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