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Èç (7.46) ïîëó÷àåì âûðàæåíèå(p)η̃1,εZN X=Qi (c)(p)G(ρ; c, di ) g1 (ρ) dρ+ [QN (c) ϕ(P, c)](p) .D(ri )i=1Åñëè c < c∗ è ïåðâûå ïðîñòðàíñòâåííûå ïðîèçâîäíûå ôóíêöèéi = 1, . . . , p + 1, ðàâíîìåðíî îãðàíè÷åíû â D̄, òîÒåîðåìà 7.11.(i){uk },(p) u(r) − Eη̃1,ε ≤ Cp ε,Äîêàçàòåëüñòâî.r ∈ D,ε > 0.(7.51)ßñíî, ÷òî ()(p) p kp−kX(−1) (c − c0 )(p) (p)(p) .[uk (rN ) − ϕk (P )]u(r) − Eη̃1,ε = E(η1,ε − η̃1,ε ) = E QN (c)p!k=0(7.52)Ñëåäîâàòåëüíî, äëÿ ïîëó÷åíèÿ íåðàâåíñòâà (7.51) äîñòàòî÷íî îáîñíîâàòü ñîîòíîøåíèå(k)EQN (c) ≤ C < +∞,k = 1, . .
. , p + 1;r0 ∈ D.(7.53)Ñ ýòîé öåëüþ ïðè c < c∗ ðàññìîòðèì çàäà÷ó âèäà∆v + cv = c, v Γ = 1,(7.54)äëÿ êîòîðîé v ≡ 1. Ñîîòâåòñòâåííî (7.50) èìååì Z−E cτN(k)(k)e dt+ EQN (c) = v (k) .ct0Îòñþäà î÷åâèäíûì îáðàçîì ñëåäóåò (7.53) è, ñëåäîâàòåëüíî, (7.51).(p)Òåîðåìà 7.12.7.11c < c∗ /2Dη̃1,ε < Cp < +∞ε>0Äîêàçàòåëüñòâî. Èç âåðîÿòíîñòíîãî ïðåäñòàâëåíèÿ ðåøåíèÿ çàäà÷è (7.54) ñëåäóåò(k)ðàâíîìåðíàÿ îãðàíè÷åííîñòü âåëè÷èí DQN , k = 1, . . . , p + 1; r0 ∈ D, â óñëîâèÿõòåîðåìû. Äàëåå äîêàçàòåëüñòâî ñòðîèòñÿ î÷åâèäíûì îáðàçîì íà îñíîâå ñîîòíîøåíèÿ(7.52).(p)Èíòåãðàëû, âõîäÿùèå â âûðàæåíèå äëÿ η̃1,ε , ìîæíî íåñìåùåííî îöåíèâàòü ïî îäíîìóñëó÷àéíîìó óçëó íà îñíîâå ïðåäñòàâëåíèÿZZG(ρ; c, d)d2G(ρ; c, d)d2p0 (r, p)g(ρ) dρ =Eg(ρ) ,G(ρ; c, d) g(ρ) dρ =2n D(r)G(ρ; 0, d)2nG(ρ; 0, d)D(r)(7.55)ãäå ρ ñëó÷àéíàÿ òî÷êà â D(r), ïëîòíîñòü ðàñïðåäåëåíèÿ âåðîÿòíîñòåé êîòîðîé ðàâíà(ïðè n > 2)2n11−2p0 (r, p) = 2nd G(ρ; 0, d) =−, |ρ − r| ≤ d.(7.56)(n − 2)d2 ωn (ρ − r)n−2 dn−2RÍåòðóäíî ïðîâåðèòü, ÷òî p0 (r, ρ)dρ = 1.
Ïðè n = 2 èìååìâñåõ óñëîâèÿõ òåîðåìû.ïðèp0 (r, p) = 4d−2 G(ρ; 0, d) =èìååì2dln,πd2 |ρ − r|äëÿ|ρ − r| ≤ d.Îòíîøåíèå çíà÷åíèé ôóíêöèè G â âûðàæåíèè (7.55) îãðàíè÷åíî, òàê êàê ôóíêöèèÃðèíà äëÿ ðàçëè÷íûõ çíà÷åíèé èìåþò â òî÷êå ρ = 0 ïîëþñà îäíîãî ïîðÿäêà. Ïîñëåòàêîé ðàíäîìèçàöèè ïîëó÷àåì îöåíêó(p)η̃˜1,ε =NX(" i−1Yi=0(p)#d2 G(ρ; c, di )s(c, dj ) g1 (ρi ) i2nG(ρ; 0, di )j=0)(p)+("N −1Y#)(p)s(c, dj ) ϕ(rN , c),(7.57)j=0(p)ïðè÷åì Eη̃˜1,ε = Eη̃1,ε .
Íåòðóäíî âèäåòü, ÷òî äîêàçàòåëüñòâî òåîðåìû 7.12 îñòàåòñÿ ñïðà(p)âåäëèâûì è ïîñëå çàìåíû η̃ íà η̃˜, ò. å. Dη̃˜1,ε < Cp < +∞ äëÿ âñåõ ε > 0.Ðàññìîòðèì òåïåðü â îáëàñòè D ⊂ Rn ïåðâóþ êðàåâóþ çàäà÷ó äëÿ íåîäíîðîäíîãîáèãàðìîíè÷åñêîãî óðàâíåíèÿ∆2 u = −g, uΓ = ϕ0 , ∆uΓ = ϕ1 .(7.58)(1)Äëÿ ïîñòðîåíèÿ ñîîòâåòñòâóþùåé îöåíêè η̃˜1,ε èñïîëüçóåì ðàâåíñòâàs(c, d) = 1/[1 −d2 c+ o(c)],2ns0 (c, d) =d2d4 c−+ o(c),2n 4n(n + 2)s(0, d) = 1,s0 (0, d) =d2,2ng1 = −g, ðåçóëüòàòå ïîëó÷àåì:#" i−1N20XXd11G(ρ;0,d)ij(1)η̃˜1,ε =−d2i g(ρi ) −−2n i=02nG(ρi ; 0, d)2nj=0ϕ = cϕ0 − ϕ1 .N−1X!d2jϕ1 (rN ) + ϕ0 (rN ),(7.59)j=0ãäå ρi - ñëó÷àéíàÿ òî÷êà ðàñïðåäåëåííàÿ â D(ri ) â ñîîòâåòñòâèè ñ ôîðìóëîé (7.56).Èçâåñòíî, ÷òî ïðè n = 3 âûïîëíÿþòñÿ ñëåäóþùèå ñîîòíîøåíèÿ:s(c, di )G(ρi ; c, d)=,G(ρi ; 0, d)s(c, di − νi )G0 (ρi ; 0, d)1 2=di − (di − νi )2 ,G(ρi ; 0, d)6Ñëåäîâàòåëüíî, èìååì"#NiXX11(1)η̃˜1,ε =−d2j + (di − νi )2 d2i g(ρi ) −36 i=06j=0N−1Xρi = ri + νi ωi .!d2jϕ1 (rN ) + ϕ0 (rN ).(7.60)j=0Cëó÷àéíàÿ âåëè÷èíà νi , ðàñïðåäåëåííàÿ â èíòåðâàëå (0, di ) ñ ïëîòíîñòüþ 6x(1−x/di )d−2iè åäèíè÷íûé èçîòðîïíûé âåêòîð ωi ìîäåëèðóþòñÿ ïðè ïîìîùè èçâåñòíûõ ôîðìóë.7.5.5.
Âû÷èñëåíèå êîâàðèàöèîííîé ôóíêöèè ðåøåíèÿ áèãàðìîíè÷åñêîãîn = 2. Êîëåáàíèÿ ïëàñòèíû â îãðàíè÷åííîé îáëàñòè D ⊂ R2 ïîä äåéñòâèåì ñëó÷àéíîãî ïîëÿ íàãðóçîê σ(r) = −g(r) îïèñûâàþòñÿ óðàâíåíèåì âèäà (7.58),ïðè ýòîì ìîæíî ó÷èòûâàòü òàêæå ñëó÷àéíîñòü ãðàíè÷íûõ ôóíêöèé ϕ0 (r) è ϕ1 (r). Ðåøåíèå ýòîé çàäà÷è òàêæå ÿâëÿåòñÿ ñëó÷àéíûì ïîëåì. Òðåáóåòñÿ îïðåäåëèòü åãî êîâàðèàöèîííóþ ôóíêöèþ v(r, r0 ) = E[u(r)u(r0 )]. Ïðåäïîëàãàåòñÿ, ÷òî Eg(r) ≡ Eϕ0 (r) ≡Eϕ1 (r) ≡ 0 è, ñëåäîâàòåëüíî, Eu(r) ≡ 0. Ñîãëàñíî ôîðìóëàì (7.47) è (7.48) èìååì√√ √1J0 (z c) N0 (d c)1√ , G(ρ; c, d) =√−N0 (z c) +.(7.61)s(c, d) =4J0 (d c)J0 (d c)óðàâíåíèÿ ïðèÈñïîëüçóÿ àñèìïòîòè÷åñêèå ïðè c → 0 âûðàæåíèÿ √ 2√ √√√z c2z cγ + lnJ0 (z c),J0 (z c) ∼ 1 −, N0 (z c) ∼2π2√ √√z22 1z2 z2z c00J0 (z c) ∼ − , N0 (z c) ∼+−γ + ln,4π 2c842(7.62)(7.63)ïîëó÷àåì äîñòàòî÷íî èçâåñòíûå ôîðìóëû:s(0, d) = 1,d2s (0, d) = ,401dG(ρ, 0, d) =ln ,2π z1G (ρ, 0, d) =8π0dd − z − z lnz222.Ñëåäîâàòåëüíî, ïðè n = 2 îöåíêà äëÿ ðåøåíèÿ óðàâíåíèÿ (7.58) èìååò âèä" i−1#N222XX1d−ν−νln(d/ν)i(1)iiη̃˜1,ε =−d2j − id2i g(νi , ωi )−16 i=0ln(d/ν)ij=01−4N−1Xj=0!d2jϕ1 (rN ) + ϕ0 (rN ) =NXi=0bN ϕ1 (rN ) + ϕ0 (rN ),Qi g(ρi ) + Q(7.64)ãäå ωi - åäèíè÷íûé èçîòðîïíûé âåêòîð, νi /di ñëó÷àéíàÿ âåëè÷èíà, ðàñïðåäåë¼ííàÿ âèíòåðâàëå (0, 1) ñ ïëîòíîñòüþ −4x ln x.Îñðåäíÿÿ (óñëîâíî, äëÿ ôèêñèðîâàííûõ òðàåêòîðèé {ri }, {ri0 }) ïðîèçâåäåíèå ñîîòâåòñòâóþùèõ îöåíîê òèïà (7.64), ïðèõîäèì ê ñëåäóþùåìó ñîîòíîøåíèþ" N N0iXX(1)(1)v(r, r0 ) = E η̃˜1,ε (r)η̃˜1,ε (r0 ) = EQi Q0j K(ρi , ρ0j )+hi=1 j=1+N Xb0 0 K1 (ρi , rN 0 )Qi QNN0 XbN K1 (ρ0 , rN ) + Q0 K0 (ρ0 , rN ) ++ Qi K0 (ρi , rN 0 ) +Q0j Qjjji=1j=1(7.65)#bN Qb0N 0 K11 (rN , rN 0 ) + QbN K10 (rN , rN 0 ) + Qb0N 0 K10 (rN 0 , rN ) + K00 (rN , rN 0 ) .+QÇäåñü ïðèíÿòû îáîçíà÷åíèÿ:K(r, r0 ) = E[g(r)g(r0 )],K00 (r, r0 ) = E[ϕ0 (r)ϕ0 (r0 )],K0 (r, r0 ) = E[g(r)ϕ0 (r0 )],K10 (r, r0 ) = E[ϕ1 (r)ϕ0 (r0 )],K1 (r, r0 ) = E[g(r)ϕ1 (r0 )],K11 (r, r0 ) = E[ϕ1 (r)ϕ1 (r0 )].Òàêèì îáðàçîì, ïðè ïîìîùè âûðàæåíèé (7.64), (7.65) ìîæíî îöåíèâàòü êîâàðèàöèîííóþ ôóíêöèþ ðåøåíèÿ v(r, r0 ), èñïîëüçóÿ òîëüêî êîâàðèàöèîííûå ôóíêöèè ñëó÷àéíîãîïîëÿ íàãðóçîê è ñëó÷àéíûõ ãðàíè÷íûõ óñëîâèé.
Ñîîòâåòñòâóþùàÿ îöåíêà èìååò äèñïåðñèþ, çàâåäîìî ìåíüøóþ äèñïåðñèè ìåòîäà äâîéíîé ðàíäîìèçàöèè (ñì. ðàçäåë 4.7),ò. ê. çäåñü îñóùåñòâëåíî ÷àñòè÷íîå àíàëèòè÷åñêîå îñðåäíåíèå.Îäíàêî ïðè ðåàëèçàöèè ïîëó÷åííîãî ìåòîäà âîçíèêàåò ïðîáëåìà, ñâÿçàííàÿ ñ îãðàíè÷åííîñòüþ ìàøèííîé ïàìÿòè.  îòëè÷èè îò ðåàëèçàöèè îöåíîê âèäà (7.60) çäåñüíåîáõîäèìî ñîõðàíÿòü âñå âåñà è êîîðäèíàòû öåíòðîâ ñëó÷àéíûõ êðóãîâ â (7.65) õîòÿáû äëÿ îäíîé òðàåêòîðèè. Èçâåñòíî, ÷òî ïðîöåññ áëóæäàíèÿ ïî êðóãàì âåñüìà áûñòðîñõîäèòñÿ ê ãðàíèöå îáëàñòè.
Ñëåäîâàòåëüíî, ìîæíî ïðè ïðåâûøåíèè êîëè÷åñòâà òî÷åê{ri } íåêîòîðîãî äîñòàòî÷íî áîëüøîãî óðîâíÿ M íå ñîõðàíÿòü èíôîðìàöèþ î ñëåäóþùèõ òî÷êàõ ñ íîìåðàìè i = M + 1, . . . , N − 1, íî â Γε èñïîëüçîâàòü ïîëó÷åííûå âåñQN è êîîðäèíàòû rN . Ïðè ýòîì Qi = 0 äëÿ i = M + 1, . . . , N − 1, è ìîæíî çàìåíèòüQM íà QM (N − M + 1). Åñëè òàêàÿ çàìåíà ïðàêòè÷åñêè íå âëèÿåò íà ðåçóëüòàò, òîîöåíêà óäîâëåòâîðèòåëüíà.
Ïîðÿäîê (ïî ε) âåëè÷èíû M ìîæíî ýâðèñòè÷åñêè îöåíèòüñ èñïîëüçîâàíèåì àñèìïòîòè÷åñêîé òåîðèè âîññòàíîâëåíèÿ; ïðè ýòîì îêàçûâàåòñÿ, ÷òîóäîâëåòâîðèòåëüíî M (ln ε)2 .7.6. ÈÑÏÎËÜÇÎÂÀÍÈÅ ÃÐÀÍÈ×ÍÛÕ ÈÍÒÅÃÐÀËÜÍÛÕÓÐÀÂÍÅÍÈÉÄàëåå ðàññìàòðèâàþòñÿ ãðàíè÷íûå èíòåãðàëüíûå óðàâíåíèÿ òåîðèè ïîòåíöèàëà, äëÿêîòîðûõ â ñòàöèîíàðíîì ñëó÷àå ñïåêòðàëüíûé ðàäèóñ ðàâåí åäèíèöå è ïîñòðîåíèå àëãîðèòìîâ ìåòîäà Ìîíòå-Êàðëî îñóùåñòâëÿåòñÿ íà îñíîâå àíàëèòè÷åñêîãî ïðîäîëæåíèÿðÿäà Íåéìàíà.
Ýòà ìåòîäèêà ðàçðàáîòàíà Ê.Ê.Ñàáåëüôåëüäîì [5].Ðàññìîòðèì óðàâíåíèå Ëàïëàñà∆u(x) = 0(7.66)â îãðàíè÷åííîé îáëàñòè G ⊂ R3 ñ êóñî÷íî ãëàäêîé ãðàíèöåé ∂G. Óäîáíî ðàññìàòðèâàòüîäíîâðåìåííî äâå êðàåâûå çàäà÷è äëÿ (7.66): âíóòðåííþþ çàäà÷ó Äèðèõëåu(t) = Ψ1 (t),t ∈ ∂G,(7.67)è âíåøíþþ çàäà÷ó Íåéìàíà∂u/∂n = Ψ2 (t),t ∈ ∂G,lim u(x) = 0.|x|→∞(7.68)Èçâåñòíî, ÷òî â (7.66)(7.68) ãðàíèöó ∂G ìîæíî çàìåíèòü íà Γ = ∂G − Γ0 ãäå Γ0 ìíîæåñòâî (ìåðû íóëü) ãðàíè÷íûõ òî÷åê, â êîòîðûõ íå îïðåäåëåíà íîðìàëü. Ðåøåíèåçàäà÷è (7.66), (7.67) èùåòñÿ â âèäå ïîòåíöèàëà äâîéíîãî ñëîÿ ñ íåèçâåñòíîé ïëîòíîñòüþµ(t) (t ∈ Γ):Z1∂µ(t) dσ(t),(7.69)u(x) =Γ ∂n |x − t|ãäå σ(t) ýëåìåíò ïîâåðõíîñòè, n(t) âíóòðåííÿÿ íîðìàëü â òî÷êå t ∈ Γ.Ñîîòíîøåíèå äëÿ ðàçðûâà ïîòåíöèàëà äâîéíîãî ñëîÿ íà Γ äàåò èíòåãðàëüíîå óðàâíåíèå äëÿ ïëîòíîñòè:ZΨ1 (t)µ(t) = − k(t1 , t) µ(t1 ) dσ(t1 ) +,(7.70)2πΓãäå k(t1 , t) = cos ϕt1 ,t /(2π|t1 −t|2 ), à ϕt1 ,t óãîë ìåæäó âåêòîðàìè n(t1 ) è (t−t1 ).
Ðåøåíèåâíåøíåé çàäà÷è Íåéìàíà èùåòñÿ â âèäå ïîòåíöèàëà ïðîñòîãî ñëîÿ:Z1u(x) =ν(t) dσ(t).Γ |x − t|Ñîîòíîøåíèå äëÿ ðàçðûâà íîðìàëüíîé ïðîèçâîäíîé ïîòåíöèàëà ïðîñòîãî ñëîÿ äàåòóðàâíåíèå äëÿ íåèçâåñòíîé ïëîòíîñòè:ZΨ2 (t)ν(t) = −k(t, t1 ) ν(t1 ) dσ(t1 ) +.(7.71)2πΓÝòî óðàâíåíèå ÿâëÿåòñÿ ñîïðÿæåííûì ê óðàâíåíèþ (7.70).Òàêèì îáðàçîì, ðåøåíèå êðàåâûõ çàäà÷ Äèðèõëå è Íåéìàíà ìîæíî ðàññìàòðèâàòüêàê ëèíåéíûå ôóíêöèîíàëû îò ðåøåíèÿ èíòåãðàëüíûõ óðàâíåíèé ñîîòâåòñòâåííî (7.70)è (7.71):cos ϕt,xu(x) = 4π(hx , µ), ãäå hx (t) =;(7.72)4π|t − x|2u(x) = (ν, gx ), ãäå gx (t) =1.|x − t|(7.73)Îäíàêî ïðåäñòàâëåíèÿ (7.72) è (7.73) åùå íå ïîçâîëÿþò íàì ïðèìåíèòü ñòàíäàðòíûåìåòîäû Ìîíòå-Êàðëî, ïîñêîëüêóðÿäû Íåéìàíà äëÿ (7.70), (7.71) ðàñõîäÿòñÿ.RÏîëîæèì Kµ(t) = − Γ k(t1 , t) µ(t1 ) dσ(t1 ) è ïðåîáðàçóåì (7.70) â èíòåãðàëüíîå óðàâíåíèå ñ ïàðàìåòðîì λ:µλ (t) = λKµλ (t) + f (t),(7.74)à (7.71) â ñîïðÿæåííîå ê íåìó óðàâíåíèå:νλ (t) = λK ∗ νλ (t) + e(t).(7.75)Ïîñêîëüêó kKk = kK ∗ k = 1, òî ïðè |λ| < 1 ðåøåíèÿ óðàâíåíèé (7.74), (7.75) ïðåäñòàâëÿþòñÿ ñîîòâåòñòâåííî â âèäå ñõîäÿùèõñÿ àáñîëþòíî è ðàâíîìåðíî ðÿäîâ:µλ (t) = Rλ f (t),νλ (t) = Rλ e(t),(7.76)ãäå Rλ = I + λK + λ2 K 2 + .
. . ðåçîëüâåíòà.Èçâåñòíî, ÷òî âñå õàðàêòåðèñòè÷åñêèå ÷èñëà èíòåãðàëüíûõ óðàâíåíèé (7.74), (7.75),ò. å. ïîëþñû ðåçîëüâåíòû, äåéñòâèòåëüíû è îòðèöàòåëüíû, à ìèíèìàëüíîå ïî ìîäóëþõàðàêòåðèñòè÷åñêîå ÷èñëî (ïðîñòîé ïîëþñ ðåçîëüâåíòû) λ1 ðàâíî −1. Òàê êàê λ = 1 íåÿâëÿåòñÿ õàðàêòåðèñòè÷åñêèì ÷èñëîì, òî óðàâíåíèÿ (7.69), (7.71) èìåþò åäèíñòâåííîåðåøåíèå, îäíàêî îíî íå ìîæåò áûòü ïðåäñòàâëåíî ðÿäîì Íåéìàíà, ïîñêîëüêó λ1 = −1.Äëÿ íàõîæäåíèÿ ðåøåíèÿ, îäíàêî, ìîæíî çàäàííóþ â âèäå ðÿäà Íåéìàíà ðåçîëüâåíòóRλ ïðîäîëæèòü çà ïðåäåëû êðóãà |λ| < 1.Íàèáîëåå ïðîñòûì ñïîñîáîì ïðîäîëæåíèÿ ÿâëÿåòñÿ äîìíîæåíèå ðåçîëüâåíòû íà(λ + 1)/2 â öåëÿõ óíè÷òîæåíèÿ ïîëþñà.
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