T,ii> chi Till hue vi\ l)i(''<i kliicn hoc, T.27, S.2 (2011), 176-186
NHLTNG THU* NGHIEM TAO CAC DAY Tl/A-NcAu-NHIEN HAI CHIEU
VII IIOAl CHir(TNG
\',ni Conij iiiihf thong Im. \'ien Khoa hoc vu Co,,'/ nghe \',el Nam
Abstract. Hiilliui/Iliiiiiiiierslrv M'qiieuccs. liic iiKwl well-known and widely u«ed multi-dimensional quasirandom sciiiicuees, air eoustiuctfd Itv combining van der ('oi]nit sequences with on another or with k/N sequence. \\V allciu])!. Io generate two-dimensional scquciicfs by combining two simpleone- dinien»ioiial scqiicnre.s (A/.V, mid-point, V. van der f'oipiil, Rlihtniyer, i)sendorandom seqiic;nces) and expcTiineiilailv receive iiiaiiy luwcsl discicpaiicv sequences.
Tom tat. Ciic d.iy Hiiltoii/HammeisJcy. nhrrng (i;iy tira-ngrUi-nhien nhfcii chieu quen thuoc nhat va thong dung nhai. dirge tao thanh bang catli ket liap car day van der Corput \i'n nhau lioac vai day k/N. Chung toi IJnV lau ra cac day bai chieu khac bang each ket litrp hai day mot chieu don gian (day k/X. day trung diem, cac da> \'. cac da\ van der Corput. cac day Richtmyer. cac day gia-ngau-nhien 7?) va bang tlitrc nghiem dii thu dirge nhieu liiiy tira-ngau-nhien co do phan ky thap han.
1. M d T D A U
Bai bao uav mof rong [13] sang tnrang ho-]) 2 chieu. Ta sir dung 21 day mot chieu bao gom 5 day each deu, 12 day van der Corput (ky hi(}u la \(1C), J day Richtmyer va 2 day gi^-ngan-nhien. xMoi day dugc ket hap v(Vi 2:5 day khac dc tao thanh day 2 chieu ds{i,j).
Thong thuang ds{i.j} 7^ d.'i{j, i). Nhir va\- co 2 1 x 23 = 552 day hai clueu dirac k h ^ sat.
Vie< ket hgp cac day nao vai nhiui de dat dirge do pli.ui ky thap nliat la van de co y nghia thuc nghiem han la Iv thuyet
Hai day vdC khac nhau ket hgp lai thaiili mot day Halton 2 c-hicu: nhu vay 12 day vdC tao thanh 12 x 11 = 132 dav Hahon. Day hmi ban k/N ket hap vai day \dC' tao thanh mot t;ip dicmi Hanimcisley Hai day hiru him {k - l)/N va {k - 0. 5)/A' ket hap \-ai 12 day vdC tao thanh 24 tap diem tu-a-Hammerslcy. Dei la ulurng d.iy tira-ngau-nhien quen thuoc, se lam ca sd- de so sanh vai cac day thi'r nghiem do chung toi de xiiat.
Cac tinh toan cu the da cho phep tim du-ac mgt so cla\- tira-ngau-nhien dan gi^n ma lai phan bo deu han tfic day Halton va cac tap diem tna-Hammersley noi tren.
Ket quk thil- nghiem cho thay: cac day tira-ngau-nhien hai chieu co do phan ky thap la su- ket hop Clia mot (va chi mgt) day each deu vai mot day tira-ngau-nhien khac. Neu ket hap hai day each deu vai nhau ta se thu dugc day tira-ngau-nhien hai chieu rat kem, khong the chap nhan dugc.
NH&NG THri- NGHIEM TAO OAC DAY TVA-NGAU-NHIEN HAI CHIEU 177 2. C A C K H A I N I E M C O B A N , C A C N G U Y E N T A C C H U N G
2.1. C a c d a y tira-ngSu-nhien
Phucmg phap Monte Carlo la phirong phap xac suat dua tren viec s i dung cac so ngau nhien (random numbers) hoae giii-ngau-nhien (pseudorandom numbers). Bai bao dau tien ve phuong phap nay la ciia N. Metropolis (1915-1999) va S. Ulam (1909-1984). eong bo nam 1949 ([5]).
Sau do chl 2 nam, b i n sao tat dinh ciia phuong phap Monte Carlo ra doi nho c4c nhi so luan. Ten goi phuong phap tua-Monte-Carlo (quasi-Monte Carlo methods) du-gc dung den lan dau tien trong mpt bio cao nghien ciru vio nam 1951 (|7]) ciia R D. Richtmyer (My, 1910- 2003).
Y tucimg Clia phuong phap tira-Monte-Carlo la tliay the cac day so ngau nhien hoae gii- ngau-nhien bang cac day so tat dinh (hoan toan khong-ngau-nhien) nhung phan bo rat deu.
Do la cac day tua-iigau-nhien (quasuandom) hoae can-ngau-nhien (subrandom) hay cfing dugc ggi la cac day co do phan ky thap (low-discrepancy sequences). Khac biet chii yen ciia phuong phap tira-Monte-Cailo la dirt khoat khong bat chuoc tinh ngau nhien, ma tang do chinh xac va giim thoi gian tinh toan bang each dac biet chii trgng tinh phan bo deu [4,10).
XJoe lugng sai so ciia phuong phap Monte Carlo co tinh chat xac suat, con ciia phuang phap tua-Monte-Carlo la tat dinh.
Cac day so tira-ngau-nhien dang ke nhat gan lien voi ten cac nha toan hoc J. van der Corput (Ha Lan, 1890-1975), J. H. Halton (My), J. M. Hammersley (Anh, 1920-2004), I. M.
Sobol (Nga, 1926- ), H. Niederreiter (Ao, 1944- ), va H. Faure (Phap).
Cac day tira-ngau-nhien phan bo deu hon cac day giirngau-nhien eho nen dai dien tot hon cho U[0,1]. TYong mgt so truong hgp chiing cho phep cii tien hieu q u i tinh toin: co the tang toe do hgi tu dac trung ciia phuong phip Monte Carlo la 0(l/iV^^^) len den gan 0 ( l / i V ^ ^ ) , trong do JV la so diem tinh toan. Cac ky thuat giim phuong sai [3] ehi eo the tac dgng den hang so an trong 0{N~^^^) chir khong the dat toi bucre nhiy dai nhu the.
Dieu dac biet la cac so giirngau-nhien do cac nha thong ke dua ra, con cic day tua-ngau- nhien lai do cac nha so luan. Cic eong eu de phat trien va phan tich cic day tira-ngau-nhien khic han vai phuang phip Monte Carlo, vi chiing gan vai ly thuyet so va dai so truu tugng hon la ly thuyet xac suat va thong ke toan hgc. Viec doi co so, sii dung cic tinh chat ciia so nguyen to, cae he so Clia da thire nguyen thu]^, v.v. chfnh la cic khii niem ciia ly thuyet so [6].
Cac day tira-ngau-nhien la hoan toan xac dinh, cho nen can phdi c6 mot d9. do su phan tan, tuang t y nhir phuang sai ciia cac so ngau nhien. Do chi'nh la do phan ky (Discrepancy).
Cac khai niem, dinh ly va each tinh do phan ky duac trinh bay can ke trong [13] cho nen bai nay khong nhac lai.
2.2. D a y van d e r C o r p u t (vdC)
Day la day so phan bo deu trong khodng (0,1), do nha toan hoc Ha Lan Johajines
178 VUHOAl CHirONG
Gualtherus van der Corput, de xuat vhn nfim 1935. Day sd nay lap day doan [0,1) mot each dieu hoa vai do phan ky nh6. Cd the coi nd \k day tuorngau-nhien mot chieu quan trong nhat, dep nhat, la yen \6 then chot ciia nhieu can tnic da chieu va Ik ca s6f de di tdi cac day tira-ngau-nhien quen thuoc khac nhu Halton, Faurc v^ Sobol.
Lay mot so nguyen to p > 1 l&,m ca so. Moi so nguyen duang k deu co phep bieu dien duy nhat theo ca so p
«;=^«,,(*0P'',
trong do aj{k) € Fp = {0,1.2, ...,p— 1} va aj{k) = 0 khi k k h i Icrn, tire la tong ndi tren hiru han.
Moi gii tri k se dugc anh xa vao mgt diem trong [0,1) bang cich dio ngugc cac he so aj{k) va tao nen phan so 0,aoaia2... (trong he dem ca so p). Chinh xic hon
.^p(*:)=53oj(«:)p-J-', *:>0.
J>1
^p{k) dugc ggi la ham ngugc goc (radical inverse function). Day van der Corput ca so p chinh la day X = {^^{0) = 0, ~pp{l), <pp{2), ^p{3),...}.
IVong [14] chiing toi trinh bay ti mi hon ve day so nay.
2.3. D a y H a l t o n (vo h a n )
Day Halton la sir md rgng day vdC ra khong gian nhieu chieu. Day Halton chuan 5 chieu dugc ghep lai bdi s day vdC mgt chieu vai cic co so nguyen to khac nhau (thuang la s so nguyen to dau tien). Ggi pi,P2, --..Pa la cic so nguyen to Cling nhau thi day Halton s chieu la day vo han Xo,Xi,...
Xk = {'Ppl{k),'Pp2{k) ipp.{k)), Vk > 0, trong do ipp = vdC{p) = vdCp la day van der Corput voi ca so p.
Vi du, voi pi = 2 va P2 = 3 thi
¥>2 = vdC2 = ( 1 / 2 , 1 / 4 , 3 / 4 , 1 / 8 , 5/8,3/8,7/8,1/16,9/16,5/16,13/16,3/16...},
<P3 = vdCa = (1/3,2/3,1/9,4/9,7/9,2/9,5/9,8/9,1/27,10/27,19/27,4/27...}.
Day Halton (2,3) se la
{V2,<P3) = {1/2,1/3), (1/4,2/3),(3/4,1/9),(1/8,4/9), (5/8, 7/9),(3/8,2/9),(7/8,5/9), ' J (1/16,8/9)...}.
Cac gia tri dau tien ciia cac day thanh phSn ((^,) phu thugc lan nhau. Sir phu thugc nay se triet tieu tai cuoi chu ky thii nhat ciia m6i day Ket q u i quan sat thirc te cho thay: khi tich phan nhieu chieu, 150 diem Halton tuang duong vai 500 diem giirngau-nhien. Tiiy vay
NHtTNG THU NGHIEM TAO c A c DAY TUA-NGAU-NMIEN HAI CUU-Ai 17!) SO chieu cang cao thi ti'nh deu ciia day Halton cang giam. Khi do uguai ta dung ciic da.y pha tron (scrambled sequences) de giM (|uycl \'aii dc nav.
Day Halton dat dugc tmh doc lap liem ian llioiig qua \-iec su dung cac so nguyon to khac nhau trong tirng chieu khac nhau.
2.4. C a c t a p d i e m H a m m e r s l e y (hiiu hau)
Cac tap diem HanuiK'r.slev la sir ciii liieii cua (iri\- Halton. Do la day hfru han -s C'liieu co N phan t u co dinh duac dinh uglua gan gicnig day Halton, chl khac a chieu dau tien
X, ={A-/.V.^-,,|(A-)...V,(A') fp^,^i){k)). A-= 0.1 ; V - 1
\'i du, vai p = 3 va .V = 11 thi tap diem Hanniiinei>-1<'\- NC la
(fc/11, ^ - 0 - { ( 0 . 1 / 3 ) . ( I / l l . 2-•:{), (2/11.1/9). (;{/l 1.4/!)). (4/11. 7/9). (5/11, 2/9), (6/11. 5/9).
(7/11.8/9). (8/11.1/27), (9/11,10/27), (10/11,19/27)}.
2.5. Day R i c h t m y e r {k.a}
Cho a = {oi\.Q2 «*•) la t.ip cac so vo li va l,Qi,a2, -,ws doc lap tuyen tmh. Day {k.a] dugc cho duoi dang
A'A = {{k.ai}, {fc.Q2},.... {k c i j ) , fc > 0, trong do {x} bieu thi phan thap phan cua sd x.
Nguoi ta thucmg chgn n, la can cua ca( so nguyen to. Trong [13] chiing toi da chon thu nghiem OLI la cac hang sd toan hgc quen biet nhu TT, AU {t^ le vang), e, C (hang so Euler-Mascheroni). G (hang so Catalan) va thu dugc ket qua kha tdt.
2.6. D a y so V
Day sd V la day tua-ngau-nhien dua tren cac' dang thuc giua moment mau vk moment ly thuyet [12,13]. Khi can tao ra N diem phan hd deu thi tach A^ thanh tong ciia h sd hang
N = Ki ^K2 + :. + Kh sao cho cac K^ khac nhau.
Tren doan (0,1) cac diem each deu nhau mgt khoang u; diem dau [Xi) each diem 0 va diem cuoi {XK) each diem 1 mgt khodng T
X^=T,X2 = T^U,X^ = T + 2U,...,XK = T + {K-1).U^\-T,
trong do T = (1 ^ 7 ^ ^ ) / 2 va u = (1 - 2T)/{K - 1) = \/y/{K-1){K+ 1).
180 VU iioAi cMiiraNG
3. C A C D A Y SO S O S A N H T H t > N G H I E M
Ca so de" ticni liaiili so s;inli tliir nghieni In lap thiy tlianh pliati gom 21 day mgt chieu vai 2 159 phiin Iu {2 1.5i! la sn iiguveii hi), trcmg dd ed 5 day cath den (ky hieu la a), 12 day vdC (ky hieu la b), 5 day Richtmyer (ky Iiieu la c) vk 2 day giA-ngail-nhien (ky hieu la R). Ky hic'U k„.ki,.k,..ki{ la sd dav trong tirng nhdm. Moi day trong lap nay ducrc ket hgp vdi 23 (lay khac dc tao thanh day 2 eliieu d,s(/,,/), 1 < i < 24, ] < j < 2-1, ( ^ j . Thong thuong ds{,.j) ^ d<<{J. i). Miir va\' huij; eong ed 24 x 23 = 552 dav hai chieu ds{i.j). trong dd cd 132 day Haltcm va 2 i la|) diem tmirHammersley de so sanh.
a) Cdl ddy nieh deu {h„ = 5) (1). •iav(A--l)/iV.
{2}: .iiiy («.•-(!. .'•,)/,Y
{3}: diiv I'l gom mgt doan duy nliat A^ = 2459.
{4}: day 1:., do 17 rtonii hgp tliiinh v.Vi do diii l i c doan la 141. .".'). 2(i.-). 35, m. 79. 32, 38.
Ki. 264. 33, s:!. 279. .'ill. 2S, Sil, 97.
{5}: liav i:, do 22 doan hop thanh vai do dai c i r doan la 21. 415. 2.!. 722. 64. 9. 2r,. 66, 424. 14. 10. 2, 29, 8, 88, 12. 5, 3, 311, 92. 93. 20.
b) Cdc ddy so van der Corj)Ut {ki, = 12)
Tir {6} den {17}: cac day so vdC vm co so la 12 so iigiiyen to dau tien: {2. 3. 5. 7. 11.
13. 17, 19, 23, 29, 31, 37}.
c) Cdc ddy Richtmyer {k.a}, {k, = 5)
{18}: day .so {k.a} voi Q = hang so Euler = c = (l..')772156G5.
{19}' day so {k ,>} voi n = r = 2, 718281828.
{20). day so {k.a} vaia = n = 3. 141592654.
{21}: diy so {A:.Q} vai a-/2= 1.414213562.
{22}. day so {k.a} vai n = ln2 = 0,693147181.
d) Cdc ddy so gid-ngdu-nhien R, {kji = 5)
{2;j}.{24} cic day so gii-ngau-nhien R, do miy tinh tao ra bang ham Random ciia Tiirbo Paacal. Nham tang tinh ngau nhien ta dat thu tuc Randomize 6 dau chuang trinh de khiii dgng bo tao so voi mgt gia tri ngau nhien.
4. C A C TIEU C H U A N SO S A N H THl)" N G H I E M
Viec tinh toin thii nghiem cic day tua-ngau-nhien hai chieu ds{i,j) dugc tien hanh theo 5 tieu chuan dudi day. De tien so sinh, moi chi so deu dugc chia cho gii tri tuong irng ciia day so (1,6), ky hieu la H2, voi X = {k - 1)/Af (day each deu) va y = iP2{k) (day vdC nhi phan). Nhu vay tat c i cic chl so ciia ds{l, 6) deu bang 1.
Trong he thuc tren, neu thay {k - 1)/Af bang kIN thi t a co day Hammersley dau tien.
Do vay cd the ggi ds{l, 6) la day tira-Hammersley.
NHCTNG T H I J N G H I E M T A O C A C D A Y T L T A - N G A U - N H I E N H A I CIIIKU 181
a) Dp phdn ky D
Day la tieu chi quan trgng uliat doi vai cac d.iv sd tua-ngau-nhien, Viee tinh toan dua vao dinh nghia va cac edng thue (;uen biet [G, 13].
b) He so tuang quan tuyen tinh R giua cdc qid In .r vd y He sd nay cang nho thi chat luc/ng cua dav sd cang civn.
c) Phep thli khi binh phucrjig \"(23. 7)
Ky hien \ - ( 2 3 . 7) cd nglna la doan [0.1] tren true A' duc/c' chia thanh 23 khoang bang nhau, va doan [0.1] tren true V dirc/c chia la Itun 7 khoang hang nhau.
d) Sat so tfnh ti'eh phdn (I)
CAv tich phau dcni gian thutrug de tinh toan vdi do cln'nli xac cao, cho nen kho so s;inh.
Ta chgn mgt tich phan kha pln're tap tir dong nhat thue
l / r ( ; ) =- [" cos{tgT' zT)cos'-'^TdT, a trang 151 cua [I], trong do r{z) la ham gamma.
Gia tri chuih xac cua tich phan m\v la Ti:/{e.r{z)), pirn thugc z. Cac tri so gan diing ciia tich phan dugc tinh bang phuang phap Monte Carlo hinh hgc [3, 11] vdi moi day sd ds{i,j) tai 4 gia tri z sau da\-
zi - 19/11, 22 - 405/91, 23 = 20/9. 24 - 21/4.
e) Sal so lim cue tri cda mpt hdm so (S)
TVong hlnh vuong dan vi (0 < x < 1. 0 < y < 1) ta xet ham so / ( x . y) = 10(.T + 0,2 -f |x - 0,8| - Ix - 0,2|) +\y-0,5\.
Theo f[9j. trang 31), gia tri ldn nhat va nho nhat ciia ham sd nay bang m a x / ( x , y) = 10,5 va lain f{x,y) = 4.
Ta tim cac gia tri cue dai fmax{i,j) va cue tieu fmm{i,j) ciia ham sd tren bang each tun kiem he thdng [8] vai cac dav sd ds{i,j) va tinh
S{i.j) = | / m a x ( i , j ) - m a x / ( x , y ) l -I- |/min(z, j ) - min/(x,y)l
= \fmax{i,i)-lQ,b\-\-\fmm{i,j)-4\.
Sau do xep hang theo S = S{i,j)/S{l,6) tang dan.
6. K E T Q U i . SO S A N H
B ^ g 1 ghi 24 day co chi sd tdt nhat va kem nhat theo 2 tieu chuan dau tien. Cac gia tri D* (dg-phan-ky-sao) va R (he sd tuang quau tuyen tinh) cd tinh ddi xung vai i vk j:Dx (i, j ) = Dx {j,i) va R{i,j) = R{j,i), cho nen moi chi so cd 552/2 = 276 cap gia tri bang nhau tung ddi mgt. B^ng 2 ghi 12 day co chi sd tot nhat va keih nhat theo 3 tieu chuan cudi.
VII I I G A I ( ' l l i r ( J N < ;
7. T O N G n a p , N H A N X E T V A K E T L U A N hi so danh chiu plia>' deu \k
nen ta ed tiie (full (hi sd tniij; hc/p 1 qiuui IrouK nliat <(') lie sd 2
i linrrig doi so vcVi da\' sd (l.G) = da\' H2, cho la'v I runt; binh ciia cliiiug, trong dd do phan ky
Z = {2D' + /?' + 2' + / ' -f- S')/G.
('a< gia Iri l(il nhat va knn iilial ciia Z dugc ghi Irong liai Bang 3 va 4, liuiHj 1. Do phiiii kv D* va lit; sd tuang cjuan liiveii ti'idi R
{n = day cadi deii. b = rlAy vdC. bb = day Halton), c = <\i\v Richlniyer. R - dav tna-ugan-nhicn)
riii'r lir
\--l :i-l r,-{i 7-8 9-10 11-12 13-14 15-16 17-18 19-20 21-22 2:}-24 Thir t u 23-24 21-22 19-2n 17-18 1.5-Ifi 13-H 11-12 y-10 7-X 1-G :i-4 1-2
I)*(i.|]/I)*ll2 CVK'fiiahi ().K.''..1I)32 (J, 8.54 (){)() 0 . 9 0 4 1 7 7 0 . 9 0 4 1 7 7 0,98.S4.'')I 1 1 . 0 8 8 9 9 8 1 , 0 8 9 0 1 5 1 , 1 1 6 3 0 0 1 , 1 1 6 3 6 6 1 , 2 1 4 4 3 7 1 , 2 4 4 9 9 1 C a o gni U i kciii 11 h a t 2 3 , 1 4 1 1 0 0 2 0 . 9 18091 30,2r)!fr,81
.'•)7,8!J.''>rj(iO .'i7,y7.''>9(ir) (i7.9')277X 0 7 . 9 9 2 7 K 7 fi8,OI2(jnJ H J . I . ' I ' d ^ l l Hib.iirrr.i i9(iii7r>M
'^(i.i)
1,1.0). (0..{|
(2.0) (0 21 (2.21). (21 2) (.i21) (21.3) (1 21] (21 1) (1.0) (0,1) (3.19). (19,3) (2.19). (19.2) (2,22), (22,2) (3,22), (22.3) (1,19). (19,1) (1.22). (22.1)
(23.21). (2].2;i) ( I ' t J I ) 121.19) (l.^>) (-..1) {•l.h). ( 5 . 2 ) (.i..-.). (:..3) ( 1 . 5 ) . ( 5 . 1 ) (3.-I). ( 1 , 3 ) ( 2 . 1 ) . ( 1 . 2 ) ( l . - l ) . ( 1 , 1 ) 12 31 ( 3 . 2 ) ( 1 . 2 ) ( 2 . 1 ) 1 1 3 ) . ( 3 . 1 )
D J U . K
a h Im .il>, l>a ilC, Cil a<. c.i a i \ (.1 ah, ha ac, ca ac, ca ac. (.a ac, fa ac, ca ac, c.i
RR, HR
<R. Rr aa, aa aa, aa aa, aa aa, aa aa, fu\
aa, HA aa, aa iui. jui aa, aa aa, aa
U = R(i,,|)/RH2
(l.OOO.'iOT O.ODSIIO 0.009271 0.02080S 0.0315211 0,047548 0,051301 0,05.5610 0.0710.59 0.0802X0 0.098 i:>2 0.0'I,S152
dh(i,j)
(6.14). (14,6) (7.11), (11.7) (16.22), (22.10) (7.12). (12 7) (7.13) II (.7) MI,2ll| (20,11) llD.lt,) (H..10) IS 2i) I2(S) 1 I 21) (21.1) l.i IS) (18.3) (J.IM. (l.-*.2)
i
19.022192 I9.8S!.".(}_»
23.689223 70.658153 70,658153 70,658153 71.934054 71.934054 71,934054 431.982440 431.982440 431.982440
1 I'l JII (21,I'D i-M.Jll 121.21) 117.211.(21.17) (1 1). (1.1) (2 ll (1.2) (3.1). 11.3) (1.51. (5.1) (3.5). (5.3) (2.:,). (5.2) (1.3). (3.1) (12) (2.1) (2 3) (3.2)
Dang
bb, bb bb, bb he, ch hb. bb hb, bb Ixr.cb bb. I>b bR. Rb ac, ca bb, bb ac, ca ac, ca
cR. Re LR, RC bR, Rb aa, aa aa, aa aa, aa aa, aa aa, aa aa, aa aa. aa aa, aa aa, aa
NHUNG THIJ NGHIEM TAO CAC DAY TlJA-NGAU-NHnJN HAI CHIEU
Thutu- 1 2 3 4 5 6 7 8 9 10 11 12 Thir tu tir durcri len 12 11 10 9 8 7 6 5 4 3 2 1
Bdng S. Gia X'' = x'/x'H2 Cac gia tri tot nhat 0,275707 0,292952 0,292952 0,292952 0,292952 0.292952 0,292952 0,327442 0,327442 0.327442 0.413G68 0.430913 Cac gia tri kem nhat 570,501285 573,519173 573,519173 678.869323 678.869323 686.284704 1717,160608 1717,160668 1717,160668 1717,160668 1719,540488 1719,540488
tri x'^, sai sd tich phan / ' ds(i,))
(14.9) bh (2.9) ab (3.9) ah (1.9) ah (14.2) ha ( N . , i ) h a (11.1) ba (2.20) ac (3.20) ac (1.20) ac (18.1) ca (18.2) ca
(1.4) aa (2.4} aa (3,4) aa (2,5) aa (3.5) aa (1,5) aa (2,1) aa (3,1) aa (1,2) aa (1.3) aa (2,3) aa (3,2) aa
l' = I{i.j)/ln2
0,180057 0,313200 0,403425 0,403425 0.161154 0.173574 0.176')78 0,17()!I78 0.185855 0.189.5.J0 0.510iHO 0,550497
11,399076 12,162913 13,948491 14,007590 14,007590 15,217713 15,217713 15,532020 15,911344 15,911344 15,989001 41,581399
va sai sd tim cue tri S' Mh})
(21.1) ra (22.1) <a (21,3) ca.
(21,2) ca (17,18) be (9,10) bb (22,2) ca (22,3) ca (17,19) be (14,7) bb (19,17) ch (19,9) ch
(24,23) RR (24,5) Ra (4,1) aa (4,3) aa (4,2) aa (2,4) aa (3,4) aa (1,4) aa (5,3) aa (5,2) aa (5,1) aa (1,3) aa
S'=S{,.,}/S„2
0,074284 0,087509 0,123449 0,176031 0,180517 0,180517 0,186562 0,238208 0,249297 0,251370 0,274945 0,300666
2,059233 2,059534 2,064020 2.376522 2,376522 2,376522 5,538212 5,538312 5,538312 5,538317 5,542798 5,542804
d«(i,j)
(8,4) ba (7,8) bb (21,19) cc (8,1) ba (8,3} ba (8,2) ba (17,19) be (6,5) ba (15,19) be (21,7) cb (6,15) he (19,12) ch
(17,3) ba (17,2) ba (17,1} ba (16,3) ba (16,2) ba (16,1) ba (2,1) aa (1,2) aa (1,3) aa (3,1) aa (2,3) aa
Bang 3 cho thay ca 24 day CO chi so tong hop tot nhat deu co mot (va chi mot) thanh phan thuoc nhom a. TVong khi do 20/24 day trong bing 4 co dang aa Dieu do co nghia la cac day each deu {a} co tam quan trpng quyet dinh. Neu ket hop day nay vai mot day vdC {6} hoae mot day Richtmyer {c} se duroc day tua-ngau-nhien hai chieu co do phan ky thap.
Nhu-ng neu ket hap hai day each deu vdi nhau tM hau q u i la rat toi te.
Nhan xet thii hai la, trong so cac day tot nhat, ty le cac day Halton (dang 66, 6 < i, j < 17) va cac day tua-Hammersley (1 < i < 2, 6 < j < 17) kha nh6. Nhu vay qua thu nghiem, chiing toi da tim duoc kha nhieu day tirarngau-nhien men tot han va sau nay co the dimg chiing trong cac thuat toan tya-Monte-Carlo.
Ttong 4 bAng tren co ghi 108 day tot nhat va. 108 day kem nhat theo 6 tieu chuan khac nhau: 24-24 day theo do phan ky fl* va he so tuong quan tuyen tinh R, 12-12-12 day theo gia tri x ^ sai so tich phan I va sai so tim ciic tri S, 24 day theo chi so tong hcjp Z. Bing 5 duoi day thong ke Ciic dang theo s6 tuyet doi (B, D) va t^ le tuang doi (C, E). Cac day dang aa, bb, cc, RR khong the cd 2 thanh phan nhu nhau, cho nen so day 2 chieu dang nay la
>'x{ki - 1), trong dd ki la so day trong nhdm.
vu noAi (iiurcjNG
n
Tlii'r t i r 1 2 3 4 ,1 li 7
s
!)
II) II r j 1.4 14 1.') IG 1 7 1 8 19 2 0 2 1 2 2 2 3 2 4my ./. (Vl Z
0,7I.VI!II 0.71.1491;
0 . 7 1 0 1 7 7 0 , 7 1 0 1 7 7 l),71HHI2 i l . 7 ( i l : ( 0 1 0,H.lti4H2 ().8409.^>3 0 , 8 . | ! ) ! K ; ( I 0 , 8 0 0 9 0 . 1 0 , 8 9 2 I I 7 0 . 9 0 0 7 7 . 1 0 , 9 0 0 7 8 0 0 , 9 1 1380 0 , 9 1 8 2 1 2 i),9,T2238 0.9.1224.1 0 . 9 0 1 2 1 0 0 , 9 0 8 6 8 9 0 , 9 7 3 0 5 7 0 , 9 7 3 9 2 9 1 1 , 0 1 6 7 1 8 1 , 0 3 8 4 6 7
• e h i s n ( i . i ) ( 2 2 . 2 1 ( 2 2 . 3 ) ( 2 1 . 2 ) ( 2 1 . 3 ) ( 2 1 , 1 ) ( 2 2 . 1 ) ( 3 , 2 1 ) ( 1 9 . 3 ) ( 1 9 , 2 ) ( 2 , 2 1 ) ( 1 , 2 1 ) ( 6 . 3 ) ( 6 . 2 ) ( 1 9 , 1 ) ( 2 . 0 ) ( 3 . 1 9 ) ( 2 , 1 9 ) ( 6 , 1 ) ( 3 , 6 ) ( 3 , 2 2 ) ( 2 , 2 2 ) ( 1 , 6 ) ( 1 . 1 0 ) ( 1 , 2 2 )
l o ' l l d , 1 D t i i i g cu c u c a c a CII CH I l C c a c a a c a c b a b a c a a b a c a c b a a b a c a c a b a c a r
r]) Z tiit 1
I M v ( i ) ( k , l M 2 1 ( k li,21 ( k , ( ( k . )
| k I ( k l i i 2 ) V I
{k.al {k.o) (k-o.n)/N
( k - l ) / N v i l C ( 2 ) v d C ( 2 ) ( k . , ) ( k - 0 . 5 ) / N V I ( k - 0 . . 1 ) / N V(1C(2) V I V I ( k - 0 , . 1 ) / N ( k - l ) / . N ( k - l ) / . \ ( k - D / . N -
l l i t t l ) i v ( i ) ( k - l l . 1 ) / N V I ( k - 0 , 1 ) / N V I ( k - l ) / N ( k - l ) / N { k . ) V I ( k - 0 , 1 ) / . N
(k. 1 (k. 1
V I ( k - 0 . 1 ) / N ( k - l ) / N v d C ( 2 ) ( k . c ) ( k e ) ( k . l ) / N - v d C ( 2 ) ( k . l i i 2 | { k . l i i 2 ) v d C ( 2 ) ( k . e ) { k . l i i 2 )
Bang 4- Cac ehi so tong hop Z kem nhat
T h i r t i r 2 4 2 3 2 2 21 2 0 19 18 17 16 1 5 14 13 12 1 1 10 9 8 7 6 5 4 3 2 1
Z 1 8 , 0 2 6 2 5 0 19,34.1818 1 9 . 7 7 2 9 9 0 1 9 , 9 3 3 7 2 8 4 6 , 1 . 1 7 5 2 9 4 9 , 2 2 3 7 7 9 5 9 , 0 4 0 8 6 4 0 9 , 1 4 4 8 9 4 6 9 , 1 4 4 8 9 0 7 8 , 8 4 0 9 1 0 7 8 , 8 4 0 9 1 4 7 8 , 8 8 0 7 3 1 1 5 7 , 0 9 2 8 8 9 1 5 7 , 6 3 4 9 9 3 1 5 7 , 6 3 4 9 9 0 1 7 3 , 9 1 7 2 1 1 1 7 3 , 9 1 7 2 1 7 1 7 5 , 4 0 6 4 8 9 5 1 2 , 9 0 9 2 1 4 5 1 2 , 9 0 9 2 1 4 5 1 3 , 0 0 8 7 8 9 5 1 3 , 0 0 8 7 9 0 5 1 3 , 4 1 9 7 9 2 5 1 3 , 4 1 9 7 9 3
( i j ) ( 5 , 1 7 ) ( 1 1 , 2 1 ) ( 1 8 , 1 7 ) ( 2 4 , 1 9 ) ( 5 , 4 ) ( 1 , 5 ) (•1.1) ( 5 , 2 ) ( 5 , 3 ) ( 4 , 3 ) ( 4 , 2 ) ( 4 , 1 ) ( 1 . 4 ) ( 3 , 4 ) ( 2 . 4 ) ( 2 . 5 ) ( 3 . 5 ) ( 1 . 5 ) ( 1 . 3 ) ( 1 . 2 ) ( 2 . 1 ) ( 3 . 1 ) ( 2 . 3 ) ( 3 , 2 )
D a i i g a b b e c b R c na i i a a a a a a a a i l a a Eia a a a a a a a a a a a a a ^ a a a a a a a a a a
D a y ( i ) V 3 v c l C ( 1 3 ) { k . c } V 3 n
V 2 V 3 V 3 V 3 V 2 V 2 V 2 ( k - l ) / N V I ( k - 0 , 5 ) / N ( k - 0 , 5 ) / N V I ( k - l ) / N ( k - l ) / N ( k - l ) / N ( k - 0 , 5 ) / N V I ( k - 0 , 5 ) / N V I
D a v ( j ) v d C ( 3 7 ) ( k . } v c l r ( 3 7 ) ( k c ) V 2 V 3 ( k - l ) / N ( k - 0 , 5 ) / N V I V I ( k - 0 , 5 ) / N ( k - l ) / N V 2 V 2 V 2 V 3 V 3 V 3 V I ( k - 0 , 5 ) / N ( k - l ) / N ( k - l ) / N V I ( k - 0 , 5 ) / N
NHLTNG THIJ NGHIEM TAO CAC DAY Tl/A-NGAU-NHIEN HAI CHIEU Bang 5. Thong ke theo dang a, h, c, R qua 6 thir tii xep hang
(**, ' = cac gia tri Idn nhat, nhi trong cot) Dang
aa ab ac aR ba bb be bR ca ch
( • (
cR R,.
Rb Rc RR Tong so
so day A 5.(5-l)-20 5.12=60 5.5=25 5.2=10 12.5=00 (day Hiilloii) 12 5^00 12 2=21
": .".- 25 5 12=00 ,^i(r.-l)=20 .'') 2= 10 2.-)-10 2 12=24 2.->-10 2 (2-l)=2 552
so day tot B
9 24 1-1 12 (12-1) ^132 7 1 J'l 1.
T:J 16 % C^IOOB/OA 2.50 10,00*
3,89 10 1,91 0,()!1 19.33**
1.C7 0,83
0,09
= 24 (24-1)
so day k6m D 80
1
0 2,02 1 1 1 3 1 1 4 3 108
T^ le % E=100D/6A 71,07**
0,28
1,67 0,28 0,69 0,28 5,00 1,67 0,69 6,67 25,00*
108
Cik ty Ie C va E cho ta lliii\-. da sd ciic diiy tdt cd diing ca hoae ac, tire la ket h(yp giii-a mot day each deu va mdt day Richtmyer. Con da sd cae day kem cd dang aa hoae RR (2 day each deu hoae 2 day gii-ngau nhien ket h<?p vdi nhau).
TAI LIEU T H A M K H A O
[l] Fazekas Ferenc. Fiev Tamiis, Operdtorszdmittts, Specidlis jttggvcnyek, Tankonyvkiado, Bu- dapest, 1957.
[2] n-ey Tamils, Vegtelen sorozatok, sorok e's szorzatok, Tankonyvkiado, Budapest, 1973.
[3] J. M. Hammersley and D. C. Handscomb. Monte Carlo Methods, Methucn k Co Ltd. London, 1964.
[4] R. Hungarowitsch, H. Tatvann, and Vu Toan Thang, "On the use ot Low Discrepancy Siiquences in Practice". Seminar for Applied Mathematics, Technische Universitat Wien, 2007.
[5] N. Metropolis and S. M. Ulam, The montc carlo metliods, J. Amer. Statist. Assoc. 44 (1949) 335-341.
|6] H. Niederreiter. Random number generation and quaai-monte carlo methods, Society for In- dustrial and Applied Marhematics, Philadelphia, Pennsylvania, 1992.
J7] R. D. Richtmyer, "On the evaluation of definite integrals and a quaSi-Monti^Carlo method based on the properties of algebraic numbers", Report LA-1342, Los Alamos Scientific Laboratory, Los Alamos, N. M., 1951.
|8] L M Sobol, On the systematic search in a hyperoube, SIAM Journal on Numerical Analysis 16 (5) (October 1979) 790-793.
186 VUHoAi GiiiraNG
[9] ] M. Sobol (V' Ti. B. Sialiiiltov, Nhung gid; phdp lot nhat tim. chung & ddu ?, Nlia xuat bdn Ziianie, Moslivn, 1982 (tieiiR Nga).
|10] Uliovai-Hiigonc Islvan, "I-ivfizlvelelien ponl,ok a Monte Carlo algoritmusok-ban". Diploma- imnika. nMIO \"IK. Budapest, 21103,
[11] Vi'gli-Tmnielia Oiioii. Kv;izi Motid- CJUIU intcgrAlifai (ecliiiikjik". TDK-dolgozat, KLTE, Debre- cen. 1999,
[12] Vu Hoai t'liinnin, Moi tliiiat toan d o n gid.n t a o day so t y a ngan nhien. Tap chi Khoa hpc vd Cong nghe 4 0 (M1 DB) (2002) '.U '.)<).
[13] Xu Hoai Clurong, Nj;ii>(''ii Conn Dieu, So .si'mli va kiem dinh d o p h a n ky ciia cac day va tap (liem tira-iif-jMi-nlheu iiii)l elii'eu, Tnp chf Tin hoc vd Dieu khien hpe 2 3 (3) (2007) 251-259.
|14| \'\\ Hui'ii Climnif^. Nnuvi'n Vaii Himg, Homing Thi C a m Thacli, Day ttra-ngau-nhien van der Corput CO sn 2, Tap rlif Nghifn eiiu Khoa hoc K'y thudt vd Cong nghe qudn su (20) (9- 2007) 106 113,
Nhdn bdi ngdy 22 - 3 • 2011
