Luận án Phép chỉnh hóa tikhonov cho một số bài toán ngược

CHuaNG 2: IAI 'fOANMGU'~e,'filtHHHONGCHiNNVA PHi, cHiNNMOA& 201HaitO1inguQevaeaevi d\}o Haibili loandu\1cgQila nguQcnhaune'unhugiathie'tcuabai loannayl~iHl ke'tlu?ncuabaitoankiavangu'Qcl~i.ChungtasegQimQtronghaibaitoannay la bai loanthu~nth1bai Loanconl~isc gQila bai loanngu'<1c,Tuy nhienthong thu'ongc6 IDQtbai LoanduQCnghienCUlltrudcnen n6 duQcxem nhuHi b~dloan thu?n. Vi du1(Bai loannhi~lnglt<;lcthaigian) Xet b~dtoanthu?n1abai toaDCauchy<.:hophU'ongtdnhnhi9t: [ au a2u 81=a;/ u(x,O)= vex) (XEIR,t>O) (xelR) Trangd6vex)HimQthamlient\lcvabi ch~ntrenIR.Ta bierdingnghi9mlien l\lcvabi ch~nu(x,t)cuabaiLoannayla t6nt<;\ivaduynhfft(xem[3]trang115)cho bdi: 1 - [ . 2 }u(x,t)= I. Jv(q)exp- (x- q) II;2"m-<0 4t Bai LoanngU'Qc: [ au a2u -=~ (x EIR,( >0)at ax u(x,l)=lex) (xelR) TIm u(x,O)=vex). Vi9Cgiai bai LoannaydiindenVi9Cgiiii m(,lphuongtdnhtichphanthuQcd~ng tich <.:h~p: 1 7 [ (x- q)2 }2;;~v(~exp- 4 1;=lex) Chlidng2: Sai toan ngt(q'c... f3 V{du 2 (Bai todnngu<Jctrongvi~cthCimdOdiGchat) X6tbiii tmlnthu?nIii Hmthiinhph~nth~ngdung/vCliahiehapd~nlsinhfabCli mQtmi~nCJdQsituh phladudim~tdilt bitt Svthayd6i elmm~tdQv~tcha't(mass density) Iii p =p(x) 0 ~x ~1(Bai roan mQtchi~u). m~tda'tx h f fv 0 1 Thea dinhlu~tha'pdancuaNewtontaco: f m ( 1 , h"" ,=r 2"" r a angsolr9ngtrLfong).r Trangtruongh<jpdu<jcxct: . m=p(x')L1x'Iakh6iIu<jngcuamQtddnvi thericht~ix'. . r =~(x- X')2+h2 IakhoangeachgiUahaidi~mx vax'. Taco: 4.fv(x)=4.f(x)cosB= rP(X')L1x'cosB= yhp(x')tu' (X-X')2 +h2 3 [(X - X')2+h2F Suyfa: I .f,,(x)=yhJ p(x')dx' 0 [( 3 X-X')2 +h2]i Bai roanngu<jctrangvi~cthamdodiachiitIab~ngdl,lllgC1,ldod~cngu'oitaeo thexacd1nhdu<jcIvvatli'dotimfa p. ChU'O'I1g2- 8ai toanngU'q'c... 14 2.2TinhkhOngchlnh. HadamardchodingmQtmohlnhtoaDhQceilamQtbili toaDV?t19'Iii chlnh (well-posed)n6unhunothaaman3tinhchillsauday: 1. T6nt<;linghi~mcilabailoan.(Existence) 2. Bai loancokhongquatnQtnghi~m.(Uniqueness) 3. Nghi~mcilabili loanph\)thuQclientl)evaodit'ki~n.(Stability) v~phuongdi~nloanhQe,slft6nt<;linghi~mcoth~d<;ltdU<;1cb~ngeachmbrQng khonggiannghi~m.N6ubai loancoquamOtnghi~mdi~udoeilngconghlala ta thi6uthongtin v~nghi~m.Trong trUongh<;1pnayb~ngeachchothemthongtin v~ nghi~mta co th~d<;ltdU<;1cHnhduy nhiltcila nghi~m.Nhu v~ytrongba Hnhchill trentinh6nc1)nhIii quantrQngb~cnhilt.N6limQtbailoanthi6utinh6ndjnhthlv~ m~thlfct6bili toandokhonggi?1idu<;1cVItabi6tr~ngbiltkymOtphepdod<;lcnao dingconhG'ngsais6d~nc16ndG'ki~neilabaitOaDloonbj nhieu.Chodilsais6tren dit'ki~ncilabili tOaDconhad6ndauthldotinhkhong6ndtnhcoth~dftnd6nsais6 rilt Wngiit'anghi~mtinhtOaDdU<;1evC1inghi~mthlfe8lfeilabai loan. Ta co th~phatbi~utinhehinhduC1id<;lngtoanhQcgall: Dink nghia2.2.1 Cho X va Y la hai khonggiandjnhehuiin.F Iii mQtanhX<;ltUX vao Y. Phuong trlnh Fx=y QU<;1egQiIii chinhn6uthaacacdi~uki~ngall: 1.T6n t<;li: \ly EY,3xEX:Fx =y 2.Duy nhilt: \Ix],X2EX,Fx] =Fx2=>Xl =x2 3.Ondjnh: \lex,,)c X,Fx" ~ Fx=>x"~x Nhungphuc1ngtrlnhkhongthaamQttrongbaHnhchilltrenduQegQiIii khong chinh(ill-posed). Vidu MQtvi dl,le6di~nv~tinhkhongehinhdl1<;1Cdl1afa b('1iehfnhHadamard:bai' toanCauchychophl1c1ngtrlnhLaplace. Chl1linq1'~B~iJoan "9U"11;... 15 TIm hams5u(x,y)thoa: I:!.u=82u+(iu =0 8X2 8y2 u(x,O)-:;:lex) 8u -(x,D) -:;:g(x) 8y (x,y) EIRx (0,+00) XEIR XEIR Nghi9mlingvoitrtfC1ngh<Jplex) -:;:0,g(x) =!sinnx du<Jchob<'1icongiliac:n u(x,y)=~-sinnxsinhnyn Ta co: 1 sup(jf(x)I+lg(x)1)=- ~ 0 oyeR n Nhting: suplu(x,y)l=~SinhllY~ 00xeR n DoV?yHnh6ndinhbi vi ph<;lm. Do eachd?t va'nd~cuaHadamardHentru'ocday ngtiC1itaChIxetcaebai loan cMnh.Caeb~tiloannayVIdu<Jcxettru'C1cneng9i Iii bai loanthu?ndftnde'ncaebai toanngu<JcthuNngIakh6ngch1nh. Nhi~ubili loanngU'<Jcdftnde'nvi(;cgiaimQtphU'dngtTlnhrichphiloIo<;1imQt. Cac toaDtli'tichphiin nay thuNngcompactvoi nhungdi~uki(;nn~lOd6 cua nhan. cacm9nhd~sanchungtophuongtr'inhF.x=y voiF Iii loantii'tuye'ntlnhcompact thuongIii kh6ngch1nh. M?nhd~2.2.2 ChoX, Y Iii cackh6nggiandjnhchuffnvaF Iii toantU'tuye'nHnhcompactthoa dime¥KerF)=00.Tc}nt~iday(x,,)c X saochoFx"~ 0 nhltngx"-40. Honnaa taco th~ch9ncho IlxlIll~00. ChlfO'ng2: Biii toaongu'q'c... 16 Changminh Tadabittkhonggianthudng'lKerFHiffi9tkhonggiandinhc{1U~nvdi chufin: II[x]II:=inf{llx+zll:zEKerF} Xet loanta F t1t"YKerF vaoY dinhngh~abdi: F([x]):==F(x) Toantanayxacdinh,1-1vacompact.Th~tv~y: D~t: \ B=={xEX:llxll<l}, B={[x]E'lKerF:II[x]lI<l} Taco: F(B) =={Fx:ll[x]lI< I} c {Fx:IIxII<I} ==F(B). Do F 13.compactliensuyfa f,(B) Hi.compactu'dngd6i. Toan ta f,-l:F(X)-'t'lKerF la khonglien Wc VI neu tnli l~i t011ntit J ==](-1](:X Iv }~ -'t X IK F se ]a tichcuamottoanfitcompactvamottoanrifIAer' Ii,er '..' . lien t\lcliennocompact.Di~unaydh de'n13 la compacttu'dngQ6ilien ~erF ph:?ticochi~uhuuh~n. p-I kh6nglientvcchota: 3([zIIDc 'lKerF: FzlI -'t 0 valI[zll]II~E> 0 ,Vn ChQn v" EKerF th6a I/z"+v,,11~ va d~tXII ==Z"+V" thl: . Fx"==Fz,,-'tOvallxlIll~; Ta co th~giii si'tFxII"*0 '\InErNva d~t: XII x,,:==Jlx"II~11Fx,,11 Khi do: Iln"I!= IIFxlIlI <~IIFx] IIxlIll~IIFxlIll- E -'to Chuang2: B1HtoanngU)fc... 17 1 --)-00 !lx"ll=~"Fxnll 0 .. M?nll d~2.2.3 Cho F: X --)-Y la loantit tuyentinhcompactgiil'ahaikhonggianHilbertv(Ji dimX =00va {x,,}la mQtdaytnfcchu§'n.Khi do: Jim }00 Changminh Tnf(Jc tieRtaco Fx" ~ 0VI: (Fx",y)=(x",F*y) -) 0 vy EY 00 «x"' F*y)Iah~s6cuamQtchu6ihQitv2)F* y,x")x,,(M~nhde1.1.4»,,=1 Neu Fx"~ 0 thlsecodaycon{u,.}cuaday{x,,}saocho: !lFu"ll~s>oVnEftJ (*) DoF compactnentu{Fu,,}nencoth~rutradaycon{Fv,,}hQitv.Taco: Fv"--)-V } 0=>v= Fv"->.0 (theotren) Dieunaymatithu~nvoi(*). 0 Trongtnfongh<JpF khongcompactneutabuQcthemdieuki~nchoRangeFta clingsecophu'dngtrlnhFx=y lakhongchlnh. Mfllhdi 2.2.4 ChoX, Y lahaikhong ianBanach.F:X --)-Y la mQtloantli'tuyentinhlien tQc,1-1vai RangeF"*Y vaRangeF=Y .Khido F-1khonglientvc. ChltO'ng2: Baitoannglt9'o... 18 Changminh Dogiathie'tveRangeFlH~nt6nt?i: y ~RangeF,Y" ERangeF:Y"~ Y Titd6: 3ex,,)c X:y" =Fx"~ Y ~RangeF N'" F -1 I ' A h1eu' lcntlJct : Ilx"II=IIF-1Y"II~MJIY"II {YJI}hQiW Denla dayCauchydlinde'n{XJl}la dayCauchytrongkhonggian Banachnen x" ~ x , Luc d6 Fx" ~ Fx=Y ERangel"mallthulin. 0 Trongtru'(jnghejpF phituye'ntac6m<%nhdesau: M?Il11dl 2.2.5 x, Y la hai khongglaDHilbert, F: X ~ Y Ja toaDtitcompact,lientvcvadong ye'u(nghlala xJI-" x,Fx"-" Y =>Y=Fx) vaX c6s6chieuvoh'.\n.Khi d6t6nt?i day {x,,}c X saocho FxlI-> FxonIningXII-fi Xo' Changminh Do dimX =r:I)Den qua du d6ng ddn vi B' la khong compact tu'dng06i, nghla la t6ntalday {z,,}c B' khongc6 dayconnaohOitlJ.Nhungqua,du B' trong khonggianHilbertl<;ticompactye'uBen{z,,}c6 dayconhQiWyell.Nhuv~yset6n t<;liday {VJI}sao cho: VII ~ X nhung VIIh x Ta kh~ngdl0h Fv"~ Fx, Th~tv~y,nc'u FvJIA-Fx nghiaIii co day con {ulI}cua{v,,}saocho: 35>O:IIFu" -Fxll? 5 {UJl}hQ1tlJyc'uDenbi ch?nvadoF compacttarutraduejcdaycon{x,,}thoa Fx"~ Y , To' (16: ChuO'ng2:Bai foal!nglitfc... 19 Fx" ~ Y } ::::>y =}x (dongye'u) x,,-"'x M~tkhac: IIFx,,-Fxllz5::::>lly-Fxllzs,mauthuiin. 0 MQtdieudangquailHim1aHnh6ndinhph\!thuQcvaokhonggiannghic;:m,ne'u tathuh(fpkh6nggiannghic;:mthlco th€ d<;ltdli<JcHnh6ndinh.Mc;:nhd~sailcho tha'ydieudo. M?nhdi 2.2.6 ChoX, Y 1ahaikhonggiandinhchuffnvaU 1akhonggianconcompactcuaX. F:U ~ Y 1amQtloanti'tlienhJc,1-1.Khido F-1lienl\lc. Chzingminh X6t ph~nta WERangeF va day {w,,}c RangeF.rheaw"~ W. f)~t: U'= F -1wva U '=F-1w, "',, Ta kh~ngdinhu"~ u. Th?tv~yne'ukhongcodi~unaytIll set6nt<,lidaycon {u,,*} rhea: Ilu" -ullz &>0 '\InErNk VI U1acompactDent6nt<,lidaycon {u"*r}hQit1,lve phftntii'ZEU.. Til'dnhlien !:\lCci'IaF ta nhi;lndu'<Jc: w =Fu ~ Fz ::::>Fz=w::::>z =u "Xl "Iq M~Hkhac: Ilu" -ullz &::::>O=llu-uliz&,mallthuiin.xl 0 ChUb'ng2: Biiitoanngu'lle... 20 2.3Saisa tru'onghQpxa'unha't(Worst-caseerror). MOtcall hc)idu'<jcd~tra la: VOlsai s6cuadil ki~nbe h<1n8 thl sai s6cua nghi~mIOnnh1tlabaonhi6u(nghIala trongtru'ongh<jpx1unh1t).M~nhd~2.2.2 chotacall tra1OjtrongtHrongh<jptoaDtU'tuyentinhcompact:sai s6co th~vo clIng lOll. Tuy nhi6nnnhhlnhnayco th~khacdi nelltachothemthongtinthuhypve nghi~mchinhxac.Cl;)th~tacodinhnghlasan: Dill" nghla2.3.1 Cho F:X ~ Y la loan t11'tuyentinh li6n t\lCgiila hai khongglaDBanach. XI eX la m0tkhongglaDconvOichuffnII'. 111m~nhh<1nchuffntrenX. Khi do ta dtnhnghlasai SO'ln((Jngh(/px([unhatcho sai s6 8 trendil ki(;n va thongtin cho tru'acvenghi~mIlxll!::;E la: ,~(5,E,II,lll ):=sup{llxll:xEX"IIFxlls:o,lIxll,S:E} Diell chungtamongmu6nla sais6tru'ongh<jp'xa'unha'tnaykhongnhu'ngchi h0itl,lv~0khi 0 ~ 0 manoconlamOtvoclIngbetheomOtb(icnaodocua8.Neu F-1 lien tl)cthl dieunayla hi~nnhienvlllxllS:IIF-111.IIFxll.Tuynhien trongtru'ong h<;1ptoaDtii'tuyen tinh compact va II . 111=II . II thl sais6 tru'ongh<jpxa'unha'tkhong hOitvnhu'm~nhd~sandaychIfa.Di~unayd~ndenvi~ctaphllidungmOtchuffn m~nhh<1n. Mfnh d~2.3.2 F: X ~ Y la mOtloantittoyeDtlnhcompactgiilahaikhongglaDdinhchuffn trongdodjm('lKerF)=00. Khi do: VE >0,3c>0,00>o:/Ifi(E,o,II.II) ~c '110E(O,Oo) (Xem [7] trang15) Trong phepchlnhhoa Tikhol1oVmata nghienCUll(j chu'<1ngsanta se xet hai tru'ong h<jp thong tin thu hyp ve nghi~m chinh xac (j d?ng XERangeF* vax ERangeF*F trongd6F* la toant11'tuyenHnhlienh<jpcuatoaD ChUb'ng2: Bai JoanngliC}'c... 21 tit tuye'ntlnhF giuahai khonggianHilbert.Dinh1ysauchotadaubgiasai s6 tru'ongh<Jpxa'unha't. Din" lj 2.3.3 ChoF la loantittuye'ntlnhcompact,1-1gifi'ahaikhonggianHilbertX, Y vdi RangeFtmm~trongtmm~trongY. (a)D~tX,:=F*(Y) vaIIxl!!:=IIF*-'xiivdix EX,. Khid6: g;(o,E,II.II,)s.J8E Hdn nuavdim6iE >0 t6nt~imQtday 0/1~ 0 saochog;(on,E,II.II,) ==~o"E . (b) D~tX2:=F* F(X) vaIIxl12:=II(F* Fr' xii vdi x EX2- Khi d6: 2 , g;(0,E,II.lb)s03E3 2 1 Hdn nuavdi m6iE >0 t6nt~imQtday 0/1~ 0 saocho g;(on,E,II.lb) ==0~E3- ChUngminh ChufinII . IIIva II . 112hoanroanxacdinhVIF* vaF*F 1a1-1.Th~tv~y: F*y=O~(F*y,x)=O,\tx EX ~(y,Fx)=O,\tx EX ~ (y,z) =0,\tz ERangeF ~ (y,z)==0,\tzERangeF(doRangeF==Y) ~y=O F* F la 1-1du'<Jcchungminhtu'dngtV- (a)'D~t x =F*v vdi IIFxlls 0 vallxlils E nghIa1a: IIFF*vlls0 va IIvllsE Khi d6: IlxW=(x,x)::::(F*v,x) =(v,Fx)sllvll.IIFxlls5E Dande'n: g;(o,E,II.Jb)s.J8E X6t (/111'x"'y,J lasingularsystemciiaF (xemdinhnghIa1.3.9).Ta c6 : Chltdng2: BiHtoilnngU'q'c... 22 . fi" >0,fill ---+0 . Fx"=fi"y",F*Y"=fi"X" Ch~m.~,,:=£.1'* y" va (),,:=p~£---+O.Khi <16: Ilx,,111=11Ey"ll= E IIFx"II=IIE.F1'* y"l\= Efi~IIY"II=8" Ilx"II=IIEfi"x"II= Ep" = .j8"E Suy fa: $(8",E,I) .111)=.j8"E (b)D~tx=F*FwvdiIIFxll:::;8vallxlb:::;EnghlalaIlwll:::;E.Khid6: IIxl12=(x,x)=(1'*Fw,x)=(Fw,F."'()sllFwll.IIFxll Ma: II Fxl12 = (Fw,Fw) = (w,F* Fw) =(w,x) :::;11wll.IIxll Suyfa: IIxl14sllFxl12.IIFwI12sllFxW .IIwll.llxll:::;82Ellxll Hay 2 I Ilxlls 83£3 V~y 2 I $(8,E,II.lb):::;o3EJ Chon x .=E' F* Fx va 5,.'.= P 3E~ _\. 0 Khi <16:. ".. " UV. 11 -, . Ilx"lb=IIEx"ll=E IIFx"II=IIE/1~y"ll=Efi~=0" 2 I Ilx"'I=IIE.F*CJl"y,Jll=EjJ~llx"ll=Ep~=8~E3 V~y 2 J {¥(o",E,II.Ib)=O~E3 0 Chu~ng2: BiHtoanngilt/coo. 23 2.4Phepcmnhboa. Trongph~naytagiasitF:X ~ Y lamQtmintit lient\lcgiuahaikhongian HilbertvaphuongtrinhFx=y comQtnghi~mlax*. Trenthlfcte',takhongbaagiC1bie'tduqcchinhxacy machIbie'tvoimQtsai86 8>0 tUGla co )i:llyO- yllso. PhU(1ngtrlnhFx==yOcoth~khonggiaiduqcVI yOkhongthuQcRangeF.Dod6 chungtadn xacdinhm(>txa'pXl XOEX chonghi~mx* saochosai s6da nghi~m khong"xauh(/n"sai s6 truC1nghqpxa'unha't.Them vaod6 ta yell cfiunghi~mxa'p Xl phaiph\!thuQclien tl,lcV8.0dil'ki~nyo. Din" nghia2.4.1 MQts(/dJ chlnhh6a(regularizationstrategy)la mQthQcaetOaDtittuye'ndnh lient\lcRa:Y~ X,a >0 saocho: limRaFx=x,'ifxEX a~O . ChQnx;:=RaYc5nhul8.mQtxa'pXlcuanghi~mchinhxac.Khi do: Ilx;-x*lIs/iRayO- RaY/l+IIRay-x*/i sllRall.IlYo- YlI+IIRaFx*-x*1I Dfinde'n: IIx; -x*lIs bllRall+11RaFx*-x*1I (2.1) 56h"mgthlihaitrongve'phaicua(2.1)d~nv~0khia ~ O.dodinhnghia2.3.1 cons6hc;lllgfulinha'ttrongtru'C1nghqpt6ngquatkhongd~nv~o.C\l th~tac6m~nh d~sau(xem[7] trang25). M?nhdi 2.4.2 ChoRa18.mQtsod6chlnhhoacuatoaDtittuye'nHnhcompactvadimX=00 thl: 3(a j ): 1.imllRa 11=00 J~ro j Chu'O'I1g2: BaifoBng~c... 24 sais6 II RaFx *-x*1I IRailo a. . Tadn ch<;ma =a(5)phVthuQcvao0d~gillchot6ngsais6(Jrml'cnhenha'tco thedu<1c.Cv th&tacodinhnghIasan: Dink ngkia2.4.3 MQtsod6chlnhoavOia =a(5)du<1cg<;il \"admissable"n{§unothoaman: . a(5) ~ 0 khi 5 ---)0 sup{lIx~(o)-xll:IIFx- yll~5}---)0 khi 5 ~ 0,'t;JxeX. ChU'O'ng2: Bai toBnng«'9'c... 25