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