Một số phương pháp hiệu chỉnh giải bài toán đặt không chỉnh

Ap dung cho phirong trinh vi phan thuong 17 CHUDNG II BAI TOAN TUYEN TINH KHONG CHINH TREN COMPACT YEU §1 Ma dau 23 §2.. 6 n g da d~e xuat khai niem bai toan dat khong chinh cho Icfp

&

NGUYEN VAN HUNG

M O T S O P H U O N G P H A P H I E U CHJNR GIAI BAI TO AN DAT KHONG CIliMH

Chuy^n nganh : Toan hoc linh (can

Ma so : 1.01.07

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LUAN AN PHO TIEN SI KHOA HOC TOAN - LY

NGUdl HUdNG DAN KHOA HOC: TIEN SI - PIIAM KY AN13

L51 noi dau 1

CHUONG I

PHUONG PHAP COMPACT THU HEP CAI BIEN

§1 Ma dau 5

§2 Cac gia thi^'t cua bai toan 5

§3 Thuat toan compact thu hep dang Robust 7

§4 Tnrcmg hgp kh6ng duy nhat nghiem 12

§5 Tnrcmg hgp ve phai va toan tir kh6ng biet chinh xac 15

§6 Ap dung cho phirong trinh vi phan thuong 17

CHUDNG II

BAI TOAN TUYEN TINH KHONG CHINH TREN COMPACT YEU

§1 Ma dau 23

§2 H6i tu y^'u trong kh6ng gian Hilbert 25

§3 Danh gia tinh 6n dinh cua nghiem trSn compact y6u 27

§4 Phuang phap khai tridn ky di chat cut 29

§5 Phuang trinh tich phan dang tich chAp 31

§6 Danh gia diam Vs trong phaang phap compact thu hep cua Gaponenko 36

CHUONG III

M O T S O PHUONG PHAP LAP - HIEU CHINH

§1 Phirang phap Gauss - Newton hi6u chinh (RGN) 40

§2 Ki^m tra di^u kien B] cua Bakushinski 41

§3 Phuang phap hieu chinh Gasse - Newton g'an dung 46

§4 Phirang phap Seidel - Newton hieu chinh va bai toan phi tuyS'n cong huang 52

I - Phuomg phap Seidel - Newton (SN) vd phuomg phap

Seidel - Newton hieu chinh ( RSN) 52

II - Syc hdi tu dia phuomg cua phuomg phap RSN 54

III - Bdi todn Men tudn hodn cho phuomg trinh Duffing - Van derpol 57

nil - Bdi todn bien tudn hodn doi v&iphuomg trinh Van derpol 65

Phan ket luan 66

Phu luc 67 1- Bai toan - Lcfi giai 67

2- Thuat giai va chuong trinh 72

Tai lieu tham khao gg

# Ngay nay, ciing v6i vice sur dung ph6 biC'n may tinh, loan hoc ngay cang

dirge ung dung rong rai trong cac linh virc khoa hoc va ihurc ti6n Vice ap dung loan

hoc mot each s^u sac da thiic day manh me sir phat trien cac phirang phap tinh toan

Trong thirc te ta thirofng gap nhiJng bai toan ma dfr ki6n ban dau chi dugc biet gan

dung, nhCmg thay ddi nho cua du* kien ban d'au c6 thd dSn dfi'n thay do! liiy y cua

nghiem, do do vi^c tun nghidm cua bai toan gap nhicu kho khan NhCmg bai toan

khong on dinh nhu vay la m6t vi du ve bai toan dat khong chinh

Khai niem bai toan dat khong chinh da dirge nha toan hoc Phap Hadamard J

dira ra rfau tien cho Idrp phirang trinh vi phan [63,64] Thoat dau nguai ta cho rang

bai toan nay kh6ng c6 y nghia toan hoc va thirc tiSn, n6n ft chu y de'n no Nhimg den

cuo'i nhung nam 50 nguofi ta phat hien ra rang, nhicu bai toan ly thuyet va hau het

Ccic bai toan trong ki thuat va ihirc 16' deu dan de'n bai loan dat khong chinh

Tikhonov A N la ngirai c6 c6ng dau trong nghi^n cuti vah de nay[49,50]

6 n g da d~e xuat khai niem bai toan dat khong chinh cho Icfp phirang trinh toan tu

trong kh5ng gian T6p6 va cho ra dofi mot loat cong trinh xung quanh v^n dc nay I<c

tCr do de'n nay, cac nha toan hoc tr6n the' gidi da tap tiling nghien cuu bai loan dat

khong chinh va da dat dirge nhung thanh tiru dang ki - Trong so do phai ke den

cac nha toan hoc Morozov B.A , Laverientiev M.M ,Vasin J.M, Ivanov V.K,

Bakushinski A.B, Gaponenko Yu L, Bertero M.P, Nashed M.Z, Groetsch C M ,

Khai niem dat chinh theo Hadamard la:

Xet phirang trinh

Ax = y (0,1) Trong do A la toan tu dua khong gian T5p6 X vao khong gian Topo Y

1 - Vai m6i y e Ytbn tai x e X

2 - Nghiem x la duy nhat

3 - Nghiem phu thu5c lien liic vao cac du kien cua bai loan

• M6t VI du didn hinh khi A la toan tu* tuye'n Ifnh hoan toan lien tuc, con X, Y

la cac khdng gian Banach v6 ban chieu, khi do:

00

i-ImA ^ YvalmA = u {Ax:||x||< n)latap pham tru thirnha't.(dieu nay CO nghia

n = l

la bai toan (0,1) giai dugc khOng phai vdi moi y e Y)

ii - Ne'u A ' : Y-> X c6 ton tai thi cung khong lien tuc Di^eu nay chung to nghiem cua bai toan (0,1) khong phu thuoc lien tuc cac du" kien ban dau

Tu* dinh nghia v'e tinh chinh thay rang: Tinh chinh cua bai toan phu thu6c bo ba {A,X,Y) Ngucri ta thucmg khac phuc tinh khong chinh bang mot trong cac phuang phap sau day

1 - Xet nghiem suy rong, do la phuang phap tira nghiem Ivanov va e - tua nghiem cua Liscovets

Tua nghiem cua (0,1) la tap M cac phan tir c6 do lech nho nha't so vai nghiem chinh

xac Phuang phap tua nghiem la di tim cue iiin phie'm ham khong khap tren lap

chap nhan dugc nghiem [41]

2 - Thu hep mfen xac dinh cua so lieu ban d'au

3 - Thay ddi kh6ng gian va t6p5 cua chung

Hai phuang phap 2 va 3 ft dugc six dung vi kh5ng dap img dugc yeu cau cua thirc te

4 - Phuang phap hieu chinh : La phuang phap thay bai toan dat kh5ng chinh bang m6t ho bai toan dat chinh phu thu5c tham s6' ma nghiem cua bai toan dat chinh sc d'an de'n nghiem bai toan dat kh6ng chinh khi tham so d'an tai khOng Phuang phap Laverenliev, phuang phap Tikhonov [54], phuang phap tua nghich dao Lattes -Lions, phuang phap lap, su dung khai tridn ky di va khai tri^n ky di chat cut la nhung phuang phap hieu chinh quen biet

Phuang phap hieu chinh ciia Tikhonov dua tren b6 de sau:

• A^ : A ( M ) ^ M lien tuc

Vao cu6'i nhung nam 50, Tikhonov[54] da de xua't khai niem phiem ham on dinh nhu sau

i, Q : X -> R ^ a phie'm ham lien tuc va khong am

ii, Xd e Xo = dom Q , Xo = X , (trong do Xd la nghiem cua bai loan (0,1))

iii, Vdfi m6i r > 0, tap K(r) = (x e Xo : fi[x] < r} la tap compact tuang doi trong X

Phuang phap hieu chinh Tikhonov la lay nghiem g^an dung cua (0,1) la didm cue tieu cii phie'm ham tran:

M "^ [x,y5 ] = p^ (Ax,y5 ) + a 0[x ] -> min

x e Xo Vai in6t s6' gia thie't nhat dinh c6 th^ chiing minh dugc rang

i/ 3 ! Xa= ArgminM''[ x,y5]

X e Xo ii/ Tham s6' hieu chinh a = a (8) chon tu cac nguyen ly khong khc5rp, nguyen

ly tua tO'i iru,v.v

iii/Xa(6)^Xd ( 6 - > 0 )

5 - Ma r6ng nglu nhien bai toan la't dinh dat kh6ng chinh

Ban luan an nay nghien citu mot so phuang phap hieu chinh giai bai toan dat khong chin Nhitng v^in d'e dugc quan tam trong luan an la:

1- Giai bai toan dat kh6ng chinh tren tap compact

2- Danh gia tfnh 6n dinh ye'u cua bai loan dat khOng chinh tren compact va compact yeu

3 - Cac phuang phap lap hieu chinh giai bai toan dat kh6ng chinh

Chuang nay trinh bay phuang phap compact thu hep cai bien va danh gia dugc toe do h6i tu cua phuang phap

Chuang II: Bdi todn tuyen tinh khong chinh tren compact yeu

Trong bai toan (0,1) xet truong hgp X,Y la cac khOng gian Hilbert, A la toan tur tuye'n tinh lien tuc vai khai tridn ki di da cho tru6c Trong chuang nay chung t6i trinh bay m6t thuat toan dang khai tri^n ki di chat cut giai phuang trinh (0,1) va danh gia tfnh on dinh ye'u cua nghiem

Chuang I I I : Mot so phuomg phap lap hieu chinh

Chuang nay trinh bay phuang phap lap hieu chinh Gauss-Newton gan dung va phuang phap Seidel - Newton hieu chinh

Phan phu luc ; Trinh bay mot vi du so giai bdi todn Cauchy bang phuang

phdp khai trien ky di chat cut

Noi dung chfnh cua luan an da dugc cong bo trong 6 bai bao dang tren cac tap chf cap truong, B5, qu6'c gia, qu6'c te' va dugc bao cao tai Xeminar toan hoe tfnh toan cua Dai Hoc Tong Hgp Ha Noi (Tien sy Pham Ky Anh chu tri) Hoi nghi khoa hoe 35 nam thanh lap khoa Toan - Co - Tin DHTHHN hoc nam 1991, Hoi nghi khoa hoc khoa toan DHSPHN 2 1992, H5i nghi khoa hoc khoa Toan - Co - Tin hoc DHTHHN 1994,

Hoi nghi quoc te' vt bai toan ngugc 1995 ( Tai thanh pho Ho Chf Minh)

$1 - Ma dau:

Xet phuang trinh

Ax = y • (LI)

6 day A la toan tir phi tuye'n, X,, Y la cac khong gian vector t6p6 Gaponenko

Yu.L [24,25] nam 1982 da de xua't phuang phap c6 ten la "compact thu hep" Ong da

xay dung dugc cac tap Vg gbm huu han phan tir sao cho:

V xg e Vg => II X5 - Xd II < diam V5 + 5 -> 0 ( 5 -> 0) Arsenin V.Ia , ([5], nam 1989 ), xet bai toan (1.1), trong do A : H ^ C[a,b], H la

kh6ng gian Hilbert Thay vi biet ys e C[a,b] chi bie't m thi hien {y'sl "'1, trong do han

mOt nua y'5 thoa man dieu kien

II / 5 - y d ||c<5 ( l < j i < m ; i = l - r ,r>m/2) (1.2) Bang each sir dung ham Robust, Arsenin da dua ra mot phuang phap hieu chinh

V6i ham M" [ x,y^5 ,.•• ,y'"8 ] = ^^ (Ax) + aC)(x),

Arsenin V la da churng minh dugc cac ke't qua sau

l/3!Xa-ArgminM"[x,y^8 y"\]

2/ 3 a = a(5) : Xa(6) -> x* (5 -> 0 )

Cung nhu phuang phap Tikhonov AN, phuang phap Robust kh6ng cho phep danh gia

sai s6' cua nghiem g'an dung

Y tuang cua Gaponenko va ky thuat cua Arsenin da dugc chung toi sir dung de xay

dung thuat toan giai bai toan (LI) va da danh gia dugc tO'c dO hoi tu cua nghiem g^an

dung

52 - Cac gia thiet cua bai toan

Xet phuang trinh (1.1), trong do A : X -> C[a,b], A la toan tir lien tuc, X la

khdng gian Banach Goi Xi la mfen xac dinh cua phie'm ham on dinh D.[x]

OO

Xi c: X va ta c6

Xi ==uK(n) (2.1)

trong do K(n) = (v e X] ,D.{v) <n )

Chung ta da bie't m6i tap K(n) la tap compact trong X

Gia suf phuang trinh (L1) vdfi ve'phai dung yd c6 nghiem duy nhat Xd

A x d - yd

Trong thue te'ta kh6ng bie't yd(t), ma chi bie't cac ihi hien cua no y'5(t) , ,y'"8 (t) va ,

y'6(t) e C[a,b] , (i = 1-^ m) sao cho:

bie'n, vai 0 < t,r < +00, 4^(t,r) -> 0 khi t ^ 0

Goi S(n,h) e K(n) (Vn > 1) la (p(h) luai huu ban cua tap compact K(n), c6

nghia la:

Vxe K(n), 3 Xh e S (n,h) : ||xh - x|| < (p(h)

Trong do 0 < (p(h), la ham dan dieu tang tren ( 0,1 ] va lim (p(h) - 0

h - > 0

Xet phie'm ham Ibi b m b

(l)6(y) = J Z P6 [ y's (t) - y(t)]dt 4 Rg (t,y(t)) dt

Ham (jjg (y) c6 tinh cha't sau (xem [5])

i/ T6n tai ygCt) e C[a,b] : (t)s(y5) = Inf {(^sCy) : y e C[a,b] )

T =To I < 1 khi to < yd (t) - Q5

< m , V T , t e [a,b], 5 > 0

S3 - Thuat toan Compact thu hep dang Robust

Gia sir ta c6 he thiic

b m

(t)6(yd) = J Z P5(y'5(t)-yd(t))dt<©(8)

a i ^ l

(3.1)

a day a)(5) la ham lien tuc, khong am hoi tu de'n 0 khi 5 - ^ 0

Nhan xet 3.1: Dfeu kien (3.1) hiin nhien duac thue hien trong tru&ng hap (1.2) thoa

man vai moi i ( i = l,m) Trong truong hap c6 cac ^§(1) khong thoa man (1.2) nhung

thoa man danh gia:

11^5-yd II < 5 , (3.2)

LPJ

Of day Pj > 1 , ^

Khi do di'eu kien (3.1) van dugc thue hien Vay (3.1) dugc thue hien ngay ca trong

truong hgp c6 nhung tG[a,b] sao cho I y'5(t) -yd(i) I c6 the Ian tuy y

De dang chiing minh dugc cac danh gia sau:

(l)5(yd)<I||y'8-yd||

i = 1 Li

||y'5-y|| < ( b - a ) | | y ^ 8 - y | | , | | y s - y | | < (b-a)^^^M|y5-y||

Li C Ll L p

d dayqj> 1 : l/qj+ 1/pj = 1, Tir he thue (2.2 ) va (3.2) suy ra he thii-c (3.1)

Gia su 5 > 0 la m6t sC c6 dinh tuy y ta chon day hn > 0 va so N = N (5) sao cho

(p(hN-0 >5>(p(hN)

So db tinh toan theo phuang phap Compact thu hep cai bien dang Robust gbm cac

buac sau day:

Budfc 1, Chon ri = 1 va dat

V, {VE S(r,,hi) : (|)5(Av) < GT((p(h,),ri) + co(5)l

Ne'u V] = (j) la'y r2 = ri +1

Ne'u Vi 9^ ([) la'y T2 = ri sau do ihirc hien cac buac tie'p theo

Birdc n < N, Vdi r„ dugc xac dinh tu budre truac

Vn = i ve S (rn, hn) : ^^ (Av) < GT((p (hn), rn) + CO (5)}

Ne'u Vn = (t) lay rn+i = rn +1

N e ' u V n ^ ^ l a ' y Tn^-i = r n

Birdfc N, VN = V5 :=: {ve S(rN, hn) : ^^(Aw) < GT((p(hN),rN) + co(5))

Ta chutig minh su hoi tu cua phuang phap nay theo luge do sau:

Bo de 3.1 Gia sir vai v e H c6 dinh thoa man he thu"e

(j)5(Av) ^ 0 (5 -> 0 )

thi I (t)8(Av) - Uyd 1 ^ 0 (5 -> 0 )

v a V = Xd

Chung minh day dii bd de nay xem trong [5 ]

Bo de 3,2: T6n tai s6' No = No (Xd) sao cho

V N > N o = > Q[Xd] <rN

Chvcng minh: Ta chung minh bang phan chung

Gia sir ngugc lai: V n 3 N > n : r N < ^[Xd]

Ta dat ni = [Q[Xd] ] + 1, khi do tlm dugc Ni va mi < Ni sao cho Vmi ^ ^

That vay, ne'u kh6ng, theo each xay dung tren ta c6 V^ = Ni > ni > n[Xd] mau thuan

vdi gia thie't

Tuang tu, vdi ni = Ni + 1, tbn tai s6' N2 sao cho v6i mi < m2 < N2, Vm2 ^ (j) v.v

Tie'p tuc tie'n hanh nhu tren ta dugc day tap hgp {Vmk )"° khac r6ng Chon day {Vn )

Ta CO (Vn ) d D := { V e Xi : Q[v] < rd) Do D la tap hgp compact trong X, nen

tbn tai day {Vk } c (Vn} sao cho Vk —> VQ Khong ma't tinh tong quat ta c6 the coi

Theo bd rfe (3.1) Vo = Xd e Xmau thuan vai (3.3)

Bd d'e dugc chung minh []

Bd de 3.3: Vdfi moi sd tu nhien N > No tap VN khong rdng va chira trong tap

compact K

Chirng minh:

Tur bd d'e (3.2) suy ra vbi moi N > No, fi[Xd] < TN va do do Xd e K{T^)

gia surxh e S(rn,hn) sao cho ||xh - Xd|| < (p(hn)

Theo tinh cha't ciia ham (t)8 va toan tii: A, ta c6 danh gia:

0 < (|)5(Axh) < I (t)5(Axh) - (1)5 (AXd) I + (|)6(Axd) < G ^ (II Xh-Xd || ,rN ) + co(5)

< G ^ ( ( p ( h N ) , r N ) + 0D(5)

Suyra: Xh e V N = V 5

Bd d'e dugc chi^ng minh []

Nhan xet 3,2: VN ^ i?, doi vdi moi N > No ta luOn c6 rw < rNo ^ No va

VN <= K (No) == K (K - Tap compact)

Bd de 3.4: Day tap compact {V5) co Ve didm Xd khi 5 -> 0

Chvcng minh: Gia sir ngugc lai di^eu do kh6ng xay ra, khi do tbn tai hai day{Vk")va

so 8 >0 dd:

| | v k " - V k - | | > s > 0 (3.4) Vai Vk* e VNk c: Ko, k = 1,2 Do Ko la tap Compact nen khCng giam tdng quat

chung ta c6 the coi rang Vk" -^ v" khi k -> o)

Khi do ta cung c6 Avk" -^ Av"*" Mat khac

(|)5 (Avk") < GT((p(hNk ),rNk) + co(5)

Tir do suy ra

0 < (1)5 (Av*) < co(5) -> 0 ( 5 -> 0)

Theo bd de (3.1) ta c6: v* = Xd e X Dieu nay mau thuan vox (3.4 )

Nhu vay ta c6 diam V5 -> 0 ( 5 -> 0)

Bd de dugc chung minh []

Ne'u ta lay x^ e V5 = VN ( N > No) tiiy y, do V5 ehira Xh:

II Xh - Xd II < (p(hN) nen la thu dugc

||x5-Xd|| < ||x6-Xh|| + ||xh-Xd|| < diam V5+ cp ( hw)

D o d o ||xs-Xd|| <diam V5 + 5 (3.5) Nhu vay ta da chung minh dugc dinh ly sau:

Djnh !y 3.1: Phuang phap compact thu ht^p dang Robuts h(3i tu va ta c6 danh gia

IIX5 - Xd 11 < diam Va + 5

Nhan xet 3.3:

Ne'u phie'm ham dn djnh Q[x] thoa man di'eu kien Q[x] > C ' | | x | Vx e X

(C-const > 0 ), thi (2.3) c6 ihi thay bang di^eu kien

V Xi,X2 e X : || Ax, -Axsjl < T ( | | x i - X2II, r) ( 3.6)

6 day r - max (|| Xi ||,1| X2II) Khi do tat ca chung minh tren van dung, ne'u ta la'y tap

hap Vn nhu sau:

Vn = ( V e S (rn,hn) : ^^ (Av) < G ^ (cp (hn),C.r ) + co(5) )

D^ y rang phuang phap compact thu hep [24] la truong hgp rieng cua phircng

phap nay khi n = 1, Q[x ]= ||x|| 1 va ||y5 - yd|| < 5, a day || || 1 la chu^n nao do irong

kh6ng gian con Xo, trii mat va compact trong X

Nhan xet 3.4: Ne'u thay ddi each xay dung tap Vn mot chut thi thuat loan cua ta khong

nhat thie't phai thue hien dung N but^c nhu da trinh bay a tren

BireJc 1: Lay r, = 1 va thanh lap tap hgp:

V i - { v : v e S ( h n , r i ) : ( | ) 8 ( A v ) < G^F(5,ri ) + co(5))

Ne'u Vi ^ (|), ta la'y tuy y xs e Vi

Khi do: || X5 - Xd || < diam Vi + cp ( hN) ^ diam Vi + 8

Thuat loan dimg a day

Ne'u Vi ^ (|), ta la'y r2 = ri + 1 va thue hien buac 2

Budc ( n < N) dugc thirc hien ne'u Vi, V„-1 = ([>

Khi do rn = n, ta thanh lap tap Vn

Vn-{v:veS(hn,rn):(l)6(Av) < GT(5,rn ) + co(5)) Ne'u Vn ^ (t>, ta la'y xs e V, tuy y, thi

II xg - Xd II < diam Vn + 5 va thuat toan dung lai

Ngugc lai Vn = (j), dat rn = n +1 va quay lai h\x6c n cho de'n khi n < N thi dung

Nhan xet 3.5: Ne'u rn > n[Xd ], thi V„ ?^ cj) khi do thuat toan a nhan xet (3.4) dimg lai

Dfeu nay chung to: Ne'u Qxd la mot so nguyen thi so bubc cua thuat loan la fifxd ] Ne'u il[Xd ] khong phai la so nguyen thi so' bu6e cua thuat toan la [Q[Xd ]]+ 1 Vay so bu(5fc ciia thuat toan khOng phu thu()c vao 5 khi 5 du be

Nhan xet 3.6: Ne'u tien nghiem bie't rang Q[Xd] < R thi ta thanh lap tap

V5={v:veS(hN,R):(|)5(Av) < G4^(6,R)+co(5)}

va thuat toan chi c6 mot bu6e

Nhdn xet 3.7: Ta c6 thd thay gia thie't (3.1) bang gia thie't i^^{y^) < (3(5), trong do

P(5) > 0 la ham lien tuc, p(8) ^ 0 khi 5 ^ 0 Khi do c6 ihi la'y tap V„ nhu sau:

• Vn = ( V e S(rn,hn) : U^v) < GT((p(hn),rn ) + Vs + co(5))

Cac ke't qua trinh bay b tren van con dung trong truong hgp nay

§4 - Truotig h(ifp khong duy nhat nghiem:

Trong muc nay, ta't ca cac gia thie't cua muc 52 trtr gia thie't (1.1) c6 nghiem duy nhat dugc giu" nguyen

Goi U = {x e Xi : Ax = Axd = yd) la tap hgp nghiem cua (1.1) Gia su trong Utbn tai duy nhat x* sao cho

Q[x*] = minf^[x]

x e U TrU(Je he't ta xet thuat toan cai bien sau:

Vain > l,dat:

V„ = {v:v e S(hN ,rn),: (f)8(Av) < G^(5,r„) + co(5)}

Ne'u Vn = ([), ta la'y rn +1 = rn +1 va thue hien buac tie'p theo

Ne'u nN < N, Vn 5^ (j) va Vn = (|), v(jfi n = 1,2 nN - 1, thi bube tie'p theo dugc lien

hanh nhu sau:

va tie'p tuc thue hien h\x6c n + 2

Tom lai thue hien thuat toan de'n buac nN+ m ta eo:

V<5i each xay dung thuat toan nhu tren ta c6 ke't qua sau

Djnh ly 4 1 : Phlln tir V5 dugc chon a tren c6 tinh cha't

a) | | v s - x * | | - ^ 0 khi 8 ^ 0 b) IIV5 - X* II < T(8) trong do T(8) ^ 0 khi 8 -> 0

Dd chung minh dinh ly ta can ba bd de sau:

Sao cho llvk"*"-Vk'll > e > 0 vai k =: 1,2,

Vi Vn e K (nk) - la compact, k - 1,2, nen khong giam tdng quat ta gia sir rang:

Theo each xay dung thuat toan ta nhan dugc u6c lugng

nN - = r < Q[x*] < r = nN + —•

• N " N ^ " ' " N * " " ' N N

Do nhan xet (3.5 ) thi nN = ^ [ x * ] ne'u n[x*] la sd nguyen hoae nN = n[x*] + 1

trong tnrbng hcDfp ngugc lai

a - Truong hcrp Cllx*] la sb nguyen thi m = 0 va

n[vo^ ] - [n[x']] - 1 - a Tijr (4.2 ) suy ra

a[vo* ] = a [ X* ] (4.3)

Til (4.1) (4.3) va tinh duy nha't cua x* nen Vo" = x* Difeu nay mau thuan vai gia

thie't phan chung

Bd de dugc chung minh []

Djnh ly 4.1: Dugc suy ra true tie'p tu: bd de nay

§5 - Truomg hgp ve phai va toan tur khong biet chinh xac

Trong muc nay chung ta gia thie't rang: Trong phuang trinh (1.1) ta khong bie't A ma

chi bie't A ^

A^ : X -> C[a,b] La toan tur lien tuc thoa man

a - II A^v - Av|| < u(|a,Q[v]) V v e Xi, u(ja,s) la ham lien luc khOng am, kh5ng

giam theo JJ, va s, vbi mdi s cb dinh u(]a,s) —> 0 khi p ^ 0 va u(o,s) = OV s > 0

b - V v i , V 2 E X i : llA^Vi A^V2|| < T ( | | v i - V 2 | | , r ) ;

Trong do r = max(n[vi ],Q[v2]), ham T(t,r) thoa man gia thie't trong S2

c - Ta van gia thie't (|)5( yj) = (t)5(AxT) < co(8)

Cac gia thie't khac giiJ nguyen nhu trong 52

Chon day (hn) kh6ng tang; 0 < h„ < 1, hn —> 0 khi n —> oo, 8 va p ed dinh,

N = N(8,ii) chon tijr he thitc (p(hN 0 > ji + 8 x p (hN )

Budc 1 : Dat ri = 1 va xay dung tap hgp

V, = (v: V G S(h,,ri) : U^^'v) < G[T((p (hi),ri) + u(vi,ri) ]+ co(8))

Ne'u W\^^ia la'y r2 = ri

Bd de nay dugc chung minh tuang tu bd de (3.2)

Bd de 5.2 : Vbi moi sb tu nhien N > No tap VN khong rdng va chu-a trong tap compact Ko nao do

Chtrng minh: Tur bd d& 5.1 suy ra v6i N > No; ^[Xd] < rN; Xd e K(rN) va do do tim

dugc Xh e S(hN,rN) sao cho

||xh -XTII < ( p ( h ) < 8

T a c o :

(t)5(A^Xh) < I (t)6(A^Xh) - ^5(A^Xd) I + I (t)5(A^Xd) - (t»6(AXd) I + (t)5(AXd) •

< GlJA'^Xh - A^Xdll+Gu(vi,rN)+ oj(S)

< G^(cp(hN),r) + Gu(p,rN) + 0(8)

< G T ((p(hN),rN) + Gu(p,rN) + co(8) Trong do r = max(n[Xh],n[Xd]) < rN

Dieu nay chihig to xi, e VN hay VN ^ (j)

Bd dfe dugc chung minh.[]

Bd de 5.3 : Day tap compact (Vg^} co ve didm Xdkhi p, 8 -> 0

Chicng minh: Gia sir 6\tw do kh6ng xay ra, khi do tbn tai hai day{ Vk" 1 <= Vn va

Dfeu nay mau thuSn vcfi (5.1) Nhu vay ta cb diam Vg -> 0 ( 8,|i -> 0 )

Bd de dugc chiing minh []

Do Vg chiia Xh : ||xh - Xd||< q)(hN) nen ta co

||x^5 - Xdll < 11x^^8 - Xh|| + II Xh - Xdll < diam V^g + cp(hN) Hay ||x^g - Xdll < T(8,p) = diam V^g + 8 + u (5.2)

Ta phat bidu ke't qua vua nhan dugc dudi dang dinh ly sau:

Dinh ly 5.1: Phuang phap compact thu hep dang robust v6i ve' phai la toan tir A bie't

g"an dung hOi tu va co danh gia (5.2)

Nhdn xet 5.1: Ne'u chi bie't g"an dung A' va phuang trinh khong cb nghiem duy nhat,

ta cb thd ke't hgp cac phuang phap nghien cun trinh bay trong §4, §5

§6- Ap dung cho phuang trinhvi phan thuong

@ - Xet bai toan Cauchy dbi vai phuang trinh vi phan tuye'n tinh bac 2

0^! HC:C QUDC GIA HA NOI

KT T'J

^MJ^ liJiBlA^niHiiSTi?i.!r^V;

Gia sit bai toan (6.1) eo nghiem duy nha't Trong thue tb ta chi bie't cac thd hien cua

ve' phai y'(x)v , y"'(x) va ditu kien bien (pi^ ,(p2^ thoa man

a-||y^^J-f|| < 6 :V6i 1< Kj < m, m / 2 < j < m

b - I (p^-(pi I < 6 : Vbi i ^ 1,2

Ta cung gia thie't rang ( xem (3.1))

(t»8(Du) - <|)g (/) < co(6)

6 day co(6) -> khi 6 -> 0 Dual day ta se sur dung cac chudn || || 2, || J ^^1 dugc xac

dinh nhu sau 2

Vbix e C'iOA]: \\ x||2 = max { || x||^|| x||^|| x||^ },aday 1| ||^aehuSn Qo.i]

v b i x e | | x | L i : | | x | L 1 =||xl| +11x11

"^2 "^2 L2[0.1] L2[0,l

@ - Tuong tu rai rac va cac tinh chat

Gia sir u(x) e W2^[0,l] la mCt ham cb dinh thoa man |lu||2 ^ S, 0 < p - const

Cho hai sb h, x e > 0 cb dinh, ta cb luoi:

(Xi,yk) : Xi = ih, i = 0,1,2 , N ; sao cho Nh=: 1

S = S (h,x,p)

yk = kT ,k = 0 , ± l , ± 2 , , ± p x - ' Goi M = M (h,x,p) la tap ta't ca cac ham luoi tren S, tiJc la cac ham lien luc tren

[0,1], tuye'n tfnh tren m6i khoang (xi 1, Xi) i = 1,2, , N, con tai cac didm

Xi (i = 0,1 ,N) chung chi nhan cac gia tri trong tap {0, ± x ,± 2x , ± px''}

Goi u"h t la ham lirori trfin S thoa man dieu kiSn

dugc goi la tuang tu rbi rac cua ham u (x) e W [0,1]

Bd de 6.1 [20]: Ne'u u(x) e W ^ [0,1], eon Uhx(x) la tuang tu rai rac ctia u(x), thi tbn tai hang sb ho = ho(u) > 0 d^ ta co danh gia

Bu6cn: ( n < N )

Vn = (v : r*(v) e Kn(rn) : ^^{Dw) < G\\f {^TK-^-ZU + 26,r„ + I (piM + ICP2H ) + co (6))

Ne'u Vn = (j), ta la'y r„ ^.i = rn + 1

Ne'u Vn ^ ^, ta la'y rn -n = rn va thue hien buac n + 1 < N

Birdc N:

Vn = V5 = { V : r Vv) e Kn (rn + Co) ; (t)6 (Dv) < GvK (36,rN + i cpi^ I + i ipi"" I) + co (6))

La'y u^ (x) e V^ tiiy y ta cb cac ke't qua sau

Bo de 6.1 : T6n tai sb No(Ud) sao cho:

Dieu nay mau thuin vai gia thie't

Tuang tu vbi n2 - Ni +1, tbn tai sb N2 sao cho vai V2 ^ N2, Vm2=?^ (j),

Tie'p tuc tie'n hanh nhu tren la dugc day tap hgp {Vmn p khac rbng Chon day

Vi hinh cau dong trong W2'[ 0,1] la tap hgp compact trong C[0,1] nen tir day

{v"n} = (an) CZ C[0,1], tbn tai day {ak } e {an)sao cho

n

ak -> ao khi n -> 00 trong C[ 0,1]

n

Kh6ng ma't tinh tdng quat la cb the' coi

an -> tto trong C[ 0,1] khi n ^ 00 suy ra Vn -^ Vo trong C^[0,I] khi n -^ 00,

Trong db

X t

Vn = J J an(Tl)dTldt + ( p , \ + (p2^

' 0 0

Vi Vne VtTin nen (t)6(DVn) < G\jy(V hn + Xn + 25, Vn + I (pi^ I + I (P2^ I) + co(6)

Tu* tinh lien tuc cua ^^ va D suy ra

(t)6(Dvo) < Gy (36,rN + I (piM + I ^Pi" I) + co (5)

Theo b6 de (3.1) thi Vo = u Nhung vi

llctnll , ^rd<||u"|| ,

nen llaoll^i = | | u " | | ^ , = II v 1 | < r.< ||u"|| ^

2 2 2

Bat dang thiJe cubi cung mau thuan vai gia thie't phan chung

Bo de dugc chiing minh.[]

Bo de 6.2: Vbi sb tu nhien N > No tap VN khong rOng va chiia trong tap compact Ko nao db

Chieng minh: Tu bd de (6.1) suy ra:

l|u"L ^ II"11 1 ^rN va||u",,|l < rN Lfo.i] "w Lro.i]

2 2 2

Theo each xay dung, tren mOi doan [Xi,Xi +1] u"ht(x)

CO dang ajX + bi, tiif da'y suy ra

^8(Du\,) < Gvj; (35, rN + I cpi^ I + I (P2M + 25 ) + co(5)

Suy ra u\ ^ eVn , hay VN ^ ^

Bd de dugc chung minh.[]

Bd de 6.3 Day tap compact {V5) co ve didm u khi 5 -> 0

ChAng minh:

Gia sur ngugc lai dieu db khong xay ra Khi do ton tai hay day {a~n 1 c: V5 va s > 0

sao cho ||a'*'n - a'n||c > £ (6.5) Vi V5 c Ko = KN (rN + Co) la tap compact, nen

khOng giam tdng quat, la cb ihi coi rang a'^n -> a*o trong C[ 0,1] khi n -> 00

(t)6(Dv*„) < Gv|;(35,rN + I (pi^ I + ICP2'' I) + co (5)

Tu bd rfe (3.1) suy ra v*n = Vo mau thuSn vdfi (6.5) Mau ihuan nay suy ra bd de

dugc chung minh.[]

Tijr cac bd de tren suy ra dinh ly sau:

Djnh ly 6.1: Vai moi 5 > 0 tbn lai sb N(5) sao choVN = V^ chiia Uhx ,vai moi u^ e V^

ta cb danh gia:

||u^ - uh^llu"^ - u \ x II2 + II u \ , - u II2 < diam V^ + 35

Nhdn x^t 6.1: Thuat loan vijra trinh bay de dang ma rOng cho truong hgp phuang

trinh vi phan bac n

D(u(x)) = /(x)

0 < X < 1 LiU - u'(0) - cpi

Trong db D : d"^[ 0,1] -> Qo ,1] la loan tur vi phan phi tuye'n bac n

Nhdn xet 6.2: Ta cung cb ihi xay dung thuat loan vdri s6' budfc khCng qua n

( n < N ), hoae bie't ||u" | 1 < R ta cung eo thd xay dung thuat loan mot budfc nhu

^2 nhung nhan xet a cac muc tru6c

CHUONG II BAI TOAN TUYfiN TINH KHONG CHINH TREN COMPACT YEU

|1 - Mo dau

Gaponenko Yu.L (1989[27]) d"e xua't phuang phap xa'p xi tuofng thich giai bai toan (0,1) trong truong hgp X = Y = L2[ 0,1] vdfi gia thie't lien nghiem | Xd(l) I ^ R V

t e [0,1]

Pham Ky Anh [71] St xua't y tuang mdfi: Six dung co sdf Fourier thay cho

Splines cb thd nhan dugc cac ke't qua tuang tu nhu [27] cho khong gian Hilbert bat

ky, han nUa khi irb lai trubng hgp L2[0,l], la giam nhe dugc dibu kien | Xd(l) | < R

Sur dung y tuang cua Pham Ky Anh [71], chung la xet tnrong hgp bie't khai

tri^n ky di cua loan tu tuye'n tfnh hoan toan lien luc, ta cb Ihd danh gia m6t sb ir6c

lugng dn dinh ye'u va de xua't mCt thuat loan kidu khai tridn ky di chat cut dd giai

bai toan dat kh6ng chinh Can noi them rang mOl sb ke't qua cua chucmg nay cung co

Ih^ nhan dugc bang ky thuat thang cac kh6ng gian Hilbert (Hilbert scales technique)

• Xet bai loan

Ax = y (1.1)

O day A la anh xa tuye'n tinh hoan loan lien tuc trong khong gian Hilbert thue

H Dd viec trinh bay dugc dan gian ta gia thie't rang (1.1) giai dugc duy nha'l Tix linh

giai dugc duy nha't cua (1.1) suy ra KerA = {0}

Trong trubng hgp Ker A^ (0) la gia thiet tbn lai loan tu ngugc suy rong A"*"

Gia sir trong phuang trinh (1.1), thue te' ta chi bie't y5 e H sao cho

||y^-y|| < 5 vdfi5>0

Ta phai tim x^ sao cho

||x5-x*|| - > 0 k h i 5 - ^ 0

a day X* la nghiem duy nha't cua (1.1)

Til gia thie't A la loan tur hoan loan lien luc nen ta cb A*A va AA* la cac loan tu

d6'i xung hoan loan lien tuc cb cung cac gia tri rieng

Gia sur ak la gia tri rieng cua A A va day gia tri rieng dugc sip nhu sau:

CTl ^ 02 ^ an ^ > 0

Goi Uk la vecta rieng cua A*A thoa man:

A*AUk = ak^Uk

< Uk, Ui> - 5ki

Trong do < , > la tich v6 hudfng cua H

(1.2) dugc goi la Ichai tridn ki di cua loan tir A( SVD cua A)

Theo dinh ly Picard: Dieu kien c'an va du dd (1.1) giai dugc la:

y e (ferA*)^

S a k'^ I < y , Vk > r < +00

k = l

§2 - Hoi tu yeu trong khong gian Hilbert

Xet mot cap khong gian (Xo,H), trong do ( Xo,|| ||o) la khong gian Banach thue lach

dugc, H la khOng gian Hilbert thue vdfi chu^n || || cam sinh bai tich vO hircfng < ,.>

Gia su Xo nhung irii mat, lien tuc trong H

Tu gia thie't ve Xo va H suy ra:

h i / V x e X o = > x e H , ||x||<Co||x||o,Co>0

Ngoai ra trong muc nay ta se gia thie't rang

h2/Xo v a H cb chung cosogbm cac vector rieng {Ui} cua A*A Hon nua gia tri

rieng On thoa man dfeu kien an ^ 0n^ > 0 ( n = 1,2, ),

trong do 0„ -> 0 ( n ^ oo) va p > 1 cb dinh ( eon goi la trubng hgp giai fan rong)

Ta xay dung khbng gian Xi nhu sau:

Ngoai ra theo dinh nghia || x || o ^ || x || i , dieu nay chung to Xi trii mat trong Y^

Bo de 2.2: Moi tap hi chan trong Xi la tap compact tuang doi trong H

Ch(mgminh:YAhitnM^{x^Xx:\\x\\x < K, K>01

Do 0i-> 0 khi i -> 00 nen vbi moi E > 0 tbn tai sb No > 0 vai moi N > No thi ON ^ e Vdi moi x e M c: H ta cb b^t dang thu*c sau:

||PNoX-xf = Z |<X,Ui>P<(E0rM<X,Ui>h)^^^(Z 0,'p |<x,u,>|-p)^^p

i=No+1 i = No+1 i =No+1

< K s ( E | < x , U i > | ' P ) " ' P < K s ( 2 | < x , u , > r - | < x , u , > p P - ' ) ' / ^ ' '

i=No+l i=No+l

•

< KE (CoK)*^(CoK)^P"'^^^ = COK'E

TIT day suy ra tap: No

MNO = {XNO = S < x,Ui> Ui, X e M )

i = 1

la CoK^E luai compact cua M va cQng suy ra M la tap compact tuang dbi trong H

Ta dinh nghia chudn bie'n phan nhu sau:

||x||v = Supl I <x,v> I ,v e Xi, IIVII1 < 11 V X e H

Dinh Iy2.1 [71] cho day {Xn} c H, x„ ^ x (n -^ oo ) khi va chi khi thoa man hai

dfeu kien sau:

X* e M ( R ) = { X G H : ||X|| < R ) , R > 0 Con ve'phai cua (1.1) thoa man ||y||^ 5, 6 > 0

Nhiing di'eu kien nay co nghia la:

II x * f = Z |<x*,Ui>P < R'

1 = 1

00

vk II Ax* f = Z Oi^ I <x*,Ui> P < 5^

*Kyhieu: cow(5,R) = S u p { | | x | | v : || Ax|| < 5 , | | x | | < R }

co(5,R,N)- Sup (IIPNXIIO: | | A X | | < 5 , ||X|| < R |

Moi u6c lugng tren cua cOw (5,R) va a)(5,R,N) dugc goi la danh gia dn dinh nghiem Djnh ly 3.1: Gia sir toan tur tuye'n tfnh hoan toan lien tuc Aco khai tridn ki di thoa

man dfeu kien cua §1 Khi do ta cd danh gia dn dinh ye'u nhu sau:

Nhdn x^t 3.1: Trubng hgp KerA^ (0) ta xet H^ la khOng gian Hilbert eon sinh bai

cac vecta rieng juili'^ ung vdfi cac gia tri rieng ai^> 0, i = 1,2

Xet trubng hgp giai fan rOng Goi x"*" la nghiem chuan cua (1.1) ta cd danh gia

||xlv<R'^ a / 5 ' ^

§4 - Phuang phap khai trien ky dj chat cut

Trong muc nay ta gia thie't ring cac gia tri Ok trong khai tridn ky di cua loan tur A va cac vector ky di Uk, Vk tuang ung da bie't

Gia sir (1.1) cb nghiem duy nha't x* va ta chi bie'l Yb'^y sao cho

l | y 5 - y M 6 Nghiem chinh xac x* trong trudfng hgp nay cb dang

X* = Z Qui , a day Q = cj;^ < y,vi> , i= 1,2

Tiir db suy ra ||XN^-X* || < ||XN^-XN* i| +1|XN*-X* || < (a+5)aN"'N^^+1| X*N- X* || (4.2)

Vdi mbi 5 > 0 du nho va cb dinh, ta chon N = N (5) tijr dieu kien

Trong trubng hgp ne'u gia thie't tien nghiem rang x* e Xo, thi ta co danh gia

IIXN''- X*|| < (N(5)-' +||x*N(6) - x*||o ^ 0 ( khi 5 ^ 0)

a day N (5) chon tir dieu kien

Nhu vay ta da chung minh dugc dinh ly sau

Djnh ly 4.1 Gia sur X'^N la nghiem xa'p xi hun ban chieu vdfi he sb Q"^ xay dung theo

(4.1) Ne'u a = a (5) -> 0 (5 ^ 0) va N = N(5) thoa man (4.3) thi

ix%6) - x* II < N(5)-'^ +||x*N(6) - x*|| ^ 0 ( 5 - ^ 0) Ngoai ra ne'u bie't x e Xo va N dugc xac dinh tir (4.4) thi ta cb danh gia

IIXN''- X*|| < N(5)'* +||X*N(6) - x*||^-> 0 ( 5 ^ 0 )

§5- Phuang trinh tfch phan dang tfch chap

Trong muc nay ta ap dung phuang phap khai tridn ki di chat cut danh gia dn

dinh ye'u cua nghiem cho phuang trinh tich phan eo dang sau:

vay he Un(s) = cos(7tns), Un(s) = sin(7ms) Uo(s) = 1 (n < 1) la he vecta rieng day du

ciia A*A tuang ung vdi cac gia tri rieng

cJ^ = k^n + k^n,n> l,a^o==k\

Aun(s) Aeos(7ms) kneos(7i:ns) +k„sin(7ms)

Dat Vn(s) = = =

^n Gn CTn

Aun(s) Asin(7ins) knSin(7ins) + kn cos(7ins) V"n(s) = = ' =

On On On

Au^o ko

Con Vo' = = = 1

ko ko Theo each dat tren ta cb

A*Auk = a \ Uk , A^AiTk = a\"iik , (k > 1) , A*Aiio = a^ouo ,

Auk ^ k Alio

Vk = ' , Vk = (k > 1) Vo =

Ok Ok Ok

Gia sur bai toan (5.1) co nghiem duy nha't x* va thay vi bie't ve' phai dung y, ta chi

bie't y6 sao cho II y - y6 || - 5

Ne'u ta chon a = a(5) = a va N(5) -> oo ( a -> 0) thoa man he thiic (4.3) co nghia la

I , , 26 < n = l , N

ll\= {x(t)tuyetdO'i lien tuc tren [-1,1], x(-l) = x(l) vax(t) e L2[-l,ll}

Chudn trong Xo duac dinh nghia bang c6ng thirc

||x||o :=max{||x||c,||i|| }

U

6 day II || II || thu-tu la c h u ^ trong khOng gian C[-1.1] va L2[-l,l]

2 " Vdi e h u ^ nay cap kh6ng gian {Xo ,H } thoa man hai dieu kien hi ,h2 ciia

54, CO nghia la Xo va H co chung ca so {1, cosTint, siuTint } , n ^ 1

V X e Xo , Ti: Z n (x „ + X n), II X II < +GO, nen co the dal

n = 1 L

2

X , - X o V a | | X 111-II X ||o

O day chudn bie'n phan duac djnh nghia nhu sau

II X ||v = sup {<x,v>: V G Xo, II V ||o ^ 1}

Ta se danh gia modul lien tuc ye'u sau

03w(6) = sup { II X ||v : II Ax II < 6 } C0w( 6,N) = sup { ||PNX||O : || AX || < 6 }

Gia sijr V e Xo, || v L ^ 1, vai V x e H, ta co

Tir dinh nghia phep chie'u

Mat khac I Xn I == I <x,Un> \ =nn\ <x, || Un || o'^ Un > I < 7tn || x || v

Tuangtu \x^A ^ 7in|| x||v, Ixol ^ || x||v

Tir do ta co ba't ding lhu*c

1 N 1 ^

II PNXIIO^ ( + 27c'Zn')|| X ||v - ( — + N(N+ 1)(2N+1) )|| x||v)

2 "=^ 2 3 B^'t horp lai ta nhan duac udc luang

Tt'

CD(6,N) < (n^ + A5"ko)'^ (1 +2N(N + 1)(2N+1) — ) 6

3

56 - Danh gia Diam V5 trong phuang phap compact thu hep cua Gaponenco

Trong phuang phap compact thu hep Gaponenko chi chung minh dugc diam

V5 ^ 0, khi 6 ^ 0, ma ehua chi ra each danh gia duong kinh tap V5.Trong muc

nay trinh b"ay each danh gia diam V5 cho mOt Idfp bai toan co dang dae biet

Xet phuang trinh

Ax = y (6.1)

Gia six trong (6.1) ta kh6ng bie't chinh xac y ma chi bie't y^

II y ' - y | l ^ 5

A: R" ^ R"" la toan tu: tuye'n tinh lien tuc, co rank A = p , p < n < m

Khi do A se co khai tri^n sau

p

Ax = Zai<x,Ui>Vi

i = i

d day ai >a2 > ^Op > 0; uj e R", Aui

<x,Ui> = x \ i , <Ui,Uj> = 6ij, i,j = 1 - n , Vi =

a

-B 6 xung them cac vec ta true chudn de he {Vk} i'" la he ca so true chuan trong R'

Vi\j = 6ij i,j = 1 - m

Goi A"*" la nghich dao suy rOng cua A

Tijf Ax == y => x"^ = A"^ y Nen ta eo danh gia

| | x 1 | ^ a p - ' ( | | y 5 | | + 6 ) < R (const > 0) Dat S = { x e R " : ||x|| < R }

Goi S5 la tap luai cua S dugc xac dinh nhu sau :

N

R'

n

N '

Vay b6 de dugc chiing minh[]