Phép chỉnh hóa Tikhonov cho một số bài toán ngược

i~eeveti6uhoamQtphiSm hamkhae. Dill" Ilghia3.1.2 Phie-mhamTikhonovJa(x) VOla >0dl1cJedtnhnghTa0011sail: .la(x)=IIFx-Ylf+allxlf, x EX (3.1) Bai loanqfeti6uhoaphiEmhamnayla conghi~mvahonnuanghi~rneuano l~ilanghi~mcliamQtphuongtr1nhehlnh. Dillh if 3.1.3 PhiC'mhamTikhonovJa(x) trenkhonggianHilbertX eocluynha'tmQteveti€u xaEX vanoelinglanghi~mcluynha'tcuamQtphuongtTlnhchinh: ax+F*Fx=F*y (3.2) ChU'O'ng3 26 Changminh TnJ'octieDtachU'ngminhtfnhchinhcuaphuongtrlnh (3.2). Xetanhx~songtuye'ntfnha:X xX ~ R nhusau: a(x,z):==a(x,z)+(Fx, Fz) Anhx~naylitlienWevabU'eVI:. Ila(x,z)lI:::;all(x,z)II+II(Fx,Fz)11 :::;allxllllzll+IIFxllllFzll :::;(a+IIFW )llxllllzll . a(x,x) ==allxW+IIFrl12~allxW Xet phi€m hamtuy€n tinh A:X ~ R nhusau: Az:=(F* y,z) AP dt;ngdinh1yLax-Milgramtadu~c: 3!xaEX:a(xa,z)=Az,\izEX Sur fa: a(xa,z) +(Fxa,Fz) ==(F*y,z) Vz EX (ax -I-F *F v 7\ ==( [7 * 1J 7) \../z c V::r ' -'a'~J' J'~ V ~.'l ax +F* Fx =F* ya a V?y phucJngtrInh(3.2)conghi~m. Nghi~mnay1aduynha"tVI: ax+F* Fx =a=>(ax+F* Fx,x)==0 =>allxW+llfxW==0 =>x=o £>6kicimtratinh6ndinhtaxetday(x,,)c X thoa: Y =(aI +F *F )x ~ 0/I /I Taco: Chuang3 1.7 (YII'X,,) = allx"W +1IFxllW sllYnIIllxlI1I ~ allx"llslly,,11 ~XII~O Cu6iclingtachungminhqtc ti6ucua(3.1)clingIii nghi~mcua(3.2)viingu'<jc I~i. .la(x)- .la(xa)=IIFx-Y112-IIFxa- yW+aCllxW-llxaW) Ap d\lOgcongthuc lIuW-llvl12=llu-vW+2(v,u-v) ta du'<;1c: Ja(x)- Ja(xa)=IIF(x- xa)112+2(Fxa- Y,F(x- xa»+allx- xaW+2a(xa'x-xa) =11F(x- xa)112+allx-xaIl2+2(F*Fxa-F* y +axa'x-xa) (3.3) Ne'uxaIa nghj~mciia (3.2)th1do (3.3)taco: .la(X)- .la(xa)=IIF(x- x~)W+allx-xaW~0 DenXuIii qtc 66ucua(3.1). Ne'uxaIii qJ'cti6uClh (3.1)taapd\lng(3.3)voi X=xa+tz,t>0: t21IFzW+2t(F*Fxa - 17*y+ axa,z) +at211zW~0 Dongianchotr6ichot ~ 0 tadu'<jc: (F* Fxa-F*y+axa,z)~O,Vz EX DiGunaydf{nde'nX"Iii nghi~melm(3.2). 0 Tv daytagqix: Iii nghi~mcuaphuongtdnh: ax+ F* Fx =17*y" Phepchlnhhoa trongtru'ongh<;1ptllye'ntinhse H{yx: UlmmQtxa'pXl cho nghi~mchinhxac x* cuaphu'ongtrlnh17x=y lingvoi saiso'trenduki~nIii 0, nghlaIii II)i - JIllso. TnioctieDtasedaubgiasaiso'chotru'ongh<jptuye'ntinhcompact.Ta giasii' F: X ~ Y Iii m()toaDt-ttuye'ntinhcompactva1-1giuahaikhonggianHilbertvo h~nchiGu.GQi (,Llj,Xj'Yj)jEf1.JIii singularsystemcllaF. f)~t: Ra:=(aI+F*Fr' F* ChU'ffng3 28 Taco Ra Iii tuye'ntinhlientvcdochungminhcuadinhIy3.1.3. M?nhd~3.1.4 Ra Iii mQtsod6chlnhhoavdi: '" flj a) Ray==I-~(Y'Yj)Xj }~Oa +Pj 1 b) IIRall~ r2 ;a VYEY (3.4) (3.5) Changminh a) Do (pj,Xj,Yj) Iii singularsystemvii F 1-1Den{Xj}Iii mQth~d~ydu.,nghlaIa: '" x== "' (x x. )x. Vx E YL.. ' ) .I' ./ j=l Xct Y EY vii d~tz=RaYtaco: (aJ+F*Fr1F*y=z hay (aJ+F*F)z=F*y IDa: '" '" 00 F* Y= ICF* y,x)Xj ==ICy,E'.:)xj ==L,uiY'Yj)Xj j=l j=l j=l ' '" (al +F* F)z ==IC(al +F* F)z,xj)Xj j=l '" =I[aCz,xj)+CF* Fz,xj)]Xj j=l '" =I[a(z,xj)+Cz,F* Fxj)]xj j=l uo ==ICa+ Jl~)(z,x)Xj j=l Tlido: (a+ Jl~)(z,Xj)==,u/Y,Yj) V~y: '" '" fl. Ray=z=I(z,x)Xj=I .I 2 (Y'Y)Xj }=I }=Ia +Jl j Chu'o'ng:3 29 b) Do a) 00 2 2" f.1. 2 IIRayll = f:t(a+~~)21(Y'Yj)1 Theoba'td~ngthucCauchy: 1r f.1j <- a+f.1~z.2-vaflj=>(a+fl~)2- 2ra (3.6) Tlid6: II RaYWs 4~~1(Y'Yj)12S 4~IIYW V~y: 1 liRalis 2J:x 0 Dinh Ii 3.1.5 Nellx*=F*z EF*(Y) voiIlzllsE thlkhichQna(c5)=~(c>O)tasec6: Ilx~- x*lls1-(-Fe+ ~)J8E2 -vc (3.7) Chungminh Tac6: 00 x* =I (x*,X)Xj j=1 -, ~ f1j (I ' * )R(J'x*=L.,;-~ 'x ,Yj Xj }=1a +f1} 00 2 ~ f.1} ( * )x=L..J 2 X ,Xi".i }=Ia+ f.1j (M9nhd~3.1.4) Tlid6: Chtro'ng3 30 00 a2 II RaF:x*-x*W=" I(x*,x .)12 f=t(a+f-l~)2 .I (3.8) =f a2 }=](a+f-l~)21(F*Z,X)12 00 2. -" a f-l2 -Lt .I }=I(a +,u~)21(z,Y)12 a ~-llzW4 (do3.6) Hay: IIRaFX*-x*IIS..Jd £ 2 (3.9) Nhu'v~y: Ilx~-x*II~IIRay8-R"Fx*II+IIRJ;X*-x*11 ~IIRalll!y8- YII+IIRaFx*-x*11 1 . -r;;s-o+-£ 2..Ja 2 (do(3.5)va(3.9» co Thay a =Ii taauqc(3.7). 0 Dinh Ii 3.1.6 .' 2 Ne'u x*=F* Fz E F* F(X) volIlzlI~E thlvOlcachCh9lla(o)={;y, c >0 taco: 1 ! ~ IIx: - x"'ll$(c+ r )£3532-..;c (3.10) Changminh Do(3.8)taco: Chuang3 3f 00 a2 II RaFx"'-x"'W=I 2 21(F'" FZ,Xj)f J=I(a+Jlj) 00 a2 Jl4. 2 =I ( ~)21(z,x)1j=1 a+Jlj 5allzW Hay: IIRaFx*-x*ll~aE Dov~y: II x; -x*115I1RaIiIIYo- yll+IIRaFx*-x*11 1 5 ---=0 +aE 2Ja . 2 ( 15 ) 3 Thay a(O')=c E ta du'<Jc(3.10). 0 Dj~uclangng~cnhicnlab~chOitt)cuaphepchlnhhoaTikhonovkhongth8cao hondu'<Jcnua.M~nhd~sanchungtodi~ud6. (Xem [7 ], trang40) M?nh de3.1.7 Cho F:X ~ Y lamQtloantittuye'ntinhcompact,1-1saochoRangeFla voh~n chi~u.XetXEX, ne'ut6nt~imQthamlient\lca:[0,+00)~ [0,+00)thoa: . a(O)=0 . 2 8 -- limllxa(8)- xIIO'3=0,0->0 Vy8:IIYO- yll~8 Khi d6 x=0. BaygiGtasedanhgiasais6euaph6pchlnhh6aTikhonovtrongtru'ongh<JpF la loantittuye'nHnhlient1,lC.Ta thudu'<Jccaeke'tquatu'ongttfnhu'djnh193.1.5va 3.1.6. ChutJ'ng3 32 Djnh Ii 3.1.8 Ne'u X*=F*ZEF*(Y) vdillzlJS:E thl: Ilx:-x*II~ aE+5fa (3.11) Changminh Taco: ax:+F* Fx; =F* y6 ax*+F* Fx*=F* y+a,t'* SHYfa: a(x; -x*)+F* F(x; -x*) =-ax*+F*(y6- y) Nhanvohu'dnghaive'cuaG~ngthli'ctrencho(x~- x*) taGu<;1c: allx; - x*W+11F(x; - x*)W=-a(F *z,x; - x*)+(y' - y,F(x;- x*)) =-a(z,F(x: -x*»+(y6 - y,F(x: -x*» ~aEIIF(x; -x*)11+51IF(x;-x*)11 =(aE +5)IIF(x: - x*)/I (3.12) Bobdts6h<;ingc1~utrongve'traicua(3.12)taco: IIF(x; -x*)II~aE+8 (3.13) Tli (3.12)va(3.13)tasoyfa: allx; - x*112~(aE+0/ Nghla la: Ilx; -x*II~ aE+0ra 0 Ghiclni: Nell tach(;ma =0 thl: II x; -x*II=0(-/5) ChUb'ng3 33 Dtn1l1j3.1.9 Ne'ux*=F*}<zE F* F(X) voiIlzll~E thl: 3 Ilx;-X*lls5+a2E rex (3.14) Changminh D fi t' .-:I? J( ) " 0 Xa a clfc tIeu clla a X nen: Ja(x;) S Ja(x*-m:) Nghla1a: IIFx;- yOW+allx;112sIIF(x*-m:)- yOI12+a!lx*-m:112 IIFx:- y"+aFz-aFzW+allx;-x*+x*W~lIy-y"-aFzW+allx*-azW Khaitrj~ncab6ns6h?ngtrongba'toiingthucireDtaou\1C: IIFx: - yO+aFzW-2(Fx:- y" +aFz,aFz)+allx:-x*W+2a(x;-x*,x*) slly- y"112_2(y-O,aFz)+allm:W-2a(m:,x*) Thayx*=F* Fz (~s6h<.lilgthli'tl1cuahaive'vas~pxe'pl;,ti: !lFx;- yO+aFzW+allx;-x*W slly-yOW+allazI12+2(Fx:- yO,aFz)- 2a(Fx; - y,Fz)- 2(y- yO,aFz) =lIy-y()W+allm:112+2a(Fx;- yO- Fx; +y- y+yO,Fz) (3.15) , v ' 0 Bo s6h?ngd~utrongve'tniieua(3.15)tadl1<jc: allx; -x*W ~52+a3E2 Sitd\1ngba'td~Dgthite~ b2sa +b (a,b>0): 2 . 1°2 +a3E2 5+a3E II ()-x* ll ~ --~ f- Ixa \ a -va 0 Ghiclul 2 Ne'utaeh9Da=53thl: 2 IIx; -x*lI= 0(53) Chlfo"lIg3 34 3.2PhepchinhhoaTikhonovchohili toanphituy~n. Ph~ncu6icuachu'cfngnaytasedaubgfasai86trongtru'ongh<JpF hituye'nva lient\1cgifi'ahaikhonggianHilbert.Ta vlinxetnghi~mchinhhoax; nh~ndu'<Jctit vi~cqtcti~uhoaphiC'mhamTikhonov: .la(x)=IIFx-yW+allxW Bai toaDqfc ti6uhoanaykhongphaikhi naGclingeonghi~mvanghi~mne'u corungkhongch~cduynhift.Themvaodotoantii'F dU'<Jex tphai"gdn"v(1imQt toaDtii'tuyC'nttnhlien t\1ctheemQtnghlanaod6. Trongphgnnaytadu'aracaegiathiC'tsau: (AI) T6nt?iclfcti~ux~. (A2) Tan t?i tmlntii't1JyC'nHnhlien t1;1cG:X ~ Y va 86L >0 thoa: L /lFx- F.x:*-G(x - x*)/lS;-/lx- x*W2 'i/xeX TntoclientatrlnhbayIDQtso'di6uki~ndud~co.(AI) va(A2). M?n/I de3.2.1 Ne'uF ladongyC'u(nghlalanC'ux"-?-x vaFx"~ y thly=Fx) thl(AI) thoa. Chungminh f)~t J=~~fJa(x). Tan t1;liday {XII} sao rho J~(x.J~J d§n de'n IIFx,,11va IIx,,11bI eh?n.Do tfnhcompacty€u cuacaehinhc£ud6ngtrongkhong gianHilbertva F la d6ngyC'utarutdU'<Jcdaycon {u,.}cua{x,,}thoa: u -' x vaFu -' FX" . II Tir ttnhmlalient\ICdU'oiy€u cuachu~ntac6: /lxllS;limin~lu,,/i IIFx - yllS;limin~IFul!- yll Suyfa: ChcM1g3 3S J ::;11FX - YI12+allxI12::;liminfClIFun- yl12+allunW) =liminfJa(u,,)=J V~y: .laCx)=J 0 Mfllh d€ 3.2.2 Ne'uF khavi FnkhetvacoL >0 thoa: IIF\ _F'X, II::;Lllx, -x211 VXI,X2EX thl(A2)lhoavoi G=F'x. Changminh Theo d!nh19giatr!trunggiantaco: 1 F(x) - F(x*) =J r;'x.+1(-<-x.)(x - x*)dt 0 Tli'do: I F(x) - F(x*) -:F'x.(x- x*)=J (F'x.+t(X-~.)F'x. )(x-x*)dt 0 Suy fa: I IIF(x)-F(x*)-F'x.(x-x*)II::; J Lllx-x*I!2tdt 0 I =Lllx-x*W J tdt 0 L =21Ix-x*W 0 Ta sechungminhtrongtru'ongh<Jpdu'c;1cxctcacsai56v~Iighi~mclingcob~c tu'dngtvnhu'trongtHronghc;1ptuye'ntinh. Djllh Ij 3.2.3 Ne'u x* =:G *Z E G * (Y) voi IIzll::;~thl:L 5+allzll IIx~- x*ll::;ra~1-Llizil (3.15) ChU'Cfng3 36 Chungminh Dox; 1actfctieuciia Ja(x) nen: Ja(x;) ~Ja(x*) Nghia1a: JJFx~- yOIl2+allx:W~IIFx*-yOW+allx*W IIFx: - yO+az-azI12+allx:-x*+x*W ~IIFx*-yOW+allx*W Khai trieDvarutgc.m: IIFx:- y'"+azW+allazW-2a(Fx:- yd+az,z)+allx:-x*112=2a(x: -x*,x*) :$IIFx*-yOW Bo s5h9-ngd~ucuavritraivasii'dl;mgx*=G *z taco: allx; -x*W ~IIFx*-y"112+llazW+2a(Fx;-yO -G(x~-x*),z) , v ' =IIFx*-y" +azI12-2a(Fx*-yO,z)+2a(Fx;- yO~G(x;-x*),z) =IIFx*-y" +azW+2a(Fx;- Fx*-G(~; - x*),z) ~IIFx*_y8+azW+2allzIlIIFx:-Fx*-G(x; -x*)11. ~IIFx*_y8+azW+2allzll~lIX:-x*W2 (doA2) Suyra: a(l- LlizlDllx;-x*W ~IIFx*-yO+azW :$(1IFx*-yOIl+llazll)2 ~(o+allzlli V~y: o+allzll IIx;-x*ll:$Fd.J1-Ll!zll 0 Ghi clUJ . Nriuchqna =0 thl: IIx; - x*ll=D(JJ) ChtW'ng3 37 Djnh Ij 3.2.4 Ne'u x*=G*GwEG*G(X) vdiz=Gw,llzlI::;;! thl:L L 3 0+--llawll~+a21Iwll-Jl+Llizil IIXC)- x*ll< 2-a - - ra-JI-L/lzil (3.16) Changminh Dox; la qic ti0uclia Ja(x) nen: J( 8 ) < /( * \ a xa -. a X' -aw) NghTala: IIFx: - y"W+allx~W~IIF(x*-aw)- y"W+allx*-awW (3.17) Taco: IIFx; - y"W=IIFx;- y"+aGw-aGwW =IIFx~- y" +aGwW+11aGwW-2(Fx;- y"+aGw,aGw) =11F.: - y" +aGwW-lIaGwW-2(Fx:- y",aGw) Ilx~W==:llx~-x*+x*W =llx:-x*W+llx*W+2(x:-x*,x*) =/1x; -x*W+llx*W+2(x~-x*,G*Gw) =llx~-x*1I2+lIx*W+2(G(x;- *),Gw) Ilx*-awI12=llx*W+llawI12-2(aw,x*) =llx*W+llawW-2(aw,G* w) =lIx*W+llawW-2aIIGwW (3.18) (3.19) (3.20) Thay(3.18),(3.19),(3.20)VaG(3.17)varutg9ntadu'<Jc: II.Fx:- y8+aGwW+allx:-x*W ~IIF(x*-aw)- y8W-lIaGwW +a3I1wW+2a(Fx~- y" -G(x~-x*),Gw) Bo soh~ngd~ucuav€ trai,tachtichvohu'dngvasird\mg(AI): allx: -x*W :::;IIF(x*-aw)- y8W-lIaGwW+a3I1wI12. +2a(fx; - y-G(x; -x*),Gw)+2a(y- yO,Gw) ::;;11F(x * -aw) - y"W-IJaGwjf +a311wjf +2a~lIx~-x*WllzlI+2a(y-yO,GW) . ChuO"n~3 38 Hay: a(l- Lllz)l)llx;-x*WS;IIF(x*-aw)-y8W+a31IwW-(lIaGwW-2(Y-y8,aGW)], , Si'td\Ingd~ngthU'cIlull2_lIvW=lIu-vW+2(v,u-v) taco: ' a(l- Lllzll)llx:-x*W ~IIF(x*-aw)- ydW+a31IwW-lIy-d-aGwW+IlY-y8112 ==IIF(x*-aw)- y8112_IIY-y8-aGwW +a31IwI12+lIy-8W , v ' ~IIF(x*-aw)- Fx*-G( -aw)W +2a(y- y8-aGw,F(x *-aw)-Fx*-G(-aw»+ a311w112+82 Sir d\lOg(AI) m('>t]~nnua: [2 L a(1- LllzlI)lIx~-x*W ~~llawI14+(25+allz/I)-lIawW+a3I1wW+52 4 . 2 . L . =(5+-lIawW)2 +a3l1wW(1+LllzlI)2 . Suy fa: L (5+-llawWi +a3l1wW(1+Lllzll) Ilx:-x*II~11 2 a(l-Lllzll') . Tft ba'tdiingth(fc-J02+b2~0+b (o,b>0) tasuyfa: l ~ 8+~llawI12+a21Iwll~1+Llizil Il x8-x* ll< 2 'a - ra~I-Llizll 0 Ghichll 2 Ne'uchQna =53 thl: 2 Ilx~-x*lI= 0(53) Chu'a'ng3 39 ._.