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Òàêèì îáðàçîì,D(X̄2 + X̄1 χD (ξ (1) ))D̄ − X̄2(4π/3)((a + h)3 − (a − h)3 )X̄ − X̄26h(a + h)2==≤≤.(4π/3)(a − h)3(a − h)3D(X̄χD (ξ))D̄X̄2Ïîñëåäíÿÿ âåëè÷èíà èìååò àñèìïòîòèêó 6h/a ïðè h → 0.3.8. ÌÅÒÎÄ ÏÐÎÒÈÂÎÏÎËÎÆÍÎÉ ÏÅÐÅÌÅÍÍÎÉÏóñòü òðåáóåòñÿâû÷èñëèòü îäíîêðàòíûé èíòåãðàë I0 = a g(x) dx ïî êîíå÷íîìó èíòåðâàëó a < x < b.Åñëè âçÿòü f (x) ≡ 1/(b−a) ïðè x ∈ (a, b), òî ñîãëàñíî ôîðìóëå (3.2) ïîëó÷àåì I0 = Eζ (0) ,ãäå ζ (0) = (b − a)g(a + (b − a)α) è α ñòàíäàðòíîå ñëó÷àéíîå ÷èñëî. Ðàññìîòðèì òåïåðüg (1) (x) = (g(x) + g(a + b − x)) /2 è çàìåòèì, ÷òîZ bg (1) (x) dx = Eζ (1) , ãäå ζ (1) = (b − a)g (1) (a + (b − a)α).I0 =3.8.1.
Ïðîñòàÿ ñèììåòðèçàöèÿ. Óìåíüøåíèå òðóäîåìêîñòè.Rbñèììåòðèçîâàííóþ ôóíêöèþaìåòîäîì ïðîòèâîïîëîæíîé ïåìåòîäîì ñèììåòðèçàöèè ïîäûíòåãðàëüíîé ôóíêöèè'antithetic variates'Àëãîðèòì 3.1 ñ îöåíêîé ζ (1) (âìåñòî ζ (0) ) íàçûâàåòñÿèëè(â àíãëîÿçû÷íîéëèòåðàòóðå äëÿ ýòîãî ïðèåìà èñïîëüçóåòñÿ òåðìèí). Çàìåòèì, ÷òî(1)(0)Dζ ≤ Dζ , òàê êàêZZ bb−a b 2(0)(1)2(g (x) + 2g(x)g(a + b − x) + g 2 (a + b − x)) dx =Dζ − Dζ = (b − a)g (x) dx −4aaZ bZb−ab−a b22=(g (x)−2g(x)g(a+b−x)+g (a+b−x)) dx =(g(x)−g(a+b−x))2 dx ≥ 0,44aaRb 2Rb 2çäåñü èñïîëüçîâàíî î÷åâèäíîå ðàâåíñòâî a g (x) dx = a g (a + b − x) dx.
Îäíàêî äëÿðàñ÷åòà îäíîãî çíà÷åíèÿ ζ (1) íàäî âû÷èñëèòü äâà çíà÷åíèÿ ôóíêöèè g(x). Ïîýòîìó òðóäîåìêîñòü àëãîðèòìà 3.1 ñ îöåíêîé ζ (1) áóäåò ìåíüøå òðóäîåìêîñòè ìåòîäà ïðîòèâîïîëîæíîé ïåðåìåííîé ñ îöåíêîé ζ (0) òîëüêî òîãäà, êîãäà âåëè÷èíà Dζ (1) ïî êðàéíåéìåðå âäâîå ìåíüøå, ÷åì Dζ (0) . Îêàçûâàåòñÿ, äëÿ ìîíîòîííûõ ôóíêöèé g(x) ýòî âñåãäàâûïîëíåíî.Óòâåðæäåíèå 3.7.(a, b)g(x)(1)(0)Dζ ≤ (1/2)DζÄîêàçàòåëüñòâî. Ñðàçó çàìåòèì, ÷òî óñëîâèå äèôôåðåíöèðóåìîñòè ôóíêöèè g(x)ÿâëÿåòñÿ èçáûòî÷íûì. Äîñòàòî÷íî ïîòðåáîâàòü êóñî÷íîé íåïðåðûâíîñòè (è ìîíîòîííîñòè) g(x), ïðàâäà, ïðè ýòîì ïðèäåòñÿ íåñêîëüêî âèäîèçìåíèòü íèæåñëåäóþùåå äîêàçàòåëüñòâî, ââîäÿ èíòåãðàëû Ñòèëòüåñà âìåñòî èíòåãðàëîâ Ðèìàíà.ðåìåííîé.Åñëè äèôôåðåíöèðóåìàÿ íàôóíêöèÿìîíîòîííà, òîÈç âûðàæåíèé äëÿ äèñïåðñèéDζ(0)Z= (b − a)bg 2 (x) dx − I02 èa2Dζ(1)Z= (b − a)bZ2g (x) dx + (b − a)abg(x)g(a + b − x) dx − 2I02aâûòåêàåò, ÷òî äîñòàòî÷íî äîêàçàòü íåðàâåíñòâîZ bg(x)g(a + b − x) dx ≤ I02 .(b − a)(3.45)aÏðåäïîëîæèì äëÿ îïðåäåëåííîñòè, ÷òî g(x) íå óáûâàåò è g(b) > g(a).
Ââåäåì âñïîìîãàòåëüíóþ ôóíêöèþZ xg(a + b − v) dv − (x − a)I0 ,G(x) = (b − a)aêîòîðàÿ îáðàùàåòñÿ â íóëü íà êîíöàõ îòðåçêà a ≤ x ≤ b. Ïðîèçâîäíàÿ ýòîé ôóíêöèèG0 (x) = (b − a)g(a + b − x) − I0 ìîíîòîííî óáûâàåò, ïðè÷åì G0 (a) > 0 è G0 (b) < 0. Çíà÷èò,ôóíêöèÿ G(x) ñíà÷àëà âîçðàñòàåò, à çàòåì óáûâàåò íà îòðåçêå [a, b], è, ñëåäîâàòåëüíî,RbG(x) ≥ 0 ïðè a ≤ x ≤ b. Òîãäà èíòåãðàë a G(x)g 0 (x) dx íåîòðèöàòåëåí. Ïðîèíòåãðèðîâàâ ïî ÷àñòÿì, ïîëó÷èìZ bZ bb Z b00G(x)g (x) dx = G(x)g(x) −g(x)G (u) du ≥ 0 èëèg(x)G0 (x) dx ≤ 0.aaaaÏîäñòàâèâ â ïîñëåäíåå íåðàâåíñòâî âûðàæåíèå äëÿ G0 (x), ïîëó÷èì ñîîòíîøåíèå (3.45).Ñëó÷àé íåâîçðàñòàíèÿ g(x) ðàññìàòðèâàåòñÿ òî÷íî òàê æå, òàê êàê ïðè ýòîì G(x) ≤ 0è g 0 (x) ≤ 0.3.8.2. Ñëîæíàÿ ñèììåòðèçàöèÿ.
Äëÿ óìåíüøåíèÿ äèñïåðñèè ðàñ÷åòîâ ìîæíîòàêæå èñïîëüçîâàòü, ïðè êîòîðîé èíòåðâàë (a, b) ðàçáèâàåòñÿíà êîíå÷íîå ÷èñëî ÷àñòåé è äëÿ êàæäîé èç íèõ èñïîëüçóåòñÿ ìåòîä ïðîòèâîïîëîæíîéïåðåìåííîé.Ïóñòü (a, b) ðàçáèâàåòñÿ íà äâå ðàâíûå ÷àñòè. Îáîçíà÷èì c = (a + b)/2. ÒîãäàZZZ cZ b1 b1 cI0 =g(x) dx +g(x) dx =(g(x) + g(a + c − x)) dx +(g(x) + g(c + b − x)) dx.2 a2 cacñëîæíóþ ñèììåòðèçàöèþ ïåðâîì èç ýòèõ èíòåãðàëîâ ñäåëàåì çàìåíó ïåðåìåííûõ y = 2x − a, êîòîðàÿ ïðåîáðàçóåò èíòåðâàë (a, c) â (a, b), à âî âòîðîì çàìåíó y = 2x − b, êîòîðàÿ ïðåîáðàçóåòèíòåðâàë (c, b) â (a, b).
Ïðè ýòîì ïîëó÷àåì ñîîòíîøåíèåZ bI0 =g (2) (y) dy = Eζ (2) , ãäå ζ (2) = (b − a)g (2) (a + (b − a)α),a 1a+y2a + b − yb+y2b + a − y(2)g (y) =g+g+g+g.42222Ïðèìåð 3.7.Ðàññìîòðèì òåñòîâóþ çàäà÷ó âû÷èñëåíèÿ èíòåãðàëàZ 11I0 =(2 − 3x − x2 ) dx = ,60òî åñòü çäåñü a = 0, b = 1 è g(x) = 2 − 3x − x2 . Íåñëîæíî ïîëó÷èòü âûðàæåíèÿ äëÿîöåíîêζ (0) = 2 − 3α − α2 , ζ (1) = α − α2 , ζ (2) = (1/8)(1 + 2α − 2α2 ).Çàòðàòû íà ïîëó÷åíèå îäíîãî çíà÷åíèÿ ζ (0) , ζ (1) èëè ζ (2) ïðèáëèçèòåëüíî îäèíàêîâû âòî âðåìÿ, êàêZ 11411241(0)Dζ =(2 − 3x − x2 )2 dx −=−=,3630 361800Z 11111(1)(x − x2 )2 dx −Dζ ==−=,3630 361800Z 11191111(2)Dζ =(1 + 2x − 2x2 )2 dx −=−==×.64 036320 36288016 1803.8.3.
Èñïîëüçîâàíèå ìåòîäà ïðîòèâîïîëîæíîé ïåðåìåííîé â ìíîãîìåðíîì ñëó÷àå. Ê ñîæàëåíèþ, ðàçëè÷íûå âàðèàíòû ìåòîäà ïðîòèâîïîëîæíîé ïåðåìåííîé, âåñüìà íàãëÿäíûå è ýôôåêòèâíûå â îäíîìåðíîì ñëó÷àå, ñòàíîâÿòñÿ ãðîìîçäêèìèïðè ïåðåõîäå ê ôóíêöèÿì ìíîãèõ ïåðåìåííûõ. Ïîýòîìó ïðè âû÷èñëåíèè ìíîãîêðàòíûõèíòåãðàëîâ ìåòîäîì Ìîíòå-Êàðëî ñèììåòðèçàöèÿ ïðîèçâîäèòñÿ, êàê ïðàâèëî, òîëüêîâäîëü âûáðàííîãî íàïðàâëåíèÿ. Íàïðèìåð, â òåîðèè ïåðåíîñà èçëó÷åíèÿ äëÿ áîëåå ðàâíîìåðíîãî ðàñïîëîæåíèÿ òðàåêòîðèé ÷àñòèö â ïðîñòðàíñòâå íàðÿäó ñ òðàåêòîðèåé, èìåþùåé ñëó÷àéíîå íà÷àëüíîå íàïðàâëåíèÿ äâèæåíèÿ ÷àñòèöû ω0 , ðåàëèçóåòñÿ òðàåêòîðèÿñ íà÷àëüíûì íàïðàâëåíèåì −ω0 .
 ðÿäå ñëó÷àåâ ýòî ïðèâîäèò ê óìåíüøåíèþ òðóäîåìêîñòè ðàñ÷åòîâ.Èçâåñòíû òàêæå ïðèìåðû ýôôåêòèâíîãî èñïîëüçîâàíèÿ ëîêàëüíîé ñèììåòðèçàöèèâ ìàëûõ ïîäìíîæåñòâàõ îáëàñòè èíòåãðèðîâàíèÿ âäîëü ñëó÷àéíî âûáðàííîãî íàïðàâëåíèÿ ïðè èñïîëüçîâàíèè ðàññëîåííîé âûáîðêè íà êëàññàõ ãëàäêèõ ïîäûíòåãðàëüíûõôóíêöèé (ñì. äàëåå ïîäðàçä. 3.9.5).3.9. ÌÅÒÎÄ ÐÀÑÑËÎÅÍÍÎÉ ÂÛÁÎÐÊÈ3.9.1. Âûáîðêà ïî ãðóïïàì. Ðàññìîòðèì ïðåäñòàâëåíèå (3.2) èíòåãðàëà IÇàïèøåì âåëè÷èíó I â âèäåI=M ZXm=1=RXg(x) dx.q(x)f (x) dx,Xmãäå Xm ïîäîáëàñòè X , èìåþùèå ïîïàðíûå ïåðåñå÷åíèÿ ìåðû íóëü, ïðè÷åì X = X1 ∪. . .
∪ XM . Ââåäåì îáîçíà÷åíèÿZZf (x)pm =f (x) dx, Im =q(x)f (x) dx, fm (x) =pmXmXmïðè x ∈ Xm . Ïðåäïîëîæèì, ÷òî f (x) = 0 ïðè x 6∈ X . Òîãäà p1 + . . . + pM = 1. Êðîìåòîãî, I1 + . . . + IM = I è Im = E pm q(ξ (m) ) , ãäå ñëó÷àéíûé âåêòîð ξ (m) ðàñïðåäåëåí âXm ñîãëàñíî ïëîòíîñòè fm (x).Àëãîðèòì 3.5.Im3.1nmàëãîðèòìóÏðèáëèæåííî âû÷èñëåíèåì çíà÷åíèÿñ ÷èñëîì èñïûòàíèénmpm X(m)Im ≈q(ξ im )nm i =1mè ïîëàãàåìI≈ζ̄n(M )ñîãëàñíî ñòàíäàðòíîìóMnmXpm X(m)=q(ξ im ),nm=1 m i =1m(3.46)çäåñü n = n1 + . .
. + nM .ìåòîä ðàññëîåííîé âûáîðêèâûáîðêó ïî ãðóïïàìÀëãîðèòì 3.5 îïðåäåëÿåòèëè. Ýòîòàëãîðèòì îòëè÷àåòñÿ ïðè M = 2 îò ìåòîäà èíòåãðèðîâàíèÿ ïî ÷àñòè îáëàñòè èç ðàçä.3.7, òàê êàê ïîñëåäíèé ïðåäïîëàãàåò, ÷òî èíòåãðàë I2 èçâåñòåí (â òî âðåìÿ êàê â àëãî(2)ðèòìå 3.5 ýòîò èíòåãðàë âû÷èñëÿåòñÿ ïðèáëèæåííî ïî âûáîðêå {ξ i2 }).3.9.2. Ìèíèìèçàöèÿ äèñïåðñèè ìåòîäà ðàññëîåííîé âûáîðêè. Ñðàâíèì äèñïåðñèþ Dζ̄n = Dζ/n ñòàíäàðòíîãî ìåòîäà 3.1 âû÷èñëåíèÿ èíòåãðàëà I (çäåñü ñëó÷àéíûå(M )òî÷êè ξ âûáèðàþòñÿ âî âñåé îáëàñòè èíòåãðèðîâàíèÿ X ) è äèñïåðñèþ Dζ̄n ìåòîäàðàññëîåííîé âûáîðêè ïðè óñëîâèè, ÷òî ôèêñèðîâàíû ÷èñëî èñïûòàíèé (äëÿ âûáîðêè ïî ãðóïïàì ñóììàðíîå ÷èñëî èñïûòàíèé) n è ðàçáèåíèå îáëàñòè èíòåãðèðîâàíèÿX = X1 ∪ .
. . ∪ XM . Ïî àíàëîãèè ñ ôîðìóëîé (3.4) èìååì2 XM nmMXXpmp2m Dq(ξ (m) )(m)(M )Dζn =,(3.47)Dq(ξ im ) =nnmmm=1m=1i =1mãäåDq(ξ(m)1)=pmZ2q (x)f (x) dx −XmImpm2;(3.48)(m)çäåñü èñïîëüçîâàíà íåçàâèñèìîñòü ñëó÷àéíûõ òî÷åê {ξ im }.(M )Óòâåðæäåíèå 3.8.Dζ̄n!2qMX1pm Dq(ξ (m) ) .d2n =n m=1Ìèíèìóì âåëè÷èíûðàâåí(3.49)Ýòà âåëè÷èíà íå ïðåâîñõîäèò Dζ̄n è ðåàëèçóåòñÿ ïðè,qnm = npmDq(ξ(m))MXqpmDq(ξ (m) ).(3.50)m=1Äîêàçàòåëüñòâî.Âåëè÷èíà d2n èç (3.49) ïðåäñòàâèìà â âèäå2srM(m)XDq(ξ )nm d2n = pm×.nmnm=1Èñïîëüçóÿ íåðàâåíñòâî ÊîøèÁóíÿêîâñêîãî, ïîëó÷àåìd2nMMMXXp2m Dq(ξ (m) ) X nmp2m Dq(ξ (m) )≤×=,nnnmmm=1m=1m=1ãäå ñïðàâà ñòîèò âûðàæåíèå (3.47). Íåñëîæíî ïðîâåðèòü, ÷òî ïðè ïîäñòàíîâêå ðàâåíñòâà(3.50) â ôîðìóëó (3.47) ïîëó÷àåòñÿ ñîîòíîøåíèå (3.49).Äàëåå, óìíîæàÿ (3.48) íà pm è ñóììèðóÿ ïîëó÷åííûå ðåçóëüòàòû ïî m, èìååìZMM2XXIm(m)2pm Dq(ξ ) =q (x)f (x) dx −.pmXm=1m=1Åùå ðàç ïðèìåíÿÿ íåðàâåíñòâî ÊîøèÁóíÿêîâñêîãî, ïîëó÷àåì!2!2 XMM MMM22XXXXIImIm√m2I =Im=≤×pm =.√ × pmpmppm=1 mm=1m=1m=1m=1 mÎòñþäà ñëåäóåò, ÷òî1Dζ̄n =nZq 2 (x)f (x) dx − I 2X(M )Ïîñëåäíåå âûðàæåíèå ðàâíî âåëè÷èíå Dζ̄nMX1≥n!pm Dq(ξ (m) ) .m=1ïðè óñëîâèènm = n pm .(3.51)(M )Òàêèì îáðàçîì, d2n ≤ Dζ̄n |nm =n pm ≤ Dζ̄n . ðåàëüíûõ çàäà÷àõ äèñïåðñèè {Dq(ξ (m) )}, êàê ïðàâèëî, íåèçâåñòíû è âûáîð {nm }ïî ôîðìóëå (3.50) íåâîçìîæåí.
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