High-Performance Simulations of Coherent Synchrotron ...
33
High-Performance Simulations of Coherent Synchrotron Radiation on Multicore GPU and CPU Platforms Balša Terzić, PhD Department of Physics, Old Dominion University Center for Accelerator Studies (CAS), Old Dominion University 2015 IPAC, Richmond, 4 May 2015 May 4, 2015 CSR Simulations on Multicore Platforms 1
Transcript of High-Performance Simulations of Coherent Synchrotron ...
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
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ulat
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on
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e Pl
atfo
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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
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ulat
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e Pl
atfo
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•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
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ulat
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e Pl
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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
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ulat
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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
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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
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ulat
ions
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ticor
e Pl
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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
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015
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Sim
ulat
ions
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e Pl
atfo
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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
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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
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ulat
ions
on
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e Pl
atfo
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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
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ulat
ions
on
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ticor
e Pl
atfo
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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
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ulat
ions
on
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ticor
e Pl
atfo
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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
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ulat
ions
on
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e Pl
atfo
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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
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Sim
ulat
ions
on
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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
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ulat
ions
on
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e Pl
atfo
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4, 2
015
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ulat
ions
on
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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
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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
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4, 2
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ulat
ions
on
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ticor
e Pl
atfo
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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
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ulat
ions
on
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ticor
e Pl
atfo
rms
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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
