPA
CT:
onl
y 0.
12%
of c
oeffs
SNR
q i2
i1
Ngrid q iq i
2
i1
Ngrid
q iexact
q igrid
SNRNppc
Npp
c=3
SNR=1
6.83
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
19
Perf
orm
ance
Com
paris
on: G
PU V
s. H
ybrid
CPU
/GPU
Co
mpa
rison
: 1 C
PU v
s. 1
GPU
; 8
CPU
s vs.
4 G
PUs (
one
com
pute
nod
e)
Hybr
id C
PU/G
PU im
plem
enta
tion
mar
gina
lly b
ette
r tha
n GP
Us a
lone
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
20
Brea
kdow
n of
Com
puta
tions
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
21
New
Cod
e: C
ompu
tatio
n of
CSR
Effe
cts
3 co
ordi
nate
fram
es
for e
asie
r com
puta
tion
Com
putin
g re
tard
ed p
oten
tials
:M
ajor
com
puta
tiona
l bot
tlene
ck
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
22
New
Cod
e: P
artic
le-In
-Cel
l
•Gr
id re
solu
tion
is sp
ecifi
ed a
prio
ri(fi
xed
grid
)•
N X: #
of g
ridpo
ints
inX
•N Y
:# o
f grid
poin
ts in
Y•
N grid
=NX×
N Yto
tal g
ridpt
s•
Grid
:
•In
clin
atio
n an
gleα
•Po
int-
part
icle
s dep
osite
d on
the
grid
via
dep
ositi
on sc
hem
e
•Gr
id is
det
erm
ined
so a
s to
tight
ly e
nvel
ope
all p
artic
les
Min
imizi
ng n
umbe
r of e
mpt
y ce
lls ⇒op
timizi
ng sp
atia
l res
olut
ion
X ij,Y
ij
j1,Ny
i1,Nx
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
23
New
Cod
e: F
ram
es o
f Ref
eren
ce
•Ch
oosin
g a
corr
ect c
oord
inat
e sy
stem
is o
f cru
cial
impo
rtan
ce•
To si
mpl
ify c
alcu
latio
ns u
se 3
fram
es o
f ref
eren
ce:
•Fr
enet
fram
e (s
, x)
s–al
ong
desig
n or
bit
x–
devi
atio
n no
rmal
todi
rect
ion
of m
otio
n-
Part
icle
pus
h
•La
b fr
ame
(X, Y
)-
Inte
grat
ion
rang
e-
Inte
grat
ion
of re
tard
ed
pote
ntia
ls
•Gr
id fr
ame
(X~,
Y~)
Scal
ed &
rota
ted
lab
fram
eal
way
s [-0
.5,0
.5] ×
[-0.5
,0.5
]-
Part
icle
dep
ositi
on-
Grid
inte
rpol
atio
n-
Hist
ory
of th
e be
am
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
24
Sem
i-Ana
lytic
2D
Resu
lts: 1
D M
odel
Bre
aks D
own
•An
alyt
ic s
tead
y st
ate
solu
tion
is ju
stifi
ed fo
r [D
erbe
nev
& S
hilts
ev 19
96]
•Li
, Leg
g, T
erzić,
Bis
ogna
no &
Bos
ch 2
011:
x
Rz2
1/
3
1
1D &
2D
dis
agre
e in
:M
agni
tude
of C
SR fo
rce
Loca
tion
of m
axim
um fo
rce
Mod
el b
unch
com
pres
sor (
chic
ane)
E =
70 M
eVσ z
0= 0
.5 m
mu
= -10
.56
m-1
ener
gy c
hirp
L b=
0.3
mL B
= 0.
6 m
L d=
0.4
m
⇒1D CSR
mod
el is
inad
equa
te
Prel
imin
ary
sim
ulat
ions
sho
wgo
od a
gree
men
t bet
wee
n 2D
se
mi-a
naly
tic re
sults
and
resu
ltsob
tain
ed w
ith o
ur c
ode
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
25
O
rtho
gona
l bas
is of
func
tions
com
pose
d of
scal
ed a
nd tr
ansla
ted
vers
ions
of
the
sam
e lo
caliz
ed m
othe
r wav
eletψ
(x) a
nd th
e sc
alin
g fu
nctio
n ϕ
(x):
Ea
ch n
ew re
solu
tion
leve
l kis
orth
ogon
al to
the
prev
ious
leve
ls
Co
mpa
ct su
ppor
t: fin
ite d
omai
n ov
er w
hich
non
zero
In
ord
er to
att
ain
orth
ogon
ality
of d
iffer
ent s
cale
s,th
eir s
hape
s are
stra
nge
-Sui
tabl
e to
repr
esen
t irr
egul
arly
shap
ed fu
nctio
ns
Fo
r disc
rete
sign
als (
grid
ded
quan
titie
s), f
ast
Disc
rete
Wav
elet
Tra
nsfo
rm (D
FT) i
s an
O(M
N)
oper
atio
n, M
size
of th
e w
avel
et fi
lter,
Nsig
nal s
ize
Wav
elet
s
Dau
bach
ies
4thor
der w
avel
et
ik (x
)2k
/2
(2kxi),
k,iZ
f(x)s 00
00(x
)d ik
ik
ik (x
),
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
26
W
avel
et b
asis
func
tions
hav
e co
mpa
ct su
ppor
t ⇒signa
l loc
alize
d in
spac
eW
avel
et b
asis
func
tions
hav
e in
crea
sing
reso
lutio
n le
vels ⇒signal
loca
lized
in fr
eque
ncy
⇒Simulta
neou
s loc
aliza
tion
in sp
ace
and
freq
uenc
y(F
FT o
nly
freq
uenc
y)
W
avel
et b
asis
func
tions
cor
rela
te w
ell w
ith v
ario
us si
gnal
type
s (in
clud
ing
signa
ls w
ith si
ngul
ariti
es, c
usps
and
oth
er ir
regu
larit
ies)
⇒Com
pact
and
acc
urat
e re
pres
enta
tion
of d
ata
(com
pres
sion)
W
avel
et tr
ansf
orm
pre
serv
es h
iera
rchy
of s
cale
s
In
wav
elet
spac
e, d
iscre
tized
ope
rato
rs (L
apla
cian
) are
also
spar
se a
nd h
ave
an
effic
ient
pre
cond
ition
er⇒Solv
ing
som
e PD
Es is
fast
er a
nd m
ore
accu
rate
Pr
ovid
e a
natu
ral s
ettin
g fo
r num
eric
al n
oise
rem
oval
⇒Wave
let d
enoi
sing
Wav
elet
thre
shol
ding
: If
|w
ij|<T
, se
t wij=
0.
[Ter
zić, P
ogor
elov
& B
ohn
2007
, PR
STAB
10,
034
201]
[Ter
zić&
Bas
si20
11, P
R ST
AB 1
4, 0
7070
1]
Adva
ntag
es o
f Wav
elet
For
mul
atio
n
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
27
Wav
elet
Com
pres
sion
[Fro
m Te
rzić
& B
assi
2011
, PR
STAB
14,
070
701]
Mod
ulat
ed fl
at-to
p pa
rtic
le d
istrib
utio
nFr
actio
n of
non
-zer
o co
effic
ient
sre
tain
ed a
fter w
avel
et th
resh
oldi
ng
1% 0.1%
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
28
CSR:
Poi
nt-t
o-Po
int A
ppro
ach
•Po
int-t
o-Po
int a
ppro
ach
(2D
):[L
i 199
8]
•Ch
arge
den
sity
is s
ampl
ed w
ith N
Gau
ssia
n-sh
aped
2D
mac
ropa
rtic
les
(2D
dis
trib
utio
n w
ithou
t ver
tical
spr
ead)
•Ea
ch m
acro
part
icle
inte
ract
s w
ith e
ach
mac
ropa
rtic
le th
roug
hout
his
tory
•Ex
pens
ive:
com
puta
tion
of re
tard
ed p
oten
tials
and
sel
f fie
lds
~ O
(N2 )
⇒small
num
ber N
⇒poo
r spa
tial r
esol
utio
n⇒diffi
cult
to s
ee s
mal
l-sca
le s
truc
ture
•W
hile
use
ful i
n ob
tain
ing
low
-ord
er m
omen
ts o
f the
bea
m,
Poin
t-to
-Poi
nt a
ppro
ach
is n
ot o
ptim
al fo
r stu
dyin
g CS
R
DF
Char
ge d
ensi
ty
Curr
ent d
ensi
ty
Gau
ssia
n m
acro
part
icle
f( r, v,t)q
n m( r r 0(i
) (t))
i1N
( v
v 0(i) (t
))
( r,t)q
n m( r r 0(i
) (t))
i1N
J( r,t)q
0(i
) (t)n
m( r r 0(i
) (t))
i1N
n m( r r 0(i
) (t))
12m2
exp
(xx 0
(t))
2
(yy 0
(t))
2
2m2
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
29
CSR:
Par
ticle
-In-C
ell A
ppro
ach
•Pa
rtic
le-In
-Cel
l app
roac
h w
ith re
tard
ed p
oten
tials
(2D)
:
•Ch
arge
and
cur
rent
den
sitie
s are
sam
pled
with
Npo
int-
char
ges (δ-
func
tions
)an
d de
posit
ed o
n a
finite
grid
usin
g a
depo
sitio
n sc
hem
e
•Tw
o m
ain
depo
sitio
n sc
hem
es-
Nea
rest
Grid
Poi
nt (N
GP)
(con
stan
t: de
posit
s to
1Dpo
ints
)-
Clou
d-In
-Cel
l (CI
C)(li
near
: dep
osits
to 2
Dpo
ints
)Th
ere
exist
hig
her-
orde
r sch
emes
•Pa
rtic
les d
o no
t dire
ctly
inte
ract
with
eac
h ot
her,
but o
nly
thro
ugh
a m
ean-
field
of th
e gr
idde
d re
pres
enta
tion
p(X⃗)
x⃗ k⃗
NG
P
CIC
p(x)
•–
grid
poin
tloc
atio
nx
–m
acro
part
icle
loca
tion
DF (K
limon
tovi
ch)
Char
ge d
ensit
y
Curr
ent d
ensit
y
f( r, v,t)q
( r
r 0(i) (t
))i
1N ( v
v 0(i) (t
))
( x k
,t)q
( x k x 0(i
) (t)
X)
hh
i1N
p( X
)d X
J( x k
,t)q
0(i) (t
)( x k x 0(i
) (t)
X)
hh
i1N
p( X
)d X
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
30
CSR:
P2P
Vs.
PIC
•Co
mpu
tatio
nal c
ost f
or P
2P:
Tota
l cos
t ~ O
(N2 )
•In
tegr
atio
n ov
er h
istor
y (y
ield
s sel
f-for
ces)
: O
(N2 ) o
pera
tion
•Co
mpu
tatio
nal c
ost f
or P
IC:
Tota
l cos
t ~ O
(Ngr
id2 )
•Pa
rtic
le d
epos
ition
(yie
lds g
ridde
d ch
arge
& c
urre
nt d
ensit
ies)
: O(N
) ope
ratio
n•
Inte
grat
ion
over
hist
ory
(yie
lds r
etar
ded
pote
ntia
ls): O
(Ngr
id2 ) o
pera
tion
•Fi
nite
diff
eren
ce (y
ield
s sel
f-for
ces o
n th
e gr
id):
O(N
grid
) ope
ratio
n•
Inte
rpol
atio
n (y
ield
s sel
f-for
ces a
ctin
g on
eac
h of
N p
artic
les)
: O(N
) ope
ratio
n•
Ove
rall
~ O
(Ngr
id2 )+
O(N
) ope
ratio
ns•
But i
n re
alist
ic si
mul
atio
ns:
N grid
2 >> N
, so
the
tota
l cos
t is ~
O(N
grid
2 )•
Favo
rabl
e sc
alin
g al
low
s for
larg
er N
, and
reas
onab
le g
rid re
solu
tion
⇒Impro
ved
spat
ial r
esol
utio
n
•Fa
ir co
mpa
rison
: P2
P w
ith N
mac
ropa
rtic
les a
ndPI
C w
ith N
grid
=N
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
31
CSR:
P2P
Vs.
PIC
•Di
ffere
nce
in sp
atia
l res
olut
ion:
An
illus
trat
ive
exam
ple
•An
alyt
ical
dist
ribut
ion
sam
pled
with
•
N =
N XN Ym
acro
part
icle
s(as
in P
2P)
•O
n a
N x×N Y
grid
(as i
n PI
C)
•2D
grid
: N X=N
Y=32
•PI
C ap
proa
ch p
rovi
des s
uper
ior s
patia
l res
olut
ion
to P
2P a
ppro
ach
•Th
is m
otiv
ates
us t
o us
e a
PIC
code
EXAC
TP2
P N
=322
SNR=
2.53
PIC
N=5
0x32
2SN
R=13
.89
Sign
al-to
-Noi
se R
atio
SNR
q i2
i1
Ngrid q iq i
2
i1
Ngrid
q iexact
q igrid
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
32
Inte
grat
e ov
er p
artic
le h
isto
ries
to c
ompu
te re
tard
ed p
oten
tials
an
d co
rres
pond
ing
forc
eson
eac
h m
acro
part
icle
syst
em a
t t=t
k+∆t
Adva
nce
part
icle
s by
∆t
Nm
acro
part
icle
sat
t<t k
Nm
acro
part
icle
sat
t=t k
Out
line
of th
e P2
P Al
gorit
hm
May
4, 2
015
CSR
Sim
ulat
ions
on
Mul
ticor
e Pl
atfo
rms
33