Tính toán hiệu năng cao và ứng dụng vào bài toán mô phỏng động lực phân tử

DAI HOC QUOC GIA HA N O I

TINH TOAN HIEU N A N G CAO
VA LfNG D U N G VAO BAI TOAN
MO P H O N G DONG L U C P H A N Ttf
Bao cao tong hdp de tai nghien ciiu khoa hoc cap
DHQG do Trudng Dai hoc cong nghe quan ly)

Ma so: QC.05.01
Chu nhiem de tai: Nguyen Hai Chau
OAI HOC QUOC GIA HA NQi
TRUNG TAM THONG ''iN ^HIJ VlE^

1 ' cU
Ha Noi - 2006


M u c luc
D a n h m u c h i n h ve

2

D a n h muc b a n g

3

1

D a n h sach can b o t h a m gia thUc hien de tai

5

2

Tdm t a t nhiJng ket qua chfnh cua de tai nghien ciJu khoa hpc

6

2.1
2.2
2.3

6
6
6
6
7
7
7
7

3

Ten d^ tai
Chu tri dg tai
Nhu'ng ket qua chinh
2.3.1 Ket qua vk khoa hpc
2.3.2 K^t qua phuc vu thuc t€
2.3.3 Ket qua dao tao
2.3.4 Kgt qua nang cao tilm luc khoa hoc
2.3.5 T m h hhih su* dung kinh phi

N O I D U N G C U A D E TAI
3.1 Dat va,n d l
3.2 T5ng quan ve tfnh lire tu'dng tac nhanh trong mo phong dong luc
phan tii
3.2.1 Cac thuat toan nhanh trong ITnh vuc mo phong dpng luc phan
td
3.2.2 Thuat toan FMM va cac biin t h i
3.2.2.1
Thuat toan FMM
3.2.2.2
Thuat toan cua Anderson
3.2.2.3
Thuat toan ciia Makino
3.2.3 May tinh chuyen dung song song G R A P E va iing dung . . .
3.2.4 Cai dat thuat toan nhanh tren phan ciing chuyen dung . . .
3.2.4.1
Cai dat thuat toan tree tren p h i n ciing G R A P E . .
3.2.4.2
Cai dat thuat toan FMM tren p h i n ciing MD-ENGINE
3.3 Npi dung va kit qua nghien ciiu
3.3.1 Cac kho khan cin giai quylt
3.3.2 Giai phap va kit qua ciia chung toi
3.4 Thao luan

9
9
10
11
12

12
13
14
16
19
19
19
20
20
21
26


MUC LUC

2

3.5

28

Kit luan va kiln nghi

Tai lieu t h a m khao

29

P h u luc

34

D a n h muc hinh ve
3.1
3.2
3.3
3.4
3.5

3.6

3.7
3.8
3.9
3.10

3.11
3.12

3.13
3.14

Y tudng chmh cua thuat toan tinh luc FMM
PhUdng phap cua Anderson
PhUdng phap P^M^ ciia Makino
Kiln triic cd ban ciia mpt he may tfnh G R A P E
May tmh MDGRAPE-2 (PCI) co t i c dp cue dai tUdng dUdng 48GFlops
vdi 4 chip MDGRAPE-2 (m5i chip co t i c dp cue dai tUdng duong
16GFlops)
May tfnh MDGRAPE-2 (Compact PCI) co t i c dp cUc dai tUdng

dudng 192GFlops vdi 16 chip MDGRAPE-2 (m6i chip co t i c dp cue
dai tUdng dUdng 16GFlops)
Cum may tinh G R A P E nhin tii mat trUdc [54]
Cum may tinh G R A P E nhui txt mat sau [54]
Cum may tfnh G R A P E nhin tii ben trai [54]
So sanh thdi gian tfnh luc ciia thuat toan FMM va thuat toan tinh luc
true tilp tren he I. Dudng cong cd cac hmh trdn t h i hien hieu nang
cua FMM tren may tinh MDGRAPE-2. Dudng cong vdi cac hinh ngu
giac la hieu nang ciia FMM tren may chii. Cac hmh trdn va ngu giac
to den la hieu nang tUdng iing vdi dp chinh xac cao (p = 5), khong
to den ling vdi dp chfnh xac t h i p {p = 1). Dudng cong khong cd ky
hieu vdi net liln va net diit tUdng iing la hieu nang ciia thuat toan
tfnh luc true tilp tren MDGRAPE-2 va may chu
Tudng t u nhu hmh 3.10 nhung vdi he II
So sanh thdi gian tfnh luc ciia thuat toan FMM va thuat toan tree
tren he I. Dudng cong cd cac hinh trdn t h i hien hieu nang cua FMM
tren may tfnh MDGRAPE-2. Dudng cong vdi cac hmh tam giac la
hieu nang cua thuat toan tree tren MDGRAPE-2. Cac hinh trdn va
tam giac td den la hieu nang tUdng iing vdi dp chinh xac cao (p — 5),
khong td den iing vdi dp chfnh xac t h i p {p = I)
Tudng tu nhu hinh 3.12 nhung vdi he II
So sanh hieu nang cua hai ban cai dat FMM khi sii dung cac cong
thiic (3.11) - ban 1.0 va (3.12) - ban 2.0. Dudng cong cd hhih vudng
la hieu n t o g ciia ban 1.0. Dudng cong cd hinh trdn la hieu nang cua
ban 2.0. Cac dudng cong cd hinh td den iing \'di do chinh xac cao
(p = 5), khong to den iing vdi do chfnh xac t h i p (p = 1)

12
14
15

16

16

17
18
19
20

22
23

24
26

27


D a n h muc bang
1.1

3.1

3.2

3.3

Danh sach can bp, cong tac vien, hpc vien cao hpc va sinh vien tham
gia thuc hien d l tai
Cac pha tfnh toan va cac cdng thiic tUdng iing dUdc sii dung trong

hai ban cai dat FMM. Cac p h i n in dam dUdc thuc hien tren may tfnh
GRAPE
P h a n tfch thdi gian thUc hien cac pha tfnh toan cua ban 2.0 tren he
II vdi so lUdng hat A^ = 1024 x 1024 = 1048576. Chii y ring cac pha
"Tao cay" va "Tao danh sach lan can va danh sach tUdng tac" trong
thuat toan FMM khong dUdc chiing tdi md ta trong bao cao nay vi
cae pha nay chi cd tfnh chit chuan bi cho tfnh toan va chilm r i t ft
thdi gian tfnh to^n nhu ta thiy trong bang
So sanh vdi ban cai dat DPMTA 3.1.3 cua Wrankin [49]

5

21

25
25


Danh sach can bo t h a m gia thu'c
hien de tai
Bang 1.1: Danh sach can bp, cdng tac vien, hpc vien cao hpc va sinh vien tham gia
thuc hien dl tai
STT

H o va t e n

Hpc ham
hpc vj

Cd q u a n cong tac

TS

2

Nguyin Hai Chau
(chu nhiem d l tai)
T. Ebisuzaki

TS

3

A. Kawai

TS

4

Vu Bdi H i n g

ThS

5

T r i n Manh Tudng

CN

6

Dd Thi Minh Viet

CN

7

Nguyin Thi Thuy Linh

CN

8

Pham Quang Nhat Minh

SVK47

9

Le Thi Lan PhUdng

SVK47

10

Chu Quang Thiiy

SVK47

Khoa Cdng nghe thdng tin, trudng

Dai hpc cdng nghe, DHQGHN
Trung tam tfnh toan cao cip, Vien
nghien ciiu vat ly va hda hpc Nhat
Ban
Hpc vien Cdng nghe Saitama,
Nhat Ban
Khoa Cdng nghe thdng tin, trudng
Dai hpc cdng nghe, DHQGHN.
Khoa Toan Cd Tin hoc, trudng
DHKHTN, DHQGHN.
Khoa Cdng nghe thdng tin, trudng
Dai hpc cdng nghe, DHQGHN.
Khoa Cdng nghe thdng tin, trudng
Dai hpc cdng nghe, DHQGHN.
Khoa Cdng nghe thdng tin, trudng
Dai hpc cdng nghe, DHQGHN.
Khoa Cdng nghe thdng tin, trudng
Dai hpc cdng nghe, DHQGHN.
Khoa Cdng nghe thdng tin, trudng
Dai hpc cdng nghe, DHQGHN.

1


Tom t a t nhiJng ket qua chinh
cua de tai nghien ciJu khoa hoc
2.1

Ten de tai

Tfnh toan hieu nang cao va iing dung vao bai toan md phong ddng luc phan tii.
(High performance computing and its application to molecular dynamics simulation).
M a s d : QC.05.01.

2.2

Chii tri de tai

Ngudi chu tri: TS. Nguyin Hai Chau
Cd quan: Trudng Dai hpc Cdng nghe, Dai hpc Qudc gia Ha Ndi.
Dia chi: 144 Xuan Thiiy, C i u Giiy, Ha Ndi.
Dien thoai: 04-7547813

2.3
2.3.1

NhiJng ket qua chfnh
Kgt qua ve khoa hoc

• Da hoan thanh 02 bai bao khoa hpc giM dang tap chf Tin hpc va dilu khiln
hpc va Tap chf Khoa hpc (Dai hpc Quoc gia Ha Ndi).
• Da hoan thanh 01 bao cao tai hdi thao qudc t l vl tfnh toan hieu nang cao
dUdc td chiic tai trudng Dai hpc Khoa hoc tu nhien.
• 03 bao cao chuyen d l (technical report) vl: Lap trinh song song vdi OpenMP.
nghien ciiu hieu nang ciia he thdng file song song ao PVFS va tdng quan ve
tfnh toan song song.


2. T6MTAT
2.3.2

NHIING KET QUA CHINH CUA D^ TAI NGHIEN ClfU KHOA H0C7

K i t qua phuc v u thu'c t l

Da hoan thanh bp chudng trinh cai dat thii nghiem thuat toan FMM tren may tfnh
MDGRAPEI-2. Cac kit qua nghien ciiu cua dl tai cho thiy, thuat toan do chung
tdi thilt kl va cai dat cd hieu nang cao va dp chfnh xac thoa man cac yeu ciu cua
nhilu ling dung trong linh vUc md phong dpng luc phan tii. Thuat toan cai dat nay
CO kha nang tfch hdp vdi mpt so iing dung vl MD da dUdc triln khai nhu NAMD.
May tfnh MDGRAPE-2 cd tai Ha Npi va kha nang sii dung may tfnh nay vao md
phong la kha thi.
2.3.3

K i t q u a d a o tao

• Da hudng din tot nghiep 03 sinh vien bao ve thang 6/2006 vl dl tai tfnh toan
hieu nang cao (xem cac bia luan van kem theo).
• Dang hudng din 01 hpc vien cao hpc vl dl tai tfnh toan hieu nang cao, du
kiln bao ve 12/2006.
2.3.4

K i t q u a n a n g c a o t i e m \\ic k h o a h o c

Nghien ciiu vl cac thuat toan, tlm hilu vl tfnh toan song song, tfnh toan hieu nang
cao, tfnh toan cum va tfnh toan thdng lUdng cao. Da hudng din sinh vien, td chiic
seminar va giang day vl ITnh vUc tfnh toan hieu nang cao, tfnh toan song song va
tfnh toan cum cho cac sinh vien tii nam 2004. Nam hpc 2006-2007 se giang day vl
tfnh toan song song cho hpc vien cao hpc cua khoa Cdng nghe thdng tin, trudng Dai
hpc Cdng nghe.

2.3.5

T i n h h i n h sdf d u n g k i n h phi

Da Sli dung hit kinh phf dUdc cip ciia dl tai.

CHU NHIEM DE TAI

XAC N H A N CUA DON VI

XAC N H A N CUA Cd QUAN CHU QUAN


2. T6M

TAT NHIJNG KET QUA CHINH CUA DE TAI NGHIEN CHU KHOA HOCS

Abstract
We have implemented fast multipole method (FMM) on a special-purpose computer
G R A P E (GRAvity piPE). The FMM is one of the fastest approximate algorithms
to calculate forces among particles. Its calculation cost scales as 0(-/V), while naive
algorithm scales as 0{N'^). Here, TV is the number of particles in the system. GRAPE
is a hardware dedicated to the calculation of Coulombic or gravitational force among
particles. Its calculation speed is 100-1000 times faster than that of conventional
computers of the same price, though it cannot handle anything but force calculation.
We can expect significant speed up by the combination of the fast algorithm and the
fast hardware. However, a straight forward implementation of the algorithm actually
runs on G R A P E at rather modest speed. This is because of the limited function of
the hardware. Since G R A P E can handle particle force only, just a small fraction
of the calculation procedure can be put on it. The rest part must be performed

on a conventional computer connected to GRAPE. In order to take full advantage
of the dedicated hardware, we modified the FMM using Pseudoparticle Multipole
Method and Anderson's method. In the modified algorithm, multipole and local
expansions are expressed by distribution of a small number of imaginary particles
(pseudoparticles), and thus they can be evaluated by GRAPE. Results of numerical
experiments show that G R A P E accelerates the FMM by a factor of 3-60 depending
on the accuracy. Its performance exceeds that of Barnes-Hut treecode on GRAPE
at high accuracy (root-mean-square relative force error ~ 10~^), in the case of closeto-uniform distribution of particles.
K e y w o r d s : Molecular dynamics, numerical simulation, fast multipole method, tree
algorithm, Anderson's method, pseudoparticle multipole method, special-purpose
computer.


NOI D U N G CUA DE TAI
3.1

Dat v4n d l

Md phong dpng lue phan tii la mpt trong nhiing phUdng phap phd biin dUdc sii
dung trong vat ly/hoa hpc d l nghien ciiu cac he nhilu hat. Md phdng ddng luc phan
tii dUa tren dinh luat 2 Newton ve chuyen ddng: F = ma^ trong dd F la luc tac
dung tren hat; m, a tUdng iing la khdi lUdng va gia tdc cua hat. Tuf cac thdng tin
ve luc tac dung tren mdi hat, xac dinh gia tdc cua mdi hat trong he. Giai phudng
trinh chuyin dpng d l sinh ra mdt dudng cong md ta vi trf, van toc, gia toc cua cac
hat tai cac mdc thdi gian khac nhau. Tii dudng cong nay, trang thai tilp theo hay
trang thai trUdc ciia he se dUdc du bao.
Xet tren khfa eanh tfnh toan, viec thuc hien bai toan md phong ddng luc phan
tii cd t h i dUdc md ta qua cac budc sau:
1. Chpn vi trf ban d i u cua cac hat vdi dien tfch cho trude trong he.
2. Chpn mdt tap hdp van tdc khdi tao ciia cac hat. Cac van toc nay thudng dUdc

chpn theo phan phdi Boltzmann ddi vdi mdt vai nhiet dp, sau dd dupe chuin
hda sao cho tdng ddng lUdng ciia toan he bing 0.
3. Tfnh ddng ludng cua mdi hat tu" van toc va khoi lUdng cua chiing
4. Tfnh lUc tUdng tac tlnh dien (Coulombic force) tren mdi hat.
5. Tfnh vi trf mdi cho cac hat sau mdt khoang thdi gian ngin sau dd. Khoang thdi
gian nay dUdc gpi la bude thdi gian (time step). Viec tfnh toan nay dUdc thuc
hien bing each giai phudng trinh chuyin ddng dua tren dinh luat 3 Newton.
6. Tfnh toan van tdc va gia tdc mdi cho cac hat trong he.
7. Lap lai cac budc tu" budc 3 din budc 6.
8. Lap lai qua trinh nay du lau d l cho he dat tdi trang thai can bing.
9. Khi he dat tdi trang thai can bing, ghi lai vi trf cua cac hat sau mot sd vdng
lap n h i t dinh. Cac thdng tin nay thudng dUdc ghi lai sau tii 5 din 25 vdng
lap. Danh sach cac tpa dp nay tao thanh quy dao chuyen dong ciia he hat.


3. Nd

DUNG CUA DE TAI

10

10. Tilp tuc qua trinh lap di lap lai va ghi lai dii lieu cho d i n khi cd dii dir lieu
dUdc tap hdp de dua ra cac kit qua vdi dp chfnh xac mong mudn.
11. P h a n tich cac quy dao chuyin ddng d l thu dUde thdng tin vl he.
Luc / ( r i ) va the nang (/)(ri) tlnh dien dUdc cho bdi cac cdng thiic sau:

/(n)=f:^^^

(3.1)

va
N

^(^"'^)-E7'

(3.2)

trong dd TV la s6 lUdng c^c hat, rJ va QJ tUdng iing la vi trf va dien tfch ciia hat j ,
r la khoang each giiia hat i va j dUdc dinh nghia bdi cdng thiic r — \ / | 7 \ — fj\'^.
D l dang t h i y ring d budc 4, thuat toan tfnh luc tUdng tac tmh dien ddn gian
n h i t chfnh la thuc hien tfnh luc tUdng tac giiia timg cap hat. Do dd, chiing ta se
phai thuc hien N{N — l ) / 2 phep tfnh luc dua vao phUdng trinh (3.1). Ndi each khac,
thuat toan ddn gian nay (sau day gpi t i t la thuat todn tinh lUc trUc tiep) cd dp phiic
tap tfnh toan 0{N'^).
Trong cae budc tfnh toan neu tren, budc tfnh luc tUdng tac tren mdi hat (budc
4) la nhiem vu nang n l n h i t xet vl mat tfnh toan. Bdi vay cin phai ap dung, cai
dat cac thuat toan cd dp phiic tap tfnh toan 0{N) hoac 0{N log N) va/hoac sieu
may tfnh, may tfnh song song hoac may tfnh chuyen dung tdc dp cao d l thUc hien
nhiem vu nay.
D l tai cua chiing tdi cd hai nhiem vu chfnh.
Nhiem vu thii n h i t la nghien ciiu, tim hilu cac vin d l cd sd ciia tfnh toan hieu
nang cao va iing dung ciia tfnh toan hieu nang cao vao bai toan md phdng ddng
luc phan tii. Trong p h i n cd sd tfnh toan hieu nang cao, chiing tdi nghien ciiu, tim
hieu cac vin d l cd lien quan tdi cac kiln triic, mdi trudng va cdng cu tfnh toan hieu
nang cao: Cum may tfnh P C Linux va mdt sd p h i n ciing chuyen dung cd tdc dp cao
chuyen danh cho bai toan md phong TV-body hoac md phong ddng luc phan tii.
Nhiem vu thii hai la nghien ciiu mdt iing dung cu t h i cua tfnh toan hieu nang
cao. Chiing tdi nghien ciiu va d l x u i t phUdng phap mdi d l tang tdc dp tfnh luc tinh
dien x i p xi tren may tfnh chuyen dung song song MDGRAPE-2, n h i m tang tdc bai
toan md phong dpng lue phan tii.

3.2

Tong quan ve t m h l\ic t\idng tac n h a n h trong mo
phong dong \\ic phan tiJ

Nhiem vu nghien ciiu thii n h i t ciia dl tai la tim hilu vl cd sd tfnh toan hieu nang
cao: Cd sd ly luan, mdi trudng, cdng cu va pham vi iing dung cua tfnh toan hieu
nang cao. Chiing tdi da nghien ciiu, tlm hieu va vilt bao cao chuyen d l vl tdng


3. NQI DUNG CUA DE TAI

11

quan tfnh toan hieu nSng cao, he thong file song song PVFS va giao dien lap trinh
OpenMP (Xem cdc bao cdo chuyen dl trong phu luc).
Nhiem vu thiJ hai la nghien ciiu mpt iing dung cu t h i ciia tfnh toan hieu nang
cao trong bai todn md phdng dpng luc phan tii: tang tdc dp tfnh luc tUdng tac tlnh
dien. Tinh luc tUdng tac tinh dien trong bai toan md phong dpng luc phan tii la mdt
nhiem vu nang ne ve mat tfnh toan, ddi hoi thdi gian tfnh toan r i t ldn. Do dd, tang
tdc dp tfnh lUc la vin d l cd tfnh thdi sU. Vin d l nghien ciiu nay la su kit hdp giiia
khoa hpc may tfnh va vat ly/hda hpc. Cac hudng nghien ciiu chfnh d l tang tdc dp
tfnh lUc hien nay gdm cd:
1. Sli dung cac thuat toan "nhanh" cd dp phiic tap 0{N) hoac 0{N log TV), trong
dd TV la so lUdng cac hat trong bai toan md phong dpng luc phan tur,
2. Sli dung cac may tfnh chuyen dung cd toc dp tfnh luc r i t cao, vf du MDGRAPE
(toe dp tfnh luc nhanh hdn may tfnh khdngchuyen dung vdi cimg gia tiln tii
100-1000 lin) hoac MD-Engine.
3. K i t hdp cac hudng nghien ciiu tren.

Nhiem vu nghien ciiu iing dung ciia dl tai rdi vao hudng thii ba. Chiing tdi nghien ciiu
phUdng phap cai dat thuat toan khai triln da cue nhanh (fast multipole algorithm)
tren may tfnh chuyen dung MDGRAPE-2 va d l x u i t phUdng phap mdi d l tang tdc
dp tfnh lUc x i p xi.
Sau day chiing tdi se lin ludt trinh bay cac phUdng phap tang tdc dp tfnh toan
luc trong md phong ddng lUc phan til theo ea ba hudng nhu da neu tren.

3.2.1

Cac t h u a t t o a n nhanh trong linh v\lc m o phong d o n g liic phan

tit

'

Trong cac bai toan md phdng dpng luc phan tii cd diln (tiic la khdng cd cac tfnh
toan lUdng tii), cdng viec tfnh luc tUdng tac giiia cac hat chilm nhilu thdi gian n h i t
- khoang 95% tdng sd thdi gian chay chUdng trmh. Thuat toan tfnh luc ddn gian
n h i t dUdc gpi la thuat toan true tilp cd dp phiic tap 0{N'^). Nhu vay khi TV ldn, thdi
gian tfnh lUc se r i t ldn va cac md phdng ddng luc phan tii vdi hang trieu hoac hang
chuc trieu hat se ton r i t nhilu thdi gian, tham chf ngay ca khi sii dung GRAPE.
Do dd da ed nhilu nghien ciiu d l x u i t cac thuat toan vdi dp phiic tap 0(TV) hoac
0{N log TV) d l tfnh x i p xi luc vdi dp chfnh xac dilu khien dudc.
Nam 1985, A. Appel lin d i u tien d l x u i t thuat toan phan cip d l tfnh luc vdi
dp phiic tap O(TVlogTV) [3]. Dua tren kit qua cua A. Appel, nam 1986 P. Hut va
J. Barnes da phat triln thuat toan tree vdi dp phiic tap O(AMogTV) [5]. Thuat toan
nay nhanh chdng dude sii dung rpng rai trong md phong vat ly thien van do tfnh
ddn gian va hieu qua ciia nd. Nam 1987, L. Greengard va V. Rokhlin da phat triln
thuat toan khai triln da cue nhanh (fast multipole algorithm - FMM) d l tfnh luc
x i p xi trong khdng gian 2 chilu [19]. Day la mdt thuat toan r i t phiic tap, dac biet la

cac cdng thiic biin ddi khai triln da cue va khai trien Taylor va do dd, phai din 10
nam sau, nam 1997, phien ban 3 chieu d i u tien cua thuat toan mdi dUdc Greengard
va Rokhlin cdng bd [20]. Thuat toan FMM cd dd phiic tap tfnh toan 0{N). FMM


3. NOI DUNG CUA DE TAI

12

thudng dUdc suf dung cho cac bai t o i n md phong dpng lUc hpc phan tii eho cac he
cd so lUdng hat rat ldn. Sau day chiing tdi se trinh bay ky hdn vl thuat toan FMM
va cac biin t h i cua thuat toan nay.

3.2.2
3.2.2.1

Thuat toan FMM va cac b i i n t h i
Thuat toan FMM

FMM la thuat toan d i u tien cho phep tfnh luc tUdng tac tinh dien x i p xi cho bai
toan md phong dpng luc phan tuf vdi dp phiie tap tfnh toan 0{N). FMM trong
khdng gian hai chilu dUdc L. Greengard va V. Rokhlin phat triln nam 1987. Sau dd
thuat toan trong khdng gian ba chilu dUdc cdng bd vao nam 1997. Do FMM la mdt
thuat toan r i t phiic tap, ehiing tdi chi nhic lai y tudng chfnh cua FMM trong bao
cao nay. Md ta ehi tilt ciia thuat toan cd t h i tim trong cac bai bao ciia Greengard
va Rokhhn [19, 20].

M2L

Multipole expansion

Local expansion

Hinh 3.1: Y tudng chfnh ciia thuat toan tfnh luc FMM.
Chiing ta cin nhd lai la thuat toan tfnh lUc trUc tilp tfnh luc giiia timg cap hat,
ndi each khac thuat toan nay ed dp phiie tap tfnh toan O(TV^). Trong khi dd, y tudng
chfnh Clia thuat toan FMM la tfnh luc tUdng tac giua cac nhdm hat, sau dd tfnh x i p
xi lUc va t h i nang tren mdi hat bing each sii dung khai triln da cue va khai triln
Taylor. Cd 5 pha chfnh trong thuat toan FMM. Hinh 3.1 md ta y tudng cua FMM.
Trong dd, M2M, M2L va L2L la ba pha d i u tien cua thuat toan cd y nghla nhu sau.
M2M la biin ddi khai triln da euc-khai triln da cue, M2L la biin ddi khai triln da
cUc-khai trien Taylor va L2L la biin ddi khai trien Taylor-khai trien Taylor. Tiep
theo cac pha nay la pha tfnh luc tUdng tac "gin" va tfnh lUc tUdng tac "xa". Tfnh
luc tUdng tac g i n dUde thuc hien nhu thuat toan tfnh lUc true tilp. Luc tUdng tac
xa dudc thuc hien nhd viec liy dao ham rieng eua t h i nang dat dude tu* pha L2L.
Chiing ta ky hieu hai pha tfnh luc tUdng tac gin va xa tUdng iing la Fnear va FjarTrong 5 pha tfnh toan ndi tren, M2L, Fnear va Fjar la cac pha tfnh toan tdn thdi
gian n h i t .
Mac dil FMM dat dUdc dp phiic tap tfnh toan 0(N) nhung do tfnh chit plnic
tap ciia thuat toan n h i t la cac cdng thiic biin ddi trong cac pha M2M. M2L va L2L.


3. NOI DUNG CUA DE TAI

13

hi$u nang dat dUdc trong cai dat cua FMM chua cao. Thdi gian thue hien FMM chi
thuc su thap hdn thdi gian thuc hien thuat toan tfnh luc true tilp khi so hat TV kha
ldn, khoang 65535 trd len. Bdi vay, cd nhilu biin t h i ciia FMM n h i m lam giam su
phiic tap khi cai dat FMM de dat dUdc hieu nang eao hdn. Sau day la mdt sd biin
t h i diln hinh.

3.2.2.2

T h u a t t o a n cua A n d e r s o n

Anderson [1] da d l x u i t phUdng phap biin t h i ciia FMM. Uu dilm chfnh cua phUdng
phap nay la su ddn gian va hieu qua. Anderson khdng sii dung cae cdng thiic biin
ddi khai triln da cue va khai triln Taylor. Thay vao dd, dng da d l x u i t cac cdng
thiic mdi ddn gian hdn.
PhUdng phap ciia Anderson dudc dUa tren cdng thiic cua Poisson. Cdng thiie
Poisson cho phep giai bai toan gia tri bien ciia phUdng trinh Laplace. Chiing tdi tdm
tat phUdng phap ciia Poisson nhu dudi day, tren cd sd dd trinh bay phudng phap
ciia Anderson.
Cho trUdc t h i nang tren mat ciu ban kfnh a, khi dd gia tri t h i nang $ tai diem
f cd tpa dp ciu la f = (r, cf), 9) dUdc tfnh bdi cae cdng thiie:

n=0

vdi r > a va
00

^^^ = i^/^E(2^+1) ©"^" {-f) ^(^^"^^^

(^•'^)

vdi r < a.
Luu y chiing ta sii dung he tpa dp ciu trong hai cdng thiic (3.3) va (3.4). Trong
cac cdng thiic nay, ^{as) la gia tri t h i nang cho trUdc tren mat ciu. S la miln liy
tfch phan va d day S ehfnh la mat ciu ban kfnh 1 cd tam tai goc tpa dp (gpi t i t la
mat cdu dOn vi)\ P^ la da thiic Legendre.
D l cd t h i sii dung cac cdng thiic (3.3), (3.4) thay t h i cho cac cdng thiic ciia

khai trien da cue va khai trien Taylor, Anderson da d l x u i t phien ban "rdi rac" ciia
(3.3) va (3.4). Ong da rdi rac hda vl phai eiia cac cdng thiic tren b i n g each thay
tfch phan bing mpt tdng hiiu han va thay t h i miln S bing mdt tap hiiu han cac
dilm tren mat ciu. Tap dilm nay dUdc xac dinh dUa vao khai niem t-design cdu do
Hardin va Sloane d l x u i t [24]. Sau day la dinh nghla cua t-design ciu.
Mdt tap p = {Pi, ...,PK]
CO K dilm n i m tren mat ciu ddn vi Vt^ — S^~^ =
{x = (x], ...,Xd) ^ R^ : X • X ~ 1} dUde gpi la mdt t-design ciu nlu ddng n h i t thiic
K

^ '
/ /(x)dM(x)=^X]/(P,

(3.5)

dung cho moi da thiic / co bac < f (trong do IJ la do do d6ng nhat tren fi^ co t6ng
do do 1).


3. NQI DUNG CUA DE TAI

14

Cin chu y ring vdi t tong quat, ngudi ta vin chua bilt dudc tap dilm tdi uu cho
t-design ciu (tiic la tap dilm cd sd lUdng dilm K nho nhat). Tuy nhien b i n g thuc
nghi§m Hardin va Sloane vin cd t h i xae dinh dUde cae t-design ciu. Tpa dp ciia cac
t-design ciu do Hardin va Sloane tim ra ed t h i dUde tai xudng tai dia chi
h t t p : //www. r e s e a r c h . a t t . coin/''nj a s / s p h d e s i g n s / .

Khai trien ngoai

Gia tri the nang
Khai trien ngoai

Khai trien trong

Hinh 3.2: PhUdng phap ciia Anderson.
Sli dung t-design ciu, Anderson da dua ra hai cdng thiic rdi rac hda ciia cae cdng
thiic (3.3) and (3.4) nhu sau:
(3.6)
i=\ n=0

^

\

^

/

vdi r > a (khai tren ngodi) va
K

p

sv • r

^{asi)wi

(3.7)

1=1 n=0

vdi r < a (khai triln trong). 0 day Wi la cac trpng sd va p la cac sd hang khdng bi
c i t khi rdi rac hda. Sau day chiing ta gpi p la c i p khai trien. Hinh 3.2 minh hpa y
tudng phUdng phap ciia Anderson.
3.2.2.3

T h u a t t o a n cua M a k i n o

Makino [39] d l x u i t phUdng phap khai tnen da cUc gid hat (Pseudo-Particle Multipole Method - gpi t i t la P^M^hoac phUdng phap gia hat). P-^M^ sii dung cac cdng
thiic la biin the ciia cac cdng thiic khai trien da cue. Ldi fch cua phudng phap nay
la tinh ddn gian va hdn nu:a, cac cdng thiic cua P^M^ cd the dupe thuc hien tren
may tfnh chuyen dung GRAPE.


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TAI

Y tudng chinh ciia P^M^ la sii dung mdt sd ludng nhd eae gid hat d l bilu diln
cac khai triln da cue. Ndi each khac, phUdng phap nay cho ta each x i p xi t h i nang
gay ra do cac hat trong he bing t h i nang gay ra do cae gia hat, va sd lUdng cac
gia hat nhd hdn nhilu so vdi sd lUdng cac hat thUc. Mpt vf du ddn gian cd t h i thiy
dUde la cd t h i thay t h i 100 qua can 1kg b i n g hai qua can 50kg d l dat dUde eung
mdt trpng luc 100kg. Hinh 3.3 minh hpa y tudng phUdng phap P^M^ cua Makino.

L^

Gia hat
Hinh 3.3: PhUdng phap P^M^ cua Makino.
Y tudng ciia Makino kha gidng vdi y tudng cua Anderson. Ca hai phUdng phap
diu Sli dung so lUdng hiiu han cac dai lUdng rdi rac de x i p xi t h i nang do cac hat
gay ra. Dilm khac biet chfnh n i m d chd Anderson sii dung mdt sd hiiu han cac gid
tri thi ndng trong khi dd Makino sii dung mdt sd hiiu han cae gid hat
PhUdng phap P^^M"^ dUdc md ta nhu sau. Phan bd cua cac gia hat (hay vi trf
va dien tfch cua ehiing) phai dUde xac dinh sao cho phii hdp vdi eae he sd cua khai
triln da cue. Theo each tilp can ddn gian n h i t thi ehiing ta phai giai mdt he phUdng
trinh bing each nghich dao cdng thiic khai triln da cue. Cach tilp can nay chi thfch
hdp cho cdng thiie khai triln da cue cip p < 2 [30].
Tuy nhien n l u c i p khai triln p > 2 thi viec nghich dao cdng thiic khai trien da
cue la r i t khd. Trong trudng hdp nay, Makino da dUa ra mdt each giai ddn gian.
Ong da ed dinh vi trf cua cac gia hat theo t-desgin c i u va chi giai phudng trinh d l
tim dien tfch eua cac gia hat. Cach tilp can nay d i n din viec giai mOt he phUdng
trinh tuyIn tfnh mac dii so lUdng gia hat cd tang len. Do dd xet mdt each tdng the,
each tilp can eiia Makino da lam cho bai toan ddn gian di kha nhilu.
Vdi each tilp can neu tren, Makino da dua ra dupe cdng thiic khai triln ngoai
cho phUdng phap gia hat nhu sau:
N

2/-h 1

Qj'^Y.'^^Y.
7= 1

^=0

K

F/(cos 7, J

(3.8)

a

trong dd Qj la dien tfch ciia gia hat j . f^ — (r-i. 0.6] la vi tri cua gia hat ]. -jj L


3.

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TAI

gdc giu'a fi vk vector vi trf Rj ciia gia hat j . Chiing minh dl din tdi cdng thiic (3.8)
cd trong tai Heu [39].
3.2.3

M a y t i n h chuyen dung song song G R A P E va iJng dung

HOST
COMPUTER

Positions,
charges
GRAPE
Forces

Hinh 3.4: Kiln triic cd ban ciia mdt he may tfnh GRAPE,

.ijsoju ) l ' l \

Hinh 3.5: May tfnh MDGRAPE-2 (PCI) cd toc dp cue dai tUdng dUdng 48GFlops
vdi 4 chip MDGRAPE-2 (mdi chip cd tdc dp cue dai tUdng dUdng 16GFlops).
May tfnh chuyen dung song song GRAPE dUdc chl tao tai Vien nghien ciiu vat
ly va hda hpc Nhat Ban (RIKEN [59]) dl tfnh luc tUdng tac hoac thi nang giira cac
hat. Day la mdt may tfnh ed kiln triic da xu" ly pipeline. GRAPE khdng phai la mpt
may tfnh cd thi hoat dpng dpc lap. Dl sii dung GRAPE chiing ta cin mdt he may
tfnh gdm cd mpt may tfnh thdng thudng, vf du IBM-PC (sau day gpi la ma.y chu)
va mdt may tfnh GRAPE. Chiing ta gpi t i t he ma\" tfnh nay la he GRAPE.

3. Nd

17

DUNG CUA DE TAI

t^""'"'i';f'f(

a.oo.O'
O.Q'0.0

o.'a o o .
Hinh 3.6: May tfnh MDGRAPE-2 (Compact PCI) cd tdc dp cue dai tUdng dUdng
192GFlops vdi 16 chip MDGRAPE-2 (mdi chip ed toe dp cue dai tUdng dUdng
16GFlops).
Mdt he may tfnh G R A P E ddn gian n h i t bao gdm mdt may ehii va mdt may
G R A P E dUdc noi vdi nhau qua mpt dudng truyin ehing han bus PCI hoac Compact
PCI. G R A P E se thue hien toan bp eae tfnh toan l u e / t h l nang va may chii thue hien
t i t ca cac phep tfnh khac. Sd do khoi cua he may G R A P E dUdc md ta tren hinh
3.4. Hinh 3.5 va 3.6 minh hpa may tfnh chuyen dung MDGRAPEl-2 la mpt may tfnh
thupc hp G R A P E , phien ban eho bus PCI va Compact PCI.
D l tfnh lue/the nang, may ehu gui thdng tin vl vi trf va khdi lUdng (trong trudng
hdp tUdng tae h i p d i n ) hoac dien tfch (trong trudng hdp tUdng tac tlnh dien) ciia
cac hat d i n G R A P E , sau dd nhan kit qua tfnh toan l u c / t h l nang tra lai tu* GRAPE.
Tdc dp tfnh l u e / t h l nang eua GRAPE nhanh hdn cac may tinh thdng thudng cd cimg
gia tiln khoang 100-1000 lin. Vf du: may tfnh MDGRAPE-2 phien ban Compact
PCI cd toe dp tfnh lue tUdng dUdng vdi 192 GFlops. Sau day la md ta chi tilt vl
chiic nang cua G R A P E .
Chiic nang ed sd eua G R A P E la tfnh lue /(r*i) tac ddng len hat i tai vi trf f^, va
the nang (j){u) kit hdp vdi /(r^). Mae dii cd nhilu phien ban khac nhau cua GRAPE

vdi muc dfch iing dung khac nhau nhu md phdng dpng Iuc phan tii hay vat ly thien
van, nhung chiic nang cd sd ciia cac phien ban nay vin khdng thay ddi.
Luc /(r'l) va t h i nang (i){fi) dUde eho bdi cac cdng thiic:
N

m =E

9j

r, — r.

(3.9)

va.

0(n

E

(3.10)
OAI HOC QUOC GIA HA NQi
TRUNG TAN/1 THONG HN THU Vi^N

^

•

(

3. NQI DUNG CUA DE TAI

18

trong dd TV la s6 lUdng cac hat, fj va qj tUdng iing la vi trf va dien tich eua hat j , r^
la khoang each mem (danh cho cac ufng dung vat ly thien van) giiia hat i va j dUdc
dinh nghla bdi cong thiic Vs = %/'\fi — fj\'^ + e^ trong dd e la tham sd mim.
Dl tfnh luc /(fl), may chii cin gufi dii lieu cho GRAPE bao gom f,, fj, qj, e, va
A^. GRAPE tfnh luc f{fi) vdi mpi i sau dd giM kit qua tra lai may chii. Thi nang
(t>{fi) dUde tinh tUdng tu.
Doi vdi bai toan mo phdng dpng luc phan tuf, chiing ta khdng ed tham sd mim,
dilu dd cd nghia la € = 0. Khi dd cac phUdng trinh (3.9) va (3.10) trd thanh (3.1)
va (3.2).
Mdi may ehu cd thi lien kit vdi nhilu may GRAPE dl tang tdc dp tfnh luc
va cae he may GRAPE cd thi lien kit vdi nhau dl tao thanh mpt cum may tfnh
GRAPE vdi tdc dp tinh toan rit cao. Hinh 3.7, 3.8 va 3.9 minh hpa mdt cum may
tfnh GRAPE dUde dat ten la MDM [54] nhin tir mat trude, mat sau va ben trai.
Cum may tfnh nay cd tdc dp tUdng dUdng 78TFlops.

Hinh 3.7: Cum may tfnh GRAPE nhin tii mat trUdc [54]
Trong nam 2005, phien ban tilp theo cua MDGRAPE-2 la MDGRAPE-3 [61]
da dUdc che tao thii thanh cdng. Mdi chip MDGRAPE-3 ed tdc dp tUdng dUdng
165GFlops d tan so 250MHz va 200GFlops d tin so 300MHz (nhanh hdn MDGRAPE2 tii 10 din 12 lin). Thang 6/2006, cum may tfnh MDGRAPE-3 dimg dl md phdng
cac ling dung du doan ciu triic protein da dUdc hoan thanh va dUde dat ten la
Protein Explorer. Protein Explorer la may tfnh diu tien tren thi gidi dat tdc dp
tfnh toan vUdt ngudng 1 Petaflops vdi tdc dp 1.4PFlops. nhanh hdn may tfnh diing
dau trong Top500 [56] nam 2006 la IBM BlueGen khoang 3 lin. Tuy nhien Protein
Explorer la may tfnh chuyen dung nen khdng dUdc xip hang trong Top500.
Trong phin tilp theo chiing tdi trinh bay cac nghien ciiu va kit qua da cd vl
viee cai dat thuat toan nhanh tren cac phin ciing chuyen dung, tren cd sd dd thuc

hien nhiem vu nghien cxCn eiia dl tai.


3. NQI DUNG CUA DE TAI

19

Hinh 3.8: Cum may tfnh GRAPE nhin tu* mat sau [54]
3.2.4
3.2.4.1

Cai dat thuat t o a n nhanh tren p h 4 n ciing chuyen d u n g
Cai dat t h u a t t o a n tree tren phan ciing G R A P E

Nam 1999, J. Makino va A. Kawai da phat triln phUdng phap P^M'^va lin d i u
tien cai dat thuat toan tree tren may GRAPE, eho kit qua r i t tdt. Trong cac nam
2000-2004, J. Makino va A. Kawai da hen tuc cd nhiing phat triln mdi trong viec
tang tdc thuat toan tree. P h i n m i m cai dat thuat toan nay ciia hai tac gia nay da
tang tdc thuat toan tree mdt each dang kl. Vdi cac md phdng yeu ciu dp chfnh xac
t h i p , G R A P E tang tdc tree khoang 10 lin, va vdi cac md phdng yeu ciu dp ehfnh
xac cao, G R A P E tang tdc thuat toan tree tdi x i p xi 60 lin.
3.2.4.2

Cai dat t h u a t t o a n F M M tren p h i n ciing M D - E N G I N E

Nam 2003, T. Amisaki, S. Toyoda, H. Miyagawa va K. Kitamura da cai dat FMM
tren p h i n ciing tUdng tU nhu MDGRAPE-2: MD-ENGINE [2]. Tuy nhien kit qua
dat dUdc ehua hoan toan tdt. Dp tang tdc cua thuat toan FMM tren MD-ENGINE
kha han chl. Ly do chfnh la cae tac gia tren chi tang tdc dUdc phin tfnh luc trUc

3.

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20

Hinh 3.9: Cum may tfnh GRAPE nhin tii ben trai [54].
tilp Fnear eiia FMM ma khdng tang tdc dude pha M2L va pha tfnh luc xa Fjar hai trong ba pha tfnh toan tdn thdi gian n h i t ciia FMM (xem p h i n 3.2.2.1).

3.3
3.3.1

Noi dung va k i t qua nghien ciJu
C a c k h o k h a n c 4 n giai q u y e t

Nhiem vu nghien ciiu ciia ehiing tdi la tim each tang toc dp tfnh luc tUdng tac tinh
dien cho bai toan md phdng dpng luc phan tii. Nhiem vu nay n i m trong hudng
nghien ciiu thii 3 nhu da neu trong p h i n 3.2.
Trong thuat toan FMM, cac pha M2L, Fnear va Fjar la tdn thdi gian n h i t . Do
dd cin phai sii dung GRAPE dl tfnh toan n h i m tang tdc cac pha nay. Tfnh toan
eho pha Fnear tren GRAPE la hiin nhien vi pha nay sii dung cdng thiic (3.1). Nhu

vay d l tang tdc FMM ehiing tdi phai giai quylt cac nhiem vu sau day:
1. Tang tdc pha M2L. Viec tang tdc nay khdng d l dang nhu tang tdc pha Fnear
vi M2L sii dung cac cdng thiie biin ddi tii khai trien da cue sang khai triln
Taylor. T i t ca cac cdng thiic khai triln nay diu khdng t h i tfnh toan tren
GRAPE.
2. Tang toe pha F/ar- Trong thuat toan FMM gdc ciia Greengard va Rokhlin,
Ffar dude tfnh nhd liy dao ham rieng cua t h i nang dat dUdc trong pha L2L.
Cdng thiic tfnh lUc cho pha Fjar la / = -V^ vdi / la luc va $ la t h i nang.
Cdng thu'c nay ciing khdng t h i tfnh toan tren GRAPE. Tudng tu nhu vay,
cdng thu'c tfnh Fjar trong phUdng phap bien the ciia Anderson ciing khdng
t h i tfnh true tilp tren GRAPE. Makino da cd cdng thiic cd the tfnh khai trien
da cue tren GRAPE nhung chi ap dung dUde cho pha M2M va M2L cua FMM
(khi su" dung kit hdp vdi phUdng phap ciia Anderson).


21

3. NQI DUNG CUA DE TAI
3.3.2

Giai p h a p va k i t qua cua chung toi

D l giai quyet nhiem vu thii n h i t , chiing tdi ap dung phUdng phap P^M^ cua Makino
cho pha M2M, sau dd ap dung phUdng phap eiia Anderson eho pha M2L. Khi dd,
pha M2M dUdc tfnh toan tren may chii va pha M2L dUde tfnh toan tren GRAPE.
P h a tilp theo, L2L dUde ap dung nhd cdng thiie khai triln trong (3.4) cua Anderson.
P h a tfnh luc Ffar' Trong ban cai dat d i u tien ciia chiing tdi (gpi t i t la ban 1.0),
Ffar dUdc tfnh qua cdng thiic dao ham ciia t h i nang nhu sau [9]:
s, r VPn{u)
^

1=1 n=0 ^

w

.n-2

(2n + 1 ) — - ^ ( a 5 l ) u ; „
a'

(3.11

vdi u = Si • f/r.
Day la mpt cdng thiie phiie tap va khdng t h i tfnh tren GRAPE. Do dd hieu
nang eua thuat toan FMM tren GRAPE bi han chl. Do dd chiing tdi da tim each
tfnh pha Fjar tren may tfnh G R A P E t h i hien trong ban cai dat thii hai ciia chiing
tdi (gpi tat la ban 2.0) [10, 11]. D i u tien chiing tdi da tim ra mpt cdng thiic mdi
tUdng tu nhu cdng thiic khai triln trong ciia Anderson. Chiing tdi gpi day la cdng
thiic khai triln P^M^ trong:
N

Q.-E^^E
1=1

/=0

21^1
K

Pi{cosj,j).

(3.12)

Bang 3.1 liet ke chi tilt cae cac pha tfnh toan va cdng thiic tUdng iing dUdc chiing
tdi su* dung trong hai ban cai dat FMM. Cot thii ba trong bang la cae pha tfnh toan
ciia ban 1.0 va cdt thii tu la cac pha tfnh toan eua ban 2.0.
Bang 3.1: Cac pha tfnh toan va cac cdng thiic tUdng iing dUdc sii dung trong hai
ban cai dat FMM. Cac p h i n in dam dudc thUc hien tren may tfnh GRAPE
Thuat toan FMM [20] Ban 1.0 [9]
Ban 2.0 [10, 11]
M2M
M2L
L2L
^near

Ffar

P'^M^
trien da cue
P^M^
Tinh t o a n t h e nang
ddi
d o cac gia h a t gay r a
thiie M2L
PhUdng phap Anderson p2^j2
trien Taylor
T i n h t o a n lu'c d o c a c h a t gay r a
Tfnh toan
Cdng thiic (3.11)
T i n h t o a n lUc d o

khai trien Taylor
cac gia h a t gay r a
Khai
Biin
cdng
Khai

Sail dd, ap dung giai phap tfnh luc mdi nhd vao cdng thiic (3.12). chiing tdi da
tang tdc thuat toan FMM va dat dUdc cac ket qua kha quan the hien trong cac thuc
nghiem tren p h i n ciing MDGRAPE-2.
Chiing tdi da thilt lap hai he thdng may tfnh GRAPE. He thong thii nhit (goi
t i t la he I) cd mdt card MDGRAPE-2 phien ban Compact PCI (64 pipelines, tdc


3. NQI DUNG CUA DE TAI

22

10000 E
-4—'

C/3

c
O
u

1000 =

s

c
o
•f-H
• ^ - ^

o
cd

U

256K 512K
Number of particles A^
Hinh 3.10: So sanh thdi gian tfnh luc ciia thuat toan FMM va thuat toan tfnh luc
true tilp tren he I. Dudng cong cd cac hinh trdn the hien hieu nang ciia FMM tren
may tfnh MDGRAPE-2. Dudng eong vdi eae hinh ngii giac la hieu nang cua FMM
tren may chii. Cac hinh trdn va ngii giac td den la hieu nang tUdng iing vdi dp chfnh
xac cao {p = 5), khdng td den iing vdi dp chfnh xac thip {p = 1). Dudng cong khdng
cd ky hieu vdi net liln va net diit tUdng iing la hieu nang ciia thuat toan tfnh luc
true tilp tren MDGRAPE-2 va may chii.


23

3. NOI DUNG CUA DE TAI

10000 =
WD

o
o

1000 =

C
O

U

128K

256K

512K

IM

2M

Number of particles A^
Hinh 3.11: TUdng tU nhu hinh 3.10 nhung vdi he II.
dp cue dai tUdng dUdng 192GFlops) va mdt may chu Compaq DS20E (Alpha 21264,
667MHz). He thong thii hai (gpi t i t la he II) cd mpt card MDGRAPE-2 phien ban
PCI (16 pipelines, tdc dp cue dai tUdng dUdng 48GFlops) va mpt may chu Intel
Pentium 4 2.2GHz sii dung bo mach ehii Intel D850. Chiing tdi da thii nghiem thuat
toan FMM cai dat tren G R A P E vdi dp chfnh xac tfnh luc t h i p (cip khai trien p — I)
va cao (p == 5) vdi phan bd cac hat gin ddng n h i t trong khdi lap phudng. K i t qua
thuc nghiem dUde minh hpa tren cac hinh 3.10, 3.11, 3.12, 3.13 va bang 3.2. K i t qua
thuc nghiem tren he I cho tren cac hinh 3.10 va 3.12. K i t qua thue nghiem tren he
II eho tren cac hinh 3.11, 3.13 va bang 3.2.
Su: dung cdng thiic (3.12), ehiing tdi da tfnh toan dUde pha Ffar tren GRAPE.

Sli dung cdng thiic nay, ban cai dat 2.0 ciia chiing tdi cd hieu nang cao hdn ban
1.0 tu" 2 lin (vdi dp chfnh xac tfnh luc t h i p ~ 10~^) va 5 lin (vdi dp chfnh xac cao
~ 10"^). K i t qua thUc nghiem dUdc t h i hien tren hinh 3.14.
Chiing tdi da so sanh hieu nang ciia thuat toan FMM do ehiing tdi cai dat vdi
hieu nang ciia mdt ban cai dat khac do T. Wrankin thUe hien [49] (Distributed
Parallel Multipole Tree Algorithm - DPMTA ban 3.1.3 cd t h i tai xudng tir dia chi
h t t p : //www. ee . duke . edu/~wrankin/Dpmta/) tren he II. K i t qua so sanh dUdc cho
trong bang 3.3. Hieu nang cua FMM (cd GRAPE) do chiing tdi cai dat cao hdn
hieu nang ciia DPMTA khoang 10 lin, va t h i p hdn hieu nang cua DPMTA khoang
1.1-1.4 lin (nlu khdng cd GRAPE),


3. NOI DUNG CUA DE TAI

24

200

f

100

c
o
CJ

(0

G
O

" • * — '

CO

13
U

128K

256K

512K

IM

2M

4M

Number of particles A^
Hinh 3.12: So sanh thdi gian tfnh lUc cua thuat toan FMM va thuat toan tree tren
he I. Dudng cong cd cac hinh trdn thi hien hieu nang ciia FMM tren may tfnh
MDGRAPE-2. Dudng cong vdi cac hinh tam giac la hieu nang ciia thuat toan tree
tren MDGRAPE-2. Cae hinh trdn va tam giac td den la hieu nang tUdng iing vdi
dp ehfnh xae eao {p = 5), khdng td den iing vdi dp chfnh xac thip (p — 1).