lời giải chỉnh hóa của phương trình tích phân loại một, chương 3

CI1l(dng3 : Nghi~m Chinh Boa Cua Pht(dng Trlnh Tich Phan Lo1;liMQL

CHUaNG 3:

NGHIEM CHiNH HOA CUA PHl!(JNG TRINH TicH
)\

)\

PHAN LOJj.I MQT
Trong chuang nay Lase gidi thi~u mQts6 phuong phap chinh hoa cho phuong trlnh Au =g
vdi A la loan tit tuye'n tinh,li<.~n
t\lC,Compacttu X vao Y
X,Y la cac thong gian Hilbert
Cac cach chinh hoa co the' chia lam 2 lo~i:xc1pxi b~ng loan tit ho~c xc1pxi b~ng khong
gwn.
~
,?
I':
3-.1 X ap
Xl bang toaD tu' :
Ta se c6 g~ng xay dt!ng loan tit Ra :Y~X ma Ra la lien t\lCva la xc1pxi A+ rhea
?

nghIa R~ ~ u : =A+g khi a ~ 0 cho m6i g E D(A+); (j day g la dli ki~n chinh xac
Ne'u ta ky hi~u A.= A* A; rhea 2.7.2 ta co: u = A+gse thoa man A.u =A*gdo do

ne'u A.kha nghich ta duc;lcu = k1 A*g ;con khi A.thong kha nghjch ta hi vQngxc1pxi u bdi
vecto co d~ng Ra (A.)A*g(a >0) vdi Ra (t) la ham s6lien t\lCtren a (A.)~ [a, II A 112]
(xemO.8) va Ra(t) xc1pxi fer) = lit
Trang phc1nsan day ta se gi

Ra(t) -> lit khi a ---~0 cha m6i t > 0
va It Ra(t) I la bj ch~n dell

(2)
(3)

3-1-1.Dinh Iv: Gi<isit (Ra)a > 0 la hQcac ham s6 tht!clien t\lCtren [0,

II

A 112]thoa (2) ,

(3) khi do Ra (A.)A*g~ A+gkhi a ~ 0 cha m6i g E D(A+)
Clut'nJ?Ininh ..

Chung ta chti

y r~ng ne'u/~ la mQt da thti'c thl :

/~A *A) A* =A*/~AA*).
Da djnh ly xc1pxi weierstrass cha cac ham lien t\lCtren a(A*A)

= cr(AA*)

(xemO.7) ta duc;lc
R a (A*A ) A*
=>

= A*Ra.(AA* ) ., dilt. AA* = ARa(A.)A* = A*Ra(A-)

=> Ra(A.)A*g E RangeA*
Da do ne'u gQi {UI1,VI1,
Pn} la singular system cl'ta A(xemO.7)

Taco:

R

a

(A)A.*g =

(~ n.- 2 )(A *g, v n )vn
= R
2 Av }
(
n~l a ~n. )~' n n
~ R
n =1 a
00

l-

(

= ~ R ~l-2
_
n- 1 a n

g,~V

)(

v

~n n] n

( )

= ~ ~l~1Ra ~~ 2 ~, vn} n
n =1

(~~ 2)~, vn} n
R (A)A*g=lim ~ ~l ~1-2R (
)(g,V}=n n=l~ ~n ~,un } n
a
n=l n n
ann
= n=
~ 1~n~~2Ra

lim
a~O

~l-2

Trang 27


(Do (2) , (3) va do dinh ly hQit~lbi ch~n.)

.

=A+g
(do 2.7.4)
.
Trong tru'ong hQp g~D(A +) ke't qua san chi ra r~ng (Ra (A)A"g) akhong hQi t~l(ngay
d hQi t9 ye'u)
3-1-2. Dinh Iv : Ne'u g~D(A +)khi do m6i day an--+ O;(Ran(A)A"g)nla khong hQi t~
Clui'l1J?

111;1111..-

G9i Pia phep chie'u tntc giao tilY len Range(A) = Ker(A"l=Ker (A).L

w
Ra

n

(A)A'Pg = Ra (A)A'g--+z EX
n
w

=>A'Ra
=> ARa

n
n

(A)Pg--+z=>AA'Ra(A~g--+AZ
n

(A)rg --+Az

(Do A la compact xemO.9)

l~i do (2) va (3) ta co:
Til do SHYfa Pg

V~y:

w

(Ran

ARa

n

(A)Pg --+Pg

=Az => g E D(A +) mall

khi n--+oo

thuffn gia thie't g ~ D(A +)

.

(i'\jA' gIn kh6ng h~i II! y(!n

3-1-3. He Qua:
Ntu g ~ D(A~) thl
lim liRa(A) A*gll = 00
'
A
a --+0
~
,!!
Ha1 d ~nh Iy tren ch1
ra rang de t11Udu'QCslj h91 t~ cua xa
:= Ra(A)A"g thl di€u ki~n dn va du la g E D(A+)hay Pg E RangeA (P la phep
. chie'u trljc giao cua Y leu
RangeA) Bay giOta khao sat t6c dQhQit~lcua Xav€
'

'

?

A'

?

A+g

Ta hoan loan co the thay the' di€u ki~n (3) : tRaCt) bi ch~n dell bai di€u ki~n
I

tVll-tRL(t)l~w(a,v)
Voi tE

I

(4)

[o,[IAW]a day

.

w (a, v) la ham so' (chi t6c dQ clia slj hQi t~l) thoa marT

w(a ,v) --+0 khi a --+0 cho m6i v > 0
3-1-4. Eli'd~ : Range (AV) ~ Ker (A).L
Clu(l1g

l11inl1. ..

Avx

=

I vnn(x,w~.

n= I

n (xemO.7)

d day Anla gia tri rieng khac 0 ct1aA va wIllavectd rieng tu'dngling

,

-1

-

"-1

".

Do do : Wn= An A Wn= A An Awn E Range A
-v
"
.L

=> A x E RangeA = (KerA)

(xemO.6.3)

.

3-1-5.Dinh iv : .
.
Ntu AAv x =Pg VOlmQtv > 0 va x .EX nao do thl : II A+g -x
CIUfl1J?

D~t u

111;1111.

a II

~ w (a, v) II xii

.-

=A+g khi do Au = Pg =AA v x => A (ll - AVx) = 0 => II - A v X E KerA
Trang 28


U E (KerA) i va A v x E (KerA) i ~ u - AVx E (KerA) i
ma : KerA n (KerA) i {O}

- vx=O~u=A =
-v x

~u-A
Xa= Ra (A) A*g = Ra(A)A* Pg
ma: Pg= AAvx= Au
~ Ra (A) A* AAVx= Ra(A)A*Au

--

~Ra(A)A

v+l

x = Xa

do do: Ilu- x(x" = II Avx - Ra(A)Av+IXII

= II A v ( 1 -Rex(A)A)x II
II u -xu II ~

w (a,v) II xii

.

(do (4»

Bay giCl d~t ea : =A~g - Xa
3-1-6. Dinh Ii..! Ne'u PgE Range (A V)tuc Pg = A Vxvdi mQt v :2 1 khi do :

lieall ~ fw(a, v-1Xa, v)llxll
Clui:n!?minI!: f)~t u

= A+g~Au = Pg = A.vx= AA*CA.v-I)X

~A(u - A*A v-IX)= 0 ~u-

A*A.v-lx E KerA

ll1 nhu dinh 19 tren ta cling duc;Jc : u = A +g = A* A
.
19-1co : Xa= Ra (A)A*Pg = Rcx(A)A*A x = A*Ra (A)A x
lam ttlong

v-IX

~v

~

~v

~Ca = A+g - Xa = U - Xa = A* A.v-IX - A*RaCA)A. Vx = A*(I - Ra(A)A.)A. v-IX
~

II ea 112= (ea , A*(I-Ra(A)A.)A

=(Aea , (I - Ra (A)A)A v-IX) ~

~ IIcall 2 S

II

II

V-IX)

A call . II (I - Ra(A.)A)A v-Ixll

A ea II II xII w (a , v-I)

(*)

19-ico :
A ea = AA*( 1 - Ra(A.)A.)A.v-IX= A*A.v(I - RaCA)A.)x

Do do : II A ea

112

= (A ea , A ea) = (A*A ea, ea) = (A ea , eeL)
v
= (A*A
(I - Ra(A)A)x, ea)
v
= (A ( 1 - Ra(A)A)x , Aea)
~

~

~

~

~.

~

II Ae(xI12~IIAV(I-Ra(A.)A)II.IIAeall.llxll
sw(a,v).IIAcall.llxll
~IIAeallsw(a,v)llxll
NhUV?y(*)thanh:11
cuI12sw(a,v)llxll.llxll
~

.

s ..jw(a,v)w(a, v-I)

Ileal!

(do(4»
w(a,v-l)

Jlxll

.

Nhu V?y bhng dinh 19 3-1-1 va dinh 19 3-1-2 ta da chI ra duc;Jcr~ng Ra(A)A *g --+

A+g khi va chI khi g ED(A+).

Va m,;lI1h
hon nua ne'uPg E Range(A.* V) \:div :2 1th1
Ileal!= IIA+g - Xall ~ ..jw(a, v -1)w(a, v) Ilxll
~

-

v

*

Trang do A x =Pg va Xa= Ra (A)A g
Bay giClta chuySn sang xet cho traCInghc;Jpg'6la du ki<$nkh6ng chinh X3.Cthoa :
IlgJ - gll s 8
.5

;8 >0

-

* '6

E>~t: Xa = Ra (A)A g

Ntu X: hQit~ltheo mQtnghlan~lOdo v€ nghi<$m
s6 co chuffnnho nha'tA+g khi8--+0,ta noi
nghi<$mxa'p xi 0 tren la nghi<$mchinh hoa clla phuong trlnh . Noi chinh X3.Chon no la
nghi<$mchinh h6a ne'uchQnduc;Jcthalli s6 chInh hoa a ph~lthuQc8 gQila a

=a(8) sao cho :
Trang 29


* 0
hm R
0 ~ ° a( 0) A A g = A -I-g
~

()

Nhu' v~y mC)tC{ich chinh hoa bao g6m khong chi vi~c chQn ilIa ham chinh hoa Rx

ma con la vi~c chQn chugn cho khong gian va chQn a(o) cila thalli s6 chinh hoa

.

GQiC la h~ng s6 thoa ItRa (t)1 ~ C2vdi t E [°, IIAII2J ; a >0

(5)

Va rea) = max {IRa tl: t E [0,IIAII2]}

(6)

D€ Y r~ng

= 00

Jim rea)
a~O

3-1-7. I36 (1e :

va
I\A(Xa

IIA(xa

-

x~ )11 ~ oc2

Ilxa

-

x~ ~
II

ocJ$)

Ta co : A. (xa - x~ ) =A. Ra (A.)A*(g - gO)

Clu'cng l11Jnh:
~

-

= (A(xa - x~ ) , A(xa - x~.

x&)W

»

= (A. (xa - x~ ) , Xu - x~ )
- *
o

= (A Ra(A)A (g -g ) , Xa- x~ )
~

~

8

= (A Ra(A)(g -g') ,A(xa-x~»

~
.~
~

IIA(xa

-

11

Ra (A)(g -gO)IIIIA(xa

- x~)11

x~ )112 ~ C211g- gOIIIIA(Xa - x~ )11(do(S»

IIA(xa -x~)11 ~C21Ig-g011 ~c2(*)

Bay giG xet IIxa Ta co :

x~11

Xu - x~ = Ru(A.)A*cg - go)

~
Ilx a - x

~

r

=

(x a

-

x a' R a

(/\)A*

(

)
go))

g - gO)

= I X a - x a' A *R a (A ~ g

-

=1 A(Xa -Xa}Ra(A~g-gO))

~

Ilxa -x~r

~IIA(Xa-xalIIIRa(A~g-go)11
~

o2C2r(a)

Do (*) va do (6)

Bay giG ta se giiii quye't dieu ki~n dll de' O~O
Jim xoa

.

= A -I-g

Ta se giJ slYa :[0, -I-oc)-~ [0, -I-oc)la ham lien t\lCkh6ng am va a(O) =0
3-1-8. Dinh Iv :

Ne'u g E D(A+); a(o)~O va ~?r(a(o» ~O khi o~O khi do X~(6)-->A+gkhi 0 ~ 0
Trang30


Clutnf! minh :
Taco:
A + g - x ~(O) ~

IIA

II

+ g - x a + IIxa( 0) - x~(0)

~ IIA +g

II

- Xal/ + oc.Jr(a(o))

(Do b6 d€ 3-1-7)

l<;1ido dinh 193-1-1 ta co Xa«5)-+A+g khi 0 -+0 va do gia thie't

Iim 0.jr(a(5»
0-+0

Khi 0 -+ 0
=> IIA+ g - x ~(o)

0

II

-+ 0

+

.

Khi 0 -+ 0

=> xa(o) -+ A g

Bay giO ta xet de'n khai nit$m chinh hoa ye'u do cac di€u kit$n khong m<;1nhcua 02r(a(0))

Ta gQi xifp xl

{ ~(O)}
x

Iii weakly

regular

u€u eho mill day (0,)

->

0 ta c6:

v€ A+g khi 0 -+ 0
Ne'u di€u kit$n do khong tho a thl ke't lu~n nhu the' nao v€ s\f hQi tl,lcua

va g E D(A+) khi do

J

=0 la di€u kit$n dll clla s\f hQi tl,lm<;1nhclla

Ta da co do dinh Iy 3-1-8 0-+0
Iim 0.Jr(a(5»

3-1-9. Dil1hIf: Gia sU'

x :[0 n

0
xa(o)

:

X~(o)

?

.

Iim sup02r(a(0))<00
0-+0

X~(o) hQitl,lye'u.

CIUfnf! minh :
D~t u

= A +g va

gia sU' Oil -+ 0 khi do ta co :

8

u-x at8n)

8

= u-x

a(8n) +x a(8n) -x at8n)

Do di€u kit$nOil-+ 0 ta da co (xem dinh Iy 3-1-1) xa( 0n) -+ u Khi n -+
Bay giG ta phai chi ra

8
x at 0 n)

00

w
Khi n -+
-

xa( 0 n) -+

A +g

00

0

La'y Z E KerA ta co :

Trang 31

-x

0~

[

J

{g - g

(

= z, R a (i\

z,x 0:[0 n J

{

,

0n

o:lon J = ,Z,A. R 0: (A { g - g0n )) = [ Az, R0:(A,\g

8

= I 0, R a
La'y Z EORange(A")

"

JJ

0)1

,

~

(A g - g

n

{

JJ

-g

=0

JJ

(chli y (ketAl = (RangeA")

G1a su Z = A x, ta co :
'??

[ z, x a[ 8

X x:('8 J

~

( A \,

A *R a (A { g - g 8 n ) J

= [ x, AA *I< a (A { g - g 011 JJ =[ x, AI< a (A { g - g 011 JJ
~llxllllAR a

(Allg-gOn
ll

ll

~llxllc20

n

~O

.

Kht n ~
0

00

Ta co KerA EBRangeA" la tru m?t trong X

"
\fz EOKerA EBRangeA

;

[lOXaloof
Tli dosuy ra :

X:(oJ

0
xar\ 0 n )

-+ 0

w

-x an o

.

-~O

( n)

Bay gio ne'u di€u ki~n c1acho la ye'u hODnua
3-1-10. Dinh Iv:
Gdl sli' A kh6ng

co

0I~ 002a(o)R a(o) (a(o)

eha [X:[0 J

h~ng

P=

00

hUH h~1n, g

EO D(A+);

aCe)

. Khi d6 3 day o. vdi 0,,->0 va g'"thoa

th6a

di€u

ki~n

ligon - gll ~ 0 n sao

kh6nghqit~ yea

Clucnfjminh :

GQi {un,VIl,~lll}
la singular system cila A va d~HAn=~l~2 (Anla gia tri rieng cL\aA*A, Anla
day giam v€ kh6ng VIA kh6ng co d~ng huu Iwn xem 0.7 )
[0,+00) ~[O,+oo); a(O) =0 Ben 38n ~ 0 sao cho a(On)=All

Do di€u ki~n lien t~lCcl\a a:
s:
1::\ t 8n
u~1 g = g-Onlln
~

Chli y r~ng

:Xn(8n) =I~n(81l)

(A)A*g ~ A +g

khi n~

00

(do 3-1-1)

Trang 32


6

L~i co:

xa[ 0n) - x a(o n) = R a(o n /A)A * (g - g 0 n )

(A)A* 6 v
= Ra(o n )
n n
-0
-

-

-

nlln

a(o n

0 ~l-IR

II

aeon)

-xon

aeon)

-02
-

11

-2R

nlln

n

(\

n n

a(o) n \/...n f' n

2

=> x

) (AfY
~\-

-IR

R
/...
(/...n f -02/...
n n a(on ) ( nf

a(o) n

-

= o~a(on)Ra(o n ) (a(on)f ~
0
Ta co:

0

Khi n ---t

00

0

xa(o n ) = xa(o n ) -xa(o n ) +xa(o n ) ;:::Ilxa(o n )

Ma:

x

+

va

a(o n ) ~ A g

0

=>fIxal(on ) ~

00

6
x n
a(o n )

11

-

xa(o n )

0
00 =>xa(o

n ) KhonghOit~lye'u

Nhu' v~y trong dieu ki~n g EOD ( A +) va 11gB
-

gll

~

-

x a(o n ) - "x a(6 n )

+00

.

s; 0 ta cHikh':l0 sat sl! hOi t\l cua x~(o) ve

xa ph~l thuOc VaG Ra(t).
Bay giG ta xem m<)t cach chinh hoa voi mOt Ra (t) C~lth6
1,
Ra (t) = t+a

~

wc x

a

= ( A + aJ)

-1

Ag

va ham so' chi t6c dO hOi t\l khi c10la m«x, v) = av voi 0 < v s; 1
Celeh chinh hoa nhu v~y gQi la each chinh Tikhonov.
Ong dil gi
Om

cl!c ti6u clla phie'm ham Fa (x) = "Ax - gll~ + allx"~ gQi la phiS-m h~lll1Tikhonov.

3-2. Cach chlnh boa Tikhonoy :
3-2-1. Dinh If :

Gi,l stYA la 1-1 va Range A =Y; GQi A* loan ttYlien h<;Jpcua A khi do Va> 0,
phuong trlnh Aa + A*Ax = A*g luon co nghi~m duy nha't Xa, ph~l thuOc lien t\lc VaGg va Xa
la ctfc ti6u Cllaphie'm ham Fa (x) = !lAx- gll~ + alJxll~
Trang 33


ClucnR minh :
X6t roan tti a : X x X -t R.
'"

(u,v) -t a( u, v) + (A Au,v)

=a (u,v) + (Au, Av)

.

Ta tha'y a la song tuye'n tinh

.

a lien t~ICVI:

-

la(u, v)l::; al(u, v)j +1(Au,Av)1
::; allull.llvll + IIAull.IIAvll

(A lien t~IC)

s allull.livII + c21lull.livII

s

lIa(u, v)11 (a + C2 )11uI1.livII

.

a cuBng bur VI a(u,u)

=a

( u,u) + ( Au, Au)

= allul12+IIAuf

~alluf
X6t L: X -> R.
'"
u H ( A g, H).

. ta tha'y L la phie'm ham tuye'n Hnh lien t9c.
Do d6 :ip d~1DgLax Milgram ( xem 0.10) ta thu duQc
T6n t(;liduy nha't u EOX sao rho a (u,v) = L(v) \I V EOX.

Tuc a(u,v) + (A"'Au,v) = (A"'g, v) \I v EOX.
=> (au

'"
'"
+ A Au - A g,v) = 0 EO\Iv EOX.

'"

'"

=>au + A Au= A g.
Sv duy nha't : Do axa + A"'Axa= A"'Ag<=>(aI + A"'A)xa=A"'gDen ta chi dn chung
minh (aI + A"'A)la 1-1.
'"

'"

Ta co : (aI + A A)x = 0 =>(ax +A Ax,x) = 0
.

=> a IIxl12+ ( A'"Ax,x) = 0 => allxJJ2+ IIAxl12

= 0 =>

IIxii

=0

=> x = O.

Ngoai ra do aI + A'"Ala tuye'n tinh V?y aI + A'"A la 1 - 1.
Sv phi,!thuQc lien t~!C:
Ta chi dn chung minh ne'u ax + A'"Ax = z -> 0 thl x -t 0

Ta co : ax + A"'Ax= z =>(ax + A'"Ax,x)= (z, x)
2

*
2
2
=>allxll +(A Ax,x)=(z,x)=>allxll
+IIAxll-=(z,x)::;llzll.llxll

Trang 34


=>aIlxf ~ Ilzll.llxll=>allxll~ Ilzll
Cho z -+ 0 => a Ilxll -+ 0 => x -+ O.
Xu 1a c\fc lieu phie'u ham Fu.
Ta co:

-g112 +allxI12-CIIAXa
-g112 +a11Xa112

Fa(X)-Fa(Xa)=IIAXa

= flAx - gl12 -IIAX a - gl12 + aCllxf -llx a 112)
=IIAX-AXa

-I-Axa -g112-IIAxa _gll2 +acIIX-Xa

=IIACX-Xa)112

-I-2CAxa -g,ACX-Xa)-I-aIIX-XaIl2

+Xa112-IIXaI12)
-I-2a(Xa,X-xa)

= IIACX- Xa)112 -I-2CAxa - g,A(x - xa) -I-allx - xal12 -I-2a(Xa'X

- xa)

*
*
=> Fu (x) - Fu(xu) ;::2(A Axu - A g, x - xu) + 2a( Xu, x - xu).

*
*
=> Fu(x) - hxCxu);::2(A Axu -A g + axu, x - xu)
,*
*
ma : A Axu - A g + axu = 0 => Fu(x) - Fu(xu) ;::O.
NgliQc l?i ta cling tha'y Fu co day nha't mQt c\fc lieu xac dinh bdi 1"(0) =0 vdi mQi w
E X trong

do:

[(i) = Fu(xu + tw) = IIA(Xa+ tw - gl12 -I-allXa + tIff
,

*

*

[(0) = 2(A Axu - A g + a Xu , w) = 0 V w
Do do Xu tho a : A*Axu - A*g +a Xu = 0

.

.

*

=> A Axa.+ a Xu= A g
Dieu nay phli hQp vdi Xu= (A. -I-a1)-1 A*g

Vdi ham chi t6c dQ w(a,v) = av nhli da noi d tn~n
Cling \rdi cac ke't qua 3-1-5 va 3-1-6, ta dliQc

M qua

san :

3-2-2. H~ (jua :

.

Ne'uA+gE Range(A.V)vdi O
.

Ne'uA+gE Range(A*)khidoI/A+g-xa 1/=8(a1/2)

8(a v)

Trang 35

IIA+g-Xa II~o(av) chI al~oIIA+:~Xa II ~C*O

(Jd§y

IIA+g-Xa II~e(al/2)

chI al";oIIA+iaXa

II =c"o

Bay gio chung ta chuy€n qua xet x~ = (1\ +exlr1A*g8 vdi 11gB- gll:s:;0
Chu

y f~ng vdi

R (t)=

ex

t
It~(t)1 :s:;-:s:;
t+a

E
~t+ex =>r(ex)= maX{\R ex(t~,t
~

[

O,IIAI12

]}

=~
ex

1

3-2-3. Dinh Iy :

.

ChQn a = K8 ;A+g E

.

ChQnex=Ko

.

.

Range

2/(2v+l)

(A*) thlllA+ g - X~(o)11= 8(01/2)

-v

.

;A+gERange(A),O
'
II

+
0
2v/(2v+l)
A g-xex =8(0
11

)

Clucng minh.. Tli 3-1-7, ta SHYfa:

IIA+ g - X~(o)lI:s:; IIA+ g - Xex(o)11+llx~(o)

- Xex(o) II

:s:;IIA+ g - x ex(o)11+ o.I.J;(ex)
:s:;IIA+ g - x ex(o) II+

!r-a

L~i do 3-2-2 0 tnSn

.

Neu chQn a = Ko vdi K 1ah~ng du'ong va A+g E Range (A*) , ta ou'<;fc:
IIA +g - Xex(o)11

=> IIx~(o)

.

Neu chQn a

= Ko2/(2

IIA+ g

va -~ fa--

ex

--

-

= 8(01/2)

A+ gll = 8(01/ 2) (1)

V+l) va Alg E Range (1\") vdi O
Xexll = 8(02v /(2v + 1»

0

17
"K02/(2v+l)

1
02/(2v+l)
- -JK'

.
=> Ilx~(o) - A + gll = 8(02v/(2v + 1)}(2)

Trang36


Va nhu v~y t6c dQ hQi tt,lnhanh nha't rhea ht%qua nay la 8(02/3), n6 xay ra khi
A+gERange (A) va a

= A02/3

ung voi v = 1.

Ta dii giai quytt duQc va'n de t6c dQ hQi t~l( trong cach chinh h6a clla Tikhonov )
d?t duQc t6i da la 8(02/3) .
.

Bay gio ta chungto ding t6c dQ.d6khongth~ cao hdn m1'a.

3-2-4. Dinh Iv :
Gia sa g ED(A+) va A+g - x

II

= 0 va Xa= 0 \I a
ClutnJ! minh.. Gia sa u = A g

a

.
= O(a)

tuc

Jim

+g - x
.
a =0
a

IIA

II

a-+O

II

Khi d6 A+ g

+

dill e
.
a

=xa

- u. Khi d6 :

(A + aI)ea = (1\ + aI) (xa- u) = (A + aI)xa - (A + aI)u
= (A + aI) (A + aIrlA*g - (A + aI)u
*
*= A g - (A + aI)u = A g - (Au

(A + aI)ea= -au =>allullS Milea"
.

=>

Jim Iluil= 0 =>

a-~O

=>0

1

= -au

(do gia thitt )

u =0

= Au = Pg

-

=O(a)

+ au)

-

*

1

.

*

va Xa=(A + aIr A g = (A + aIr A Pg = 0
Trong 3-2-2 ntH cho v

=1, ta tha'y A+g E Range

(A) tIll IIA+ g - Xall = 8(a). NguQc

I~lihay xet djnh 19 salt :
3-2-5. Dinh Iv :

Ntu g ED(A+)va IIA+g-xu!1= O(a) khi d6 A+gERange (A).
ClUtllJ!minh.. GQi {un, VII,Jln}la singular system cua A
.

00

u =A+g , ta c6 : u = L: Jl (Pg, un)vn
.
n
n=1

E>i'it

-

-

1 *

1

*

00

.

}

Clingv~yXa= (A + aIr A Pg = (A + aIr A L: ~Pg,un n
n=1

= (A + aI)

-I

00

*

L: (Pg, un) A Vn

n=1
00

=(A + aIrl

L: ~ln -1(Pg, un)vn
n=1


-1

I

~l
n= 1 n

=

1 ~g, u n ~ n
~2
I\, +a
n

00 A; la gia t1'i 1'ieng clla A ; AIl= ~l~ 1 (Xern 0.7.3)

t1'ong

~l

00

=

(

\

n 2 \Pg, unf
n = 11 + a~l

n

L.

n

2

u
Il

-

x

a

I I (~ n

2 =
II

11

~n

-

1+ a~l 2
n
6

n=

~

00

)(Pg, u )v "

nn

(

L

12

n2 2Ipg,Un~
n=I(1+a~l n )

=a2

2

Nhung lilt- xa II = 8(a 2) (do gia thie't) =>

2

00

~l6

L.

n

n = 1(1 + a~l n

2 )2

I

(p, un)
g

1

bi ch~n khi

a~O,
00

~l6

=> L

11

11=1(1+a~l2)2
n

=>Pg ERange

(AA)

=>A+gERangeA
Ta

l<;\i tic'p

2
<00
I

(Pg,Un)
(Xern

/

0.7.5)

.

( VI A(A+g)= Pg)

tl,\C xet

vdi

du ki~n

kh6ng

chinh

xac

gO thoa

Ilg-g811S;O

v~n ky hit%ux~ = (1\ + alrlA"g6,

'

.

Ta v~n hy vQngse o(,\toU<;lct6c 0(>h(>it\l t6t nha'tla B(6), t1'uongh<;lpnay chi xay
1'acho loan tU' A co h,;\llg huu lwn cling vdi each chQn a = C6. Th~t v~y, ne'u A co h(,\ng
huu l1<,1n
thl A+ la bi ch~n (do 2-7-3) va :
u - x~ = A+g - Ra(A )A"g6
= (A~ - Ra (A )A*)g

= A+g - Ra(1\ )A"g + Ra(A )A*g - Ra(A )A*g6

+ Ra (A )A*(g - g6 )

Do A~ bi ch~\ll => Range (A) dong

( 2-7-3)

Nen Ra(A )A* h(>i t\l de'n A+ va IIA + - Ra (A)A *11 = 8(a) (Xern [1])
Do d6 ntu chQn a = C li ta ou<;1cfix

~

x~11

= 8(a)

Trang 38


? ~
,.!,.!
'
.,
5
'>.2/3
Trong [2] cling ch1rang:
) ch 0 m91 cac h ch 9n g, g
nell toc d 9 h91 t~l ] a 0 ( u
A'

A

.

thoa JIg- g 811~ 8 .Khi do A ph,h co h,;ll1g hull l1<;1n,
Trang

[4]

'>.

cling chi r~ng : Ne'u u E Range(A V)vdi 0< v ~ 1

'>.2/(2v+l)

'

A
,
',.!,
'>.2v/(2v+l»
va sa1 so co b~tC0 ( u

a ( u ) =C u

'

'

-

kh 1 d 0: A co h ~ng h UUh~n.

VI V?ysall day ta chi xet truong hQpA co Iwng vo h~n.

=K.

Trong uinh ly 3-2-3, ne'li chqn a

ta duQc t6c de) he)i t~l 0(52/3).

5213

Bay gio ta chi ra r~ng do la dip de)t6t nha'tco the duQC,
Ta v~n gia thi€t a

= a(5)

~ 0 khi 5 ~ Ova g E Range A va u

= A+g

3-2-6. n6 (1&:

N€u u -:t 0 khi do a(8) = 8(llu- x~(8) II)+ 8(8)
Clutng minh ..

= ( A + aI)u

Ta co : ( A + a(8)I)(u - x~(8»

-

-

.

-

1

- ( A + aI) x~(8)
'" 0

= ( A + aI)u - ( A + aI)( A + aIr A g

-

'" 0

= Au + a(8)u - A g
=> (A+a(8)I)(u-

=a(8)u + A'" (Au - g0)

x~(8»=a(8)U

=> a(8)u

=( A +a(8)I)(u - x~(8»

=> a(8)

= 8I1u-x~(8)11+8(8)

+ A'" (g-gO)
+ A'" (gO - g)

.

3-2-7. Dinh Iv :
Gia sti A khong co hi;tnghuu hi;tnva lIu- x~(8)11
thoa jig- g811~ 8

= 0(82/3)

khong phV thue)c gO

. Khido u = 0

Clutng minh .. Gqi {un, vn, ~ln}la Singular System Clta A
Do A khong co hi;tng huu hi;tnnen /-l" ~
~
'>.
- 3 ,8n
D at
. u n = ~ln va g

= g + u'>.n U n
8

00

khi n ~

00.

. '

, kl.11.0:d

'

- x
x 8n
-u=x .
-u+x n
8 )
(
a(8 n )
a (8 n }
a n
a (8n )

Trang 39


~

=x
=x

[ n)

ao

1

*(

u

+ (A + aT) A

-

u

+ 0 (A + aTt 1A *u
n

2

-

~

~ + 0 u )- (A + aT)
n n

-

[ n)

a 0

-

1 *
A g

n
2

2

0

20 ~

a(oXu

Do do Ilx

xa(onr u

~

0 ~

[Xa(onr u, vn J + [ 1:~:~ }

+H:I1~

( B~ng each lam Wong ttf nhu' trong dinh 1;' 3-2-5 )
2
2
2
0
20 0-1/3
0 0-1/3
- II
+

n n
- u
=x
X
+
n n
Do 0" =
=> IIx 11
- u, v
a 0
a 0
1+ ao - 2I 3 a [ 0 )
n
1+ ao - 2I 3
II

n)

[

[

n)

2
-

u

[

J

n

n

)

2

2

0

=>IIx n

[

n

= x

a[0n J

-

u

202/3

n

+

x

a[0n

1 +ao~ 2 !3[

a[ 0 n J

uv

-

'

J

02/3

+
n
n) ( 1+ ao ~ 2/3 J

2

0

=>0- 413 x n
n

11

a

o
(

-

u

n)

>

20- 2I 3
n

- 1+ao-2/3

11

.

Ne'u u *- 0, khi deSdo

x

[

-

a

+

u v
'n

o

J

( n)

-

(

1 + ao

-

n

2 13

)

2

bc1de (3-2-6), ta co :

a(o)6- n 2/3 ~ 0 khi 0 ~ O( VIaCe) cling ca'p VOL0)
0-2/3

=>02 0lim
sup 1+ ao
n -213
~ 0
n

5::

( x a (un )

-u,v

(*)
nJ

Do giil thie't Ilu- x:1(0) = 0(02/3) VOLg;; thoa Ilg- gOII~ 0 Hen trong tru'ong h<;Jpd~c bi~t
khi g = gO,ta cling phai colin - xa(o)11 = 0(02/3) Hen(*) => 0 21. Va 1;'. V~y n =o.

Bay gi(J ta lien h~ vdi 3-2-3 cho tUng tru'ong hQp v = 1, tue la t6c dQ d?t du'Qct6t
nha't.
3-2-8.Dinh ly :

Gia sli' a(o)=K.02/3
thuQc gOtho a I/g- gO /I~ 8
Clu'tng

, K la h~ng du'ong. Ne'u Ilu-x~(o)"=e(o2/3)

. Khido U E Range

khong phl,l

(A.)

minh ..
Trang 40

~lll-3


D?t {un, vn, ~ln}la Singular System cua A va gi
= 1)

Khi do :
~

00

u - x~

= n 2:

~ln (g, Un) 112 (g 8 , un)
=1
. 1+ a~l
~lnn(g,

n=I
00

n = 1l
00

n
2
1+ a~l 11

(g U )v

' n

I= 1~~(a~~
-6)2
(1+ a~l2) 2

11

-

n 2 (g, un) v n

(a-o~-2)2

2:

n

=1

1+ a~l n

]

n

leg,un)1

11

00
11

o~

n 2 )(g, un) -

n

11= 1

~

(~n -

n

1+ a~ n
2
~ (a~l -0)

= I:

Illl-x~112

)-

~l

.

= I:

=>

U

1

~ln
n 2 (g + og, Un ) v n
1 + a~
J

00

= I:

(do 2-7-4)

vn

2

'

6

2

1

(1 + a~l 211) 2 ~ n (g, un )1

Do liLt- Xa(o)11 = e( 02/3) Ben 3M thoa :
2
K-01l3

u-xo

M2
1I

a

20 -4/3-

-

I1

I:

2:

11=1l
2

.

N

c 81/3 -2
~ln

n = 1[ J;a~l2

.
~11
1+CI.~l2
11

6

n

1 ~lnl(g, Un)!

N

2

n=l
Tli do g E Range (AA)
=> AgE

Nhu' vh

RangeA

n

1

6
f'n(g,unJI
1

1

.'

2

Cho 0 -+ 0 ta co L C2~l6 (g, u )

+

2

00

.

2

-2

n

~M

\iN

1

(Xem 0.7.5 )

cae ke't qUll 3-2-7 va 3-2-8 kh~ng dinh ca'p dO tdt nha't cua st,I'hQi t\l (

t1'ongt1'u'ongh9P nhan khong SHYbien) la 8(02/3).
3-3. Nguyen It \Jhan ky :
.

.

Trang ph~n nay, ta gia sa gOIa dO'ki~n do d~c thoa Ilg- gOIlS 8 S IIg811 (1)


Trang do g E RangeA

D?l D( a,g8) = IIAX~- g8\1 trong do X~ = (1\+alylA"go.
3-3-1. BiBh Iv :
Giii sa g, gOlh6a (1), khi do ham s6 a --+D(a,gO)la ham lien t\le, tang va t6n t?i a
d€ D( a,gO)

=8

CIUl:nJ.?minh:

GQi {un, VII'~ln}la Singular System eua A

Taco'
xa
8

=>

= (A + aI t 1A *g 8

Ax~ _g8 =A(A+alt1A*g<>
2
00

= I

- a~l

(
n

_g8 =A(A+alt1A*g8

_g8

8

)

n2 g8,Un un -Pg

n = 11 + a~l

(j day P la phep ehie'u tn,iegiao clla y len (RangeA)1Khi do:

a~~

D(a,g8)2

I ( l+a~2 n J \go,ui

=IIAX~_g8112 = n=1

D~ng thue tren chung t6 a --+D(a,gO) lien tve va tang do

y r~ng

L<;lichli

Jim
a--+oo

+ IlrgO112
a~2
n tang.
1+ a~ 2n

:
D(a,g8)2

=11(I-p)g8112 +llpg8112 =llg8112>82
.

Cling tu g E Range(A) va PIa phep chie'u tnjc giao lIen (RangeA)1-

aJ:oo D(a,g8)=llpg811ma a --+D(a,gO) lien We Hen t6n t<;lia d€ D(a,gO)
Dinh ly tIen cho ke't qua la t6n t<;lia d€ D(a,gO)
stj t6n t<;lia thch

D( a,gO)

.

=8

=o va do a

--+ D(a;gO) la tang nen

= o la duy nha't.

Bay giGc1?tr(a,go):= g8,-Ax~
=> D( a,go)=llr(a,gO)1I

(2)


va
A \{ a,gO) = A *(gO - AX~)

= A *gO

=A*go - A(A + aI )-1
~

~

*

- A *AX~

A g

o

=(I-1\(A+alt1)A*gO.
A*{ a,gO) = a(1\ +alt

Khi do sai s6 giua u thoa Au =g ( chli

lilt - x ~

r=

+ IIX ~ 112 -

!lu112

(3)

1A*gO = ax~
Y

g E RangeA) vdi xO
a :

2(u,x~ )

ma do (3), ta co : x~ = ~ A *{ a,gO) nen
Ilu-x~112 =11uiI2+llx~112 -

~( A*{a,gO),u)

=llul12+llx~112- ~({a,gO),g)
= !lu112+ Ilx~ 112 -

= lilt
Ilu-x~r

f+

Ilx

~ ( {a, gO), g - gO+ gO)

~ 112 + ~ ( {a, gO), gO - g)

~lluf+llx~112 - ~({ex,gO),gO)+

-

~ ({ex, gO), gO)

~D(ex,gO)

Tu' day v6 san ta d~t :
E( a,gO)=lluf

+llx~112-

~({ a,gO),gO)+~ D(a,gO)..

3-2-2. Dinh IS":

=o

. Ne'u g, gOthoa (1), khi do E(a, gO)la be nha't khi va chi khi D(a, gO)
Clutl1g

Chli

111[1111:

y r~ng

D(a, gO)> 0 'l/a > 0, m~t khac :

(a,gO)=AX~

-gO =0 chomQta>O

Nhu'ngbdi (2), ta co : xO
ex = (1\+alt 1A*gO = 0
Do do gOE Ker (A*)

=> 02

=(RangeA)J.; tu g E RangeA

= jig o - gr = jigo1/2

+ Ilgf > Ilgf + 02

Trang43


Man thnfin.
d
Ta cling tinh dl(QC-E(a,gO)
da
Thlias6
Do ham

1-0
= 2(

II(A+aI)-l/2r(a,gO)1I2

0 )
D(a,g )

a

11(1\+ aI)-1/2 r(a,gO)1I2la 1uan dudng
a ~ D(a,gO)

tang Hen

dd E (a, gO) < 0
a
dd E(a,gO) >0
a

voi

D(a,gO) < 0

voi

D(a,gO) > 0

.

Do d6 E(a,gO) d~t min khi va chi khi D(a,go) = 0
Bay giG ta thie't l~p nghiem chinh h6a ki€u discrepancy

3-3-3.Binh 19 :
Ne'u g va gOthoa IlgO- gll ~ 0 ~ IlgO11va a

Khi d6

x~«S) ~

u

= a(o)

thoa D(a(O);go)

=0

Khi 0 ~ 0

Clu~ngminh.. Tli x~(O) la etfe ti€u CURphie'm ham Tikhonov
Fa(o)(o;go) tae6:
"

Ilr(a(o)ogOf +

+~(ot = Fa(o) (X~(O» ,; Fa(o) (n)
=llg-gOI12 +alluf
~ 02 + allul12

Nhu'ng dollr(a(o),go)11= D(a(O);gO) = 0

=> +~(ol

+ 02

S

02 +

"Ilof

=> Ilx~(o)11

S 11011 \16>0

Do d6 m6i day (on)h0i t~lve 0 t6n t~i day con ta v~n ky hi~u la (on)sao cho :
0
Xn

w ~y,y EX
a (0 n )

(4)


0
(xem 0.9)

L,;lido A la compact nen Axa(o n ) hQit\l m:;tnhve Ay
L:;ti c6 :

0

0

D(a(On),gon) = IIAXa(8n) - g nil = on --+ 0

va

g 0n

--+ g (do IIg8 n - gl,

V~lY Ay

0
Tli

s 8n)

=g
*

*

x a(o n ) E Range (A ) ta SHYra y E Range(A

~
)

= (KerA)

d d6 y la nghi~m c6 chuiin be nha't , tuc y = u
0
Nhu'v~y m6i day con Clla

(xa(o n » chua day con hQit\l ytu ve u va tu d6

w

On

~u

n --+00

xa(o n)
Ngoai ra ham chuiin la mia lien t\ICdu'oiye'u nen :
0

0

lIullslim inf IlxaCon ) slim sup xa(o n )

(xem 0.11)

s Ilull

0
Do d6:

Ilxa(o n )11--+IJuII

x la khong gian Hibert
0
Ta da chung minh dU9C

n

xa(o n )

W

n--+oo

.

) U

II 0

xa(o n )11 --+ Ilull

n--+oo

0
V~y xa(o n ) hQi tlJm:;tnhve u khi n -> 00 (xem 0.11)
Va do d6 x~(O) -> u khi 0 -> 0

.

Djnh ly du'oi c1aycho mQt ch~n tren cua a(o) khi chQntheo ki€u discrepancy.
3-3-4. Dinh Ii..; Ntu a(o) thoa D(a(o),gO)= 0

Thl a(o) s 01lAI12
1(llg811-0)
Clu~ng mirth "

Trang 45


Taco:
IlgOII-O=llgOII-llr(a(o),gO)II~

IlgO-r(a(o),gO)11

=>llgOII- 0 ~ IlgO- (gO - Ax~(o))11

==IIAX~(o)11

L?i do (3) ta co :
a(O)IIAx~(o)11 = \lAA* r(a(o),gO)\I

=> IIAX~(o)11~ IIAf Ilr(a(o),gOll/a(o)
=> IlgO - 0 ~ IIAf Ilr(a(o),gOll/a(O)
=> IIgO - 0 ~ IIAf o/a(o)

.

=> dieu ph~li chung minh.

Bay giOta xet de'n t6c dQhQit\l. Ta gicisu nghi~m co ChUaHnha nha't u E Range(A*)
3-3-5. niHil Iv :

No'lIlI E Range A * ; khi d611n
Clurng minh .. do U E Range(A*),

- x~(o)11=8(15)
gici su u=A *w

Taco'

H(8)

-

f +~(8t

- 2(X~(8)'uHluI12

~ 211uf - 2(X~(o)' u)

IIX~(8)-

tf s

=>IIX~(i;)

-

2(U-X~(8)'U)

f

~ 2(g

~

(do(4))
2(U-X~(8)'A

~ 2(A(u - x~(O)), w)
-

~

2(g

-

'w)

AX~(6)' w)

gO,w) + 2(gO - AX~(O)' w)

~ 4ollwll

Ne'u A hUll h~ll1chien ta co th€ thu du'<;Ict6c dQ 8(0) .
Th~t v~y trong tru'ong h<;lpnay, A+ la bi ch?n (do 2.7.3)

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