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 fν (r, s)t pν,t(r, s) = ρt(r, s) + c[dtν,t(r)−dtν,t(s)] + ǫν,t(r, s) φν,t(r, s) = ρt(r, s) + c[δtν,t(r)−δtν,t(s)] +λν[ϕν,0(r)−ϕν,0(s)] + λνNν(r, s) + εν,t(r, s) ρt(r, s) s t−τ r t λν Nν(r, s) (ν, t)r s dtν,t(r) dtν,t(s)δtν,t(r)δtν,t(s) c ϕν,0(r)ϕν,0(s) r s ǫν,t(i, j) εν,t(i, j) φt(r, s) = ρt(r, s) + c[δtt(r)−δtt(s)] +λ[ϕ0(r)−ϕ0(s)] + λN(r, s) + εt(r, s) a:= b a b r1 r2s1, s2,...,sn ϑj:= ϑ(r2, sj)−ϑ(r1, sj) ϑ φj t=ρj t+c[δtt(r2)−δtt(r1)] +λ[ϕ0(r2)−ϕ0(r1)] + λaj+εj t aj:= Nj aj  sk ϑj k:= ϑj−ϑk(i=j) ϑ j=k φj t;k=ρj t;k+λaj k+εj t;k(aj k∈Z) aj k ϑj k ϑkϑj ✲ r 0 ϑj k ϑj ϑk ϑj 0:= ϑj−ϑ0 ϑ0ϑj ϑ0:= 1 n n j=1 ϑj r ϑ0 ϑj 0 ϑj ✲ r 0 ϑj 0 ϑj ϑ0 s0 k n j=1 |ϑj 0|2≤ j=k |ϑj k|2 ϑj 0 φj t;0 =ρj t;0 +λaj 0+εj t;0 (aj 0∈Q) aj 0 ϑd(ri, sj) :=0i= 1j=k; ϑj k ϑ(ri, sj) ϑr(ri, sj) :=0i= 1; ϑj 0 ϑ(ri, sj) ϑ(1) c(ri, sj) = ϑ(ri, sj)−1 2 2 i=1 ϑ(ri, sj) = (−1)i1 2[ϑ(r2, sj)−ϑ(r1, sj)] = (−1)iϑj 2 ϑ(2) c(ri, sj) = ϑ(1) c(ri, sj)−1 n n j=1 ϑ(1) c(ri, sj) = (−1)i1 2ϑj−1 n n j=1 ϑj = (−1)iϑj 0 2  ϑc(ri, sj) := (−1)iϑj 0 2 ϑ(r, s) n n ϑ E:= R2nϑ E Eψ ϑEψ =p ψ=φ VψVψ EVψ ψ E+ ψ ϑ|ϑ′E+ ψ:= (ϑ·V−1 ψϑ′)E E+≡E+ ψ E0ϑ(ri, sj) ϕ(sj)−ϕ(ri) En + 1 E0E Ec E0E+E+ c EcE+ c 2n−(n+ 1)=n−1Ec ϑc E ϑc E0 2 i=1 n j=1 [ϕ(sj)−ϕ(ri)](−1)iϑj 0= n j=1 ϑj 0ϕ(sj) 2 i=1 (−1)i − 2 i=1 (−1)iϕ(ri) n j=1 ϑj 0 2 i=1(−1)i= 0n j=1 ϑj 0= 0 ϑc n−1 Pc E Ec ϑcϑ Ec ϑc:= Pcϑ Pc b:= {ej}n j=1 Rn ϑ:=n j=1 ϑjejϑj F:= Rn E+F Sϑ := ϑ(Sϑ)j:= ϑj S† F E (S†ϑ)(ri, sj) := (−1)iϑj  ϑ∈F ϑ′:=S†ϑ/2 Sϑ′=ϑS Vψ ψ:=Sψ Vψ FVψ Vψ=SVψS† S S∗=VψS† SS∗=Vψ S∗F E+ ϑ′∈E+ϑ∈F (Sϑ′·ϑ)F=ϑ′|S∗ϑE+ (Sϑ′·ϑ)F= n j=12 i=1 (−1)iϑ′(ri, sj)ϑj (Sϑ′·ϑ)F= 2 i=1 n j=1 ϑ′(ri, sj)(S†ϑ)(ri, sj) (Sϑ′·ϑ)F=(ϑ′·S†ϑ)E=ϑ′·[V−1 ψVψ]S†ϑE (Sϑ′·ϑ)F=ϑ′|VψS†ϑE+ S∗=VψS† F0ϑ∈F ϑj F0F Fr:= ϑ∈F:n j=1 ϑj= 0 F0Frn−1 Q0QrF F0 Fr (Q0ϑ)j=ϑ0(Qrϑ)j=ϑj−ϑ0 ϑ0ϑj ϑr:= Qrϑ ϑrbn ϑj 0Fr ϑcϑr ϑc=S†ϑr/2 ϑrϑEr n j=1 |ϑj 0|2= n j=1 |ϑj−ϑ0|2= inf ϑo∈R n j=1 |ϑj−ϑo|2 ZnFrn−1 Lrb LrarLr n aj 0 ϑr E Er FrE k F Fd:= {ϑ∈F:ϑk= 0} FdRn−1Qd F FdF0 Qd (Qdϑ)j=ϑj−ϑk ϑd:= Qdϑ bd:={ej}j=kFd ϑdbdn−1 ϑj kFd ZnFdn−1 Ldbd adLd n−1aj kj=k Ld=QdLdLd⊂QdZnLd ZnQdZn⊂Ld Ld=QdZnLr:= QrZnQr=QrQd Lr=QrLd ϑd E Ed FdE E+Fr Sr:= QrS Sr S Sr ker SrE0  E0⊂ker Srdim E0=n+ 1 dim(ker Sr)=dim E−dim Fr=2n−(n−1) = n+ 1 E+Fd Sd:=QdS SrSdker Sd=E0 FdFr QrFdFdFr Rϑd:= Qrϑd FdFrLdLr FrFd Dϑr:= Qdϑr DR ϑk→ϑ0→ϑk (DRϑd)j= (ϑj k−ϑ0)−(ϑk k−ϑ0) = ϑj k erj:= Rejj=k Fr Lrbrd:= Rbd ϑrFr ϑj kϑd=Dϑr ϑr=Rϑd=R j=k ϑj kej= j=k ϑj kRej= j=k ϑj kerj arLrn−1 aj kad=Dar F Lr arn aj 0 TF Fd Frϑ′Frϑ Fd(ϑ′·ϑ)F= (ϑ′·Rϑ)F= (Tϑ′·ϑ)F TRF R†=T (R†ϑr)j=ϑj 0(∀j=k); (R†ϑr)k= 0 DR D†R† (D†ϑd)j=ϑj k(∀j=k); (D†ϑd)k=− j=k ϑj k Vψd bdψdVψr ❄ R ✻ D ❅ ❅■ 0 ◦ϑ←ϑ1 ,...,ϑn ◦ϑd←ϑ1 k,...,ϑn k ϑr←ϑ1 0,...,ϑn 0 ◦ F:= Rn Fd ekFr F0 rrr s s s ad ar s→Ld r→Lr ❅ ❅ ❅ ❅■T ekRn ϑk k= 0 Pn j=1 ϑj 0= 0R F FrFdR D F FdF0FrD R F FdFrR†=T D D† LdLr b ψrV ψdFd VψdV ψrFr VψrQrQr b Vψr=QrVψQT r=QrVψQr V ψrFr V ψrϑ=QrV ψϑ(ϑ∈Fr) FdFr ϑ′ d|ϑdFψ;d+ := (ϑ′ d·ϑd+)Fϑd+:= V−1 ψdϑd ϑ′ r|ϑrFψ;r+ := (ϑ′ r·ϑr+)Fϑr+:= V−1 ψrϑr E+≡E+ ψ ψ Fd+≡Fψ;d+DD ψ Fr+≡Fψ;r+RD ψ V ψd=DV ψrD† V−1 ψd=R†V−1 ψrR ϑd+=R†ϑr+ ϑr+=D†ϑd+  ϑ′ d|ϑdFd+ = (ϑ′ d·R†ϑr+)F ϑ′ d|ϑdFd+ =(Rϑ′ d·ϑr+)F ϑ′ d|ϑdFd+ =ϑ′ r|ϑrFr+ ϑ′ r=Rϑ′ dϑr=Rϑd ϑd2 Fd+ =ϑr2 Fr+ ϑr=Rϑd ϑ E ϑ+ cϑ E+ c ϑ+ c:= P+ cϑ P+ c ϑr:=Srϑ ϑ Srϑ′=ϑrE0 ϑ+ cE+ ϑrϑ+ c SrS+ r ϑ+ c=S+ rϑr ϑd=Dϑrϑ+ c=S+ dϑd ϑ+ cϑrϑd E+S+ rS+ d ϑc+ ϑc+:= S†ϑr+ ϑ+ c=Vψϑc+ ϑ+ c2 E+= (ϑ+ c·ϑc+)E=ϑr2 Fr+ Sr S+ r=S∗ r(SrS∗ r)−1 ϑFrSr=QrS S∗ rϑ= (QrS)∗ϑ=S∗Q∗ rϑ=S∗Qrϑ=S∗ϑ S∗=V ψS† SrS∗ rϑ=QrSS∗ϑ=QrV ψϑ=V ψrϑ S+ r=S∗ rV−1 ψr ϑ+ c=S∗ rV−1 ψrϑr=S∗V−1 ψrϑr=V ψS†V−1 ψrϑr ϑ+ c=Vψϑc+ ϑ+ c2 E+=ϑ+ c|ϑ+ cE+ =ϑ+ c|Vψϑc+E+= (ϑ+ c·ϑc+)E ϑcϑ+ cEc (ϑ+ c·ϑc+)E= (ϑc·ϑc+)E = (ϑc·S†ϑr+)E = (Sϑc·ϑr+)F Sϑc=ϑr ϑ+ c2 E+= (ϑr·ϑr+)F=ϑr2 Fr+ ψ Vψ=η(ri, sj)σ2 ψ σ2 ψη(r, s) ψ:=Sψ Vψ= (ηjσ2 ψ)ηj:=η(r1, sj) + η(r2, sj) ϑj r+ϑj r ϑr+ϑr ϑj r+=1 ηjσ2 ψ (ϑj r−δϑ) (ϑj r≡ϑj 0) δϑ:= n j=1 µjϑj rµj:= 1 ηj n j=1 1 ηj ϑ+ c=ησ2 ψϑc+=ησ2 ψS†ϑr+ ϑr+:= V−1 ψrϑr V ψrFr V ψrϑ′=ϑrFrV ψrϑ′V ψϑ′ F0 ϑ′ϑr ηjσ2 ψϑ′j=ϑj r−δ  δR ϑ′j=1 ηjσ2 ψ (ϑj r−δ) ϑ′Frn j=1 ϑ′j= 0 δ≡δϑ η(ri, sj)ηj= 2 µj= 1/nj δϑ= 0 ϑrFr+ ϑr2 Fr+ = n j=1 1 ηjσ2 ψ (ϑj r−δϑ)2 ϑr2 Fr+ = (ϑ+ c·ϑc+)E = 2 i=1 n j=1 η(ri, sj)σ2 ψϑ2 c+(ri, sj) = n j=1 2 i=1 η(ri, sj)1 η2 jσ2 ψ (ϑj r−δϑ)2 = n j=1 [η(r1, sj) + η(r2, sj)] 1 η2 jσ2 ψ (ϑj r−δϑ)2 η(r1, sj) + η(r2, sj) = ηj ϑr[ψν,t] [ψν,t]Fr+ [ψν,t]E+ c Fr+Fd+Er+Ed+Ec+ E+ c η(ri, sj) ϑ+ c=ϑc=1 2S†ϑr E+ c Ec 0 ✟✟✟✟✟✟✟✟✟✟✟✟✟ ✟ ✘✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ◦ϑr ◦ϑ+ c ◦ ϑ ◦ϑd ϑc ◦ ϑc+ ◦ E Ed Er E+ c Ec E0 ❄ Sr ❄ Sd ✚✚ ✚ 0 ◦ ϑ ◦ϑd ϑr ◦ F Fd Fr F0 ◦ϑd+ ❅ ❅◦ ϑr+ ◦ Srϑc+ E0 E SrSd ϑcϑ E0E Ecϑc=Pcϑ ϑcϑc ϑr #rϑd #d SrFrE+ c E E+ϑ+ c=S+ r#r=P+ cϑ ϑ+ c=V ψϑc+ϑc+:= S†#r+#r+:= V−1 r#r Srϑc+=Sϑc+=SS†#r+= 2#r+ S+ dFdE+ cϑ+ c=S+ d#d ϑ+ cϑc E+ cEc ϑ+ c2 E+=ϑr2 Fr+ =ϑd2 Fd+ = n j=1 1 2σ2 ψ |ϑj r|2 E+E+ c  SdC η(ri, sj) k= 1 V−1 ψd κj,j′=1 2σ2 ψ ×1 n n−1j′=j −1j′=jj, j′∈{2,...,n} ϑFd ϑ2 Fd+ = (ϑ·V−1 ψdϑ)F= n j=2 ϑj k(V−1 ψdϑ)j j= 2,...,n (V−1 ψdϑ)j=1 2σ2 ψϑj k−1 n n j=2 ϑj k =1 2σ2 ψ(ϑj−ϑk)−1 n n j=1 (ϑj−ϑk) =1 2σ2 ψϑj−1 n n j=1 ϑj =1 2σ2 ψ ϑj 0 n j=2 ϑj k(V−1 ψdϑ)j=1 2σ2 ψ n j=2 (ϑj−ϑk)ϑj 0 =1 2σ2 ψ n j=1 (ϑj−ϑk)ϑj 0 =1 2σ2 ψ n j=1 (ϑj 0−ϑk 0)ϑj 0 n j=1 ϑj 0= 0 ϑ2 Fd+ = n j=1 1 2σ2 ψ |ϑj 0|2(ϑj 0≡ϑj r) t ξt ρj t ξ2;tr2ξ2;t= ξ2;t+ξt ρj t=ρt(r2, sj)−ρt(r1, sj) ρj t ρj t=ρj t+ (dj t·ξt)R3 dj t sj→r2tJt jth dj tJt ρt= ρt+Jtξt ρt;r = ρt;r +Jt;rξt(Jt;r := QrJt) t xt xt:= (α, ξt)T α≡arFr t1, t2,...tn X:= (α, ξ1, ξ2,... ,ξn)T ξn≡ξtnα t ytt yt:= pt;r− ρt;r φt;r − ρt;r yt=Atxt+ At:= 0Jt;r λIαJt;r Y= (y1, y2,... ,yn)T yn≡ytn Y=AX +  A A:= ·J1;r ·· · ·· λIαJ1;r ··· ·· ·· J2;r · · ·· λIα·J2;r · ··· · · ·· · ·· · ····· · · · · ·· · ·· · ····· · · ····· ·Jn;r λIα··· · ·Jn;r x≡(α, ξ)T tn xn|n=xn|n−1+Knvn vn=yn−Anxn|n−1 xn|n:= (αn, ξn)xn|n−1:= (αn−1,0) Kntnvn α Vbα ˇα Lr α Frf(ϑ) := (ϑ·V−1 bαϑ)F ˇα= argmin α∈Lr α−αV−1 bα Lr brd k α αd:= Dα bd αbrd αd:= Dαbd α brd DVbαD∗VDbα=Vbαd Lr Ld ˇαd= argmin αd∈Ld αd−αdV−1 bαd ˇα=Rˇαd k= 1 α Vbα αd=DαVbαd= DVbαD∗n−1 αd αj k=αj−αk(j= 2,...,n) Vbα (n−1)2Vbαd ˇαd ˇα=Rˇαd ˇαj= ˇαj k−ˇα0 kˇα0 k:=1 n j=k ˇαj k ˇα ξn  wn:= yn−Anxn|n=Hnvn Hn:=I−AnKn In wpwφ w w2:= wp2 Fp;r+ +wφ2 Fφ;r+ ψ=p φ wψ2 Fψ;r+ = n jψ=1 cjψ cjψ:= 1 ηjσ2 ψ (wj ψ−δwψ)2δwψ:= n j=1 µjwj ψ w2 β= jp∈Ωp βjpejp, jφ∈Ωφ βjφejφ ΩpΩφ {1,...,n} ρj+c[dt(r2)−dt(r1)] + ǫj= pj−βjpj∈Ωp pj ΩpΩφ βjpβjφ w w=Hδv=H δy y ejpejφ zjpzjφ yset =y−zjψ zjp:= (erjp,0) zjφ:= (0,erjφ) aset =a+b a a+b erj ej′ rj=−1/n j′=j ej rj=1−1/n wejpejφ fjpfjφ wset =w−Hzjψ fjp:= Hzjp fjφ:= Hzjφ w β Mβ := jp∈Ωp βjpfjp+ jφ∈Ωφ βjφfjφ Mβ wM fjpfjφ Ω := Ωp∪Ωφ θ θ0 TLOM :=w2/mm= 2(n−1)−3 tLOM := Fθ(m,∞,0) θ Fm,∞ TLOM< tLOM r= 1 Ω = Π =∅ Π jψ/∈Ωcjψw2 cmax := max jψ/∈Ωcjψ κ≤1 Πr:= {jψ/∈Ω: cjψ≥κcmax}  r rr r 3p5p3φ5φ c5φ cjψκ=0.5 n= 7Ω = ∅ 3p5p3φ5φ 5φ jψ∈Πr jψ/∈Π fjψ:=H· (erjp,0) ψ=p (0,erjφ)ψ=φ gjψ:= fjψΠset ={jψ}Π= ∅ Π∪{jψ} Πjψ fjψ r= 1 {g◦ q}q<r q<r ςq,jψ:=g◦ q|gjψ := ψ′=p,φ g◦ q;ψ′|gjψ;ψ′Fψ′;r+ ψ′pφ g◦ q;ψ′ g◦ qgjψ;ψ′ςq,jψ gjψ set =gjψ−ςq,jψg◦ q ςq,jψ=g◦ q|fjψ gjψg◦ q q<r w gjψhjψ|whjψ hjψ:=gjψ/gjψ |hjψ|w| γjψ:=gjψ|w/̺jψ̺jψ:=gjψ gjψ|w:= ψ′=p,φ gjψ;ψ′|wψ′Fψ′;r+ gjψ2:= ψ′=p,φ gjψ;ψ′2 Fψ′;r+ ¯¯ ψ |γjψ| ¯¯ ψ:=arg max jψ∈Πr |γjψ| χ0θ0/2 χ0:=Nθ0/2(0,1) |γ¯¯ ψ|> χ0m>0 ωr:= ¯¯ ψΩset ={ωr}r= 1 Ω∪ {ωr}r>1 γ◦ r:= γωrg◦ r:= gωr/̺ωr ◦ Ω Ω = {ωq}r q=1 {g◦ q}r q=1 Mr q=1 γ◦ qg◦ q w β◦ r:= βωrf◦ r:= fωr |γ¯¯ ψ|< χ0TLOM >5tLOM |γ¯¯ ψ|<χ0TLOM<5tLOM r>1 |γ¯¯ ψ|>χ0m= 0 g◦ rf◦ q uq,r g◦ r= r q=1 uq,rf◦ q  uq,r= −1 ̺ωr q≤q′<r uq,q′ςq′ ,ωrq<r 1 ̺ωr q=r 1≤q≤ruq,r rth U β◦ q r q=1 γ◦ qg◦ qf◦ q r q=1 γ◦ qg◦ q= r q=1 β◦ qf◦ q [γ◦]γ◦ q q= 1r[β◦] [β◦] = U[γ◦] β◦ q set =β◦ q+uq,rγ◦ rq<r ur,rγ◦ rq=r 1≤q≤r ww2 wset =w−γ◦ rg◦ rw2set =w2− |γ◦ r|2 m mset =m−1 tLOM set =Fθ(m,∞,0) TLOM set =w2/m TLOM> tLOM rset =r+1 KΩK ω1,...,ωr xset =x−KΩ[β◦] [β◦]= U[γ◦] x Vbx set =Vbx+ [KΩU][KΩU]T {g◦ q}r q=1 [γ◦] w w w ̟0 ζ0 θ0̟0 θ0= 0.001 ̟0= 0.80 ζ0 Fθ(m,∞,0) θ  ηj= 2j w2=1 2σ2 p n j=1 |wj p|2+1 2σ2 φ n j=1 |wj φ|2 jpjφ |wj p|/σ p|wj φ|/σ φj= 1,...,n 1 −303 121238 ˇaj fν;kj= 1, . . ., nn= 9k=1 j;fνf1f2 10 0 234 868257 496 3625 263−196 104 4−2 502 896−1 419 324 512 155 3239 705 967 6−2 593 167−1 303 294 75 973 7734 346 092 89 056 7407 252 801 9−9 386 838−7 332 507 4745 j= 1,...,9 (ˇaf1,ˇaf2) k= 1 j= 1,...,n k= 1 ˇαset = ˇaf1ˇαset = ˇaf2 −303.39 120.92238.49 •2f1 j= 2 • −1f2 j= 4 TLOM 9.13 tLOM (1.26) TLOM (f2; 1p) 5.40 (f1; 4φ) 2.63 (f1; 1p) 1.63 (f1; 8p) 1.06 TLOM TLOM 1.06 tLOM 1.30 TLOM 335.09 tLOM 1.26 TLOM (f1; 2φ) 108.81 (f2; 4φ) 9.42 (f2; 1p)5.41 (f1; 4φ) 2.82 (f1; 1p)1.74 (f1; 8p)1.13  TLOM1.13 tLOM (1.34) κ= 1κ= 0 κ= 0 κ0.5 βfν;jψ n= 9 f1 jψ p−4.806 −3.304 φ0.043 f2 jψ p−8.755 βf1;2φ≃2λ1 βf2;4φ≃ −λ2 f1 jψ p−4.806 −3.304 φ0.381 0.043 f2 jψ p−8.755 φ−0.248  |