A method of regularization for operator equations of the first kind

32

A. L. Gaponenko and Yu. L. Gaponenko

8.

LADYZHENSKAYA, 0. A., The method of finite differences in the theory of partial differential equations. Usp. mat. Nauk, 12, No. 5 (77), 123-148, 1957.

9.

BIRMAN, M. Sh., and SKVORTSOV, G. E., On the quadratic summability of highest derivatives, Izv. YZJZOY Ser. m&em., No. 5, 12-21, 1962.

10. GNUNI, V. Ts., On the equations of flexible plates and shells, Tr. VIII Vses. konf: po teorii obolochek i plastinok, pp. 186-189, Nauka, Moscow, 1970. 11. AGMON, S., On the eigenfunctions and on the eigenvalue of general elliptic boundary value problems, Communs. Pure Appl. Math., 15, 119-147, 1962. 12. HARTMAN, P., Ordinary differential equations, Wiley 1964.

A METHOD OF REGULARIZATION FOR OPERATOR EQUATIONS OF THE FIRST KIND* A. L. GAPONENKO and YU. L. GAPONENKO

Moscow (Received 7 October 1974)

A METHOD of regularization, representing a modification of Ritz’s method for an ill-posed equation, is described and proved for an operator equation of the first hind. The convergence of the method is considered when the initial data are specified approximately. Operator equations of the first hind represent an important class of ill-posed problems. The problem of finding the approximate solution of an ill-posed equation can be solved by means of a regularization method. In the light of this, it is undoubtedly of interest to construct and prove regularization algorithms (see e.g., [l-6] ). In the present paper we investigate a method of regularization which may be regarded as a modification of Ritz’s method for the case of an ill-posed equation. The method enables us to find the approximate values of the Fourier coefficients of the exact solution.

1. The problem Problem I. We consider the equation of the first kind Au=f.

Let {B, g} be the approximate values of the operator A and the element frespectively: Ilf-glK6.

IIA-BllGh,

Given the pair {B, g} it is required to construct the element $h such that

lIubh-ulI+O The

as

following assumptions will be needed:

Assumption 1. The operator A is linear and self-adjoint, and D (A) =H, His a separable Hilbert space, and Ker A =0, f=R (A). Assumption

R(A)

2. For the exact solution of the problem I, it is known a priori that

*Zh. vFchi.sl.Mat. mat. Fiz., 16, 3,577-584,

1976.

=H, where

33

Regularizationfor operator equations

2. Method of solution when the element f is specified approximately 1. Expansion ofthe solution with respect to a special basis. Given the pair {A, g}, Ilg-fll<& let Assumption 1 be satisfied. Consider an arbitrary basis. {et} in the space H. In this case the sequence {Aei} will be a complete sequence of linearly independent elements in H. Using the orthogonalization process, we construct from the elements {kei} an orthonormalized basis {&} : i-1 --

& -

llzill ’

oi=Aei-

;I: k-i

(Ae{, dJ&.

We introduce the notation ~i=‘(ei-~tAe,,d,,*). IIOill

A-f

Then, for elements of the basis {di} we have &=A$<. problem I as follows:

We define an approximate solution of the

where cfb=sign (g, \li) max { 1(g, $) 1-Sll$Jl, 0). We denote by PN the operator of projection onto the subspace induced by the elements dl , . . . , dN. Theorem 1

Let Assumption 1 be satisfied. Then, for the approximate solution UN6 we have: 1) IIuNb-PdzII+O as 6+0; 2) IIu,“--ull+O as’ ho, iv-y 3) h.h+pa- UII~IIzzNa--UII, p=l, 4) Iha- ull”~llull”-ll~N’lI”.

2,. . . ;

Proof: For the exact solution, we write the expansion with respect to the basis {d{} : 0 U=

c

Cid<.

t-1

For the coefficients ci we have

whence

1Ci-_(g, $1) I= 1 I ~Sll$ill. The coefficients ci thus have the properties

I G-C? I ~2611$tll,

1CtaI G I&I,

sign cja=sign ci,

if

CiafO

Assertions (l)-(3) of Theorem 1 follow from these properties. Recalling that the basis {&} , is orthonormalized, we have

34

A. L. Gaponenko and Yu. L. Gaponenko

*=i

i=N+l

I=,

Theorem 1 is proved. 2. Expansion of the solution with respect to a given basis. Given the pair {A, g}, IIf-gilG6, let Assumptions 1 and 2 be satisfied. We consider an arbitrary orthonormalized basis {ei} in the space H. We denote by J/i” an arbitrary element of H, for which llA$re-eill
be _

UN

I2

Cibeei,

i-1

I =max{l (g, $t”) I-6ll$i”ll--ER, O}sign(g, $i’). C!

We denote by PN the operator of projection onto the subspace induced by the elements et,

. . . , eN.

Theorem 2

Let Assumptions 1 and 2 be satisfied. Then, for the approximate solution Uj$e we have: as y& 6Il$tll+o, i=l, 2,. . . N; -PNul(+o I) IIuNae as e-+0, Sll$i”ll-0, i=l, 2,.. . , N, 2) /UN8 -ull-Gl

3) 4)

IIu~+a+p-uII~IIuNdL-ull,

p=i,

N-m;

2,. . . ;

IIuNas-~~~2~l~u~~2-IIUNee~~2.

hoof. For the exact solution of problem I, we write the expansion with respect to the basis {er} :

For the coefficient ci we have ci=
eJ=
et>-(u,

A$i”>f(u,

A$t)-(g,

$i’)+(g,

9:),

whence

1cr- (g, $3 I= 1(u, e-- A$“)+(f-g,

$9 ~~eR+Gll~i*ll.

Hence the coefficients qsf have the properties [G-C?

]G2(611$?lI+eR), )CPIGIG(, cp*+o. if sign cP”=sign ci, Assertions (l)-(4)

of Theorem 2 follow from these properties. The theorem is proved.

35

Regulariaztionfor operator equations

3. Method of solution when the element f and operator A are approximately specified 1. Expansion of the solution with respect to a specialized basis. Given the pair {B, g} , IIA-Bll
The sequence {Aei} consists of linearly independent elements and is complete in space H. The sequence {Bei} can contain linearly dependent elements, and may not be complete in space H. Denote the maximum number of linearly independent elements in the sequence {Bei} by N(B, h). We put N(h)=

min

{N(B,h)}.

B:IIB-AII
Since l(Bei-AerJIBh, we have N(h)++ 00 as h-to? and the sequence (Be0 will be complete in the limit in Has h + 0. On choosing the first N linearly independent elements from the system {Bei} !nd orthogonalizing them, we obtain the orthonormalized elements

i-i hr=Bei-

C (Bei, bk) bk. k=i

We introduce the notation i--1 cpi=l II&II

h=l

We compare the elements bi and & with the elements di, $i defined in Section 2. We introduce the notation

A.dr=d<-bi,

A$i=$-qi,

i=l, 2,. . . , N,

i=l, 2,. . . , 8,

t--i yi (h) =ih +

i=l, 2y.--,N-

(I+l(Bei,b,)I)IIAd~II,

z

h-1 Lemma The following a posteriori estimates hold for the elements

$aresin&),

II&II

G

’ llhill (IlAiIl-r*)

i=d,

2, . . . , M,

I

{ llhtll E kp~ll @+llA4Al)+A~~l k=i

i-i

+~i(Iei-~(Bei, h)~~ll}, k=,

Adi, A+ :

i=l,2

,...‘,M,

36

A. L. Gaponenkoand Yu. L. Gaponenko

where M is an arbitrary integer, M < N, satisfying the condition (YM/I~~LM~~<~. The proof follows immediately from the orthogonalization expressions. We define the approximate solution of problem I as N u.v

Ci6hbi,

611 _ ;r:

where

1(g, Cpi)I-

cibh=max{

(M?+6)

Ilqkll,0) sign (g, CFJ.

We denote by PN the operator of projection onto the subspace induced by the elements dl , . . . , dN. 7Ireorem 3 Let assumptions 1 and 2 be satisfied. Then the following hold for the approximate solution UN

bh

. ’ 1)

IIUNbh --PNull-+o

a

2) )~zz.~~~--uII-+O as

3) 4)

6-d,

d-+0,

Ilu~+h,,--ull~llUNah-UII, bNbh-

N-w; 2,. . . ;

UII~‘~U~/2--IIUN~h~~2.

&)=(u,

=(&-Au,

h-0, p=i,

bi). We have, for the coefficient c/I:

Roofi We introduce the notation Cih =(u, C;=(~,

h-+0;

cpi>

Bqd=@u>

cp,)f(Au--f,

fpi)+
%)+(gT (d’~

whence

1ci”-(g, cpi)1G (hf?+d) Ikpill. Hence the coefficients Ci6h have the properties 1C,ah-

CihI~2(hR+6)2Il@Il,

sign ci”=sign

Cih,

ICibhIQihl,

CibhZO.

if

Assertions (3) and (4) of the theorem follow from these properties. For the exact solution of problem I, we write the expansion with respect to the basis {dt}: I u=

For the element Cidi we can write

c

Cidi.

i-i

C&= (cidi-C
31

Regukarizationfor operator equations

In view of the lemma, it follows from the last inequality that IICiLEi-Cibhbill--tO as h + 0,6 + 0, whence we have 11~~‘-Pdl+O ~ as h-+0, 6+0. Since (uNbh-u, c&)+0 as h+O, 6+0 and in addition, IluN”“ll~llull, we have Ilun6”-ull+O as&-O, h+-0, N+m.The theorem is proved. 2. Expansion of the solution with respect to a given basis. Given the pair {B, g}, Ilg-f liG6, where B is a linear bounded self-adjoint operator; let assumptions 1 and 2 be satisfied. We consider an arbitrary orthonormalized basis {ei} in the space H. Let p=p (h) , O
p(h) --too as

P(l) =O,

hp( h) +O

h+O,

as h+O.

We consider the following auxiliary problem I’: given E>O, h>O and a fured element e=H to find the element @e, satisfying the conditions IIB$he-ell=GE, ‘Il$““ll
If the parameter h is sufficiently small, the problem I’has a solution. For, introducing the notation

we find that, for any h, OCk 0 and any integer N> 1, there exists h=h (E, IV), such that, for any h, O
i=l,

2,. . . ,N,

ll$:ellGp(h),

i=l,

2,. ..,N.

Notice that the elements 9:” can be found as e-approximations with respect to the functional for the corresponding variational problem. We define the approximate solution of problem I as N

bhs UN =

c

bhs

Ci

pi,

f-1

where Ci“‘=sign

(g, grh”>max{ I (g, $9

I-

(hRS6)

II$Pi”ll-d?,

0).

We denote by PN the operator of projection onto the subspace induced by the elements e,, . . . , ea. 77reorem 4 Let assumptions 1 and 2 be satisfied. Then, the following hold for the approximate solution G2

1)

ll~~~~~-J’N~ll-tO

as

2)

IIw~~~-~II-+O

as

3)

IIUN”h+ep-UIMlU~-Ull,

4)

IIUN*ha-Ul12~llul12-llU~l12.

h-+0, h-+0,

p=l,

6p(h)+O, 6p(h)-+O,

e-+0; e-+0,

N+-;

2,. . . ;

hoof: For the exact solution of problem I we write the expansion with respect to the basis {ei}

aS

IE =

z 5-l

CtPi.

A. L. Gaponenko and Yu. L. Gaponenko

38

For the coefficient ci we have cI=(u, eJ=(u, Ipiha)y whence jci-( jGR&+(hR+G)

eJ -_(u, B$$“>+(u, BqiV-_(g, 114ihell~R&+(hR+G)p(h).

$ih”)+(g,

The coefficients c phc thus have the properties tci-~:~i<2R~+2(hR+6)p(h), bhs

sign ci Assertions (l)-(4)

= sign Ci,

CPRIso.

if

follow from these properties. The theorem is proved.

Note 1. The conditions on the operator A, stated in assumption 1, can be weakened. The two methods of solution can be extended without serious modifications to the case when: Assumption?. A is a linear bounded operator, D(A) is dense in H, R(A)sH, separable Hilbert space, and j=R(A) .

where H is

Note 2. If we know that u=B, where B is Banach space, Bd?, then the approximate solution in the metric of B can be obtained in certain cases by means of a “smoothing” operator (see e.g., [7-91).

4. On improvement of the approximate solution When supplementary information is available about the problem I, it is natural to ask whether the approximate solution might be improved. Suppose that several different approximate solutions ul,. . . , b, have been obtained by one of the above methods: N Vk =

criai,

k=l,2

,...,

c

i=i

M.

We introduce the “averaging” operator uM=S( {uk}, l
= 2 ciac, i=l

where ci is the coefficient of maximum modulus of the set {cii, . . . , CMi). For the constructed element uM we have IluM-ull
k=l,

2, . . . . we use

N

b-

V Nk -

c

Cktbdiy k=i,2,...

.

<=I

We apply to the resulting approximate solutions vNk6 the “averaging” operator uNMa=s ( {uNkB), lGk
39

Regubrization for operator equations

lleorem 5 Let assumption 1 be satisfied, and let the sequence (gk} be dense in the sphere Sp [I, 6]= {g: Ilg-fll G6). Then: as M---; 1) lIUNM6--P,Ull-+O ull+O as M-tc=, N-m. 2) II&+IM6Proof. We consider the sequence of elements {&} : gk=f +

sign
ll$kll

’‘k’

k=l,

2,. . . .

For these elements we have &=Sp[fr I(&,,

61,

$k)(=j(u,

k=l,

2,.

. . ,

&)‘1+6~~~kll,

k=l,

2,..

. .

When the coefficient ckk6 is constructed with the aid of the element &, we get cUa=(u, 4.). The theorem thus holds for the sequence {&} Since the sequence {gk} is dense in the sphere Sp V; S] , the theorem also holds for this. The theorem is proved. In conclusion, the authors thank the seminar participants V. A. Vinokurov, A. V. Con&r&ii, and A. G. Yagola, for useful discussions and valuable criticism. Translated by

D. E. Brown

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TIKHONOV, A. N., 153.

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TIKHONOV, A. N., IVANOV, V. K., and LAVRENT’EV, M. hf., Ill-posed problems, in collection: Partial differential equations (Differents. ur-niya s chastnymi proizvodnymi), pp. 224-238, Nauka, Moscow, 1970.

3.

IVANOV, V. K., On the approximatesolutionof operator equationsof the first kind, Zh. vjchisl. Mat. mat. Fiz., 6, No. 6, 1089-1094, 1966.

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MOROZOV, V. A., Methods for solving unstable problems (text of lectures) (Metody resheniya neustoichi , kh zadach (teksty lektsii), Izd-vo MGU, Moscow, 1967.

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BAKUSHINSKII, A. B., Selected topics in the approximate solution of ill-posedproblems (text of lectures) (Izbrannye voprosy priblizhennogo rbsheniya nekorrektnykh zadach (teksty lektsii)), Izd-vo MGU, Moscow, 1968.

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STRAKHOV, V. N., On a method for the approximate solution of linear ill-posed problems, Zh. vychisl. Mat. mat. Fiz., 10, No. 1, 204-210,197O.

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BUDAK, B. M., BERKOVICH, E. M., and GAPONENKO, Yu. L., On the construction of a strongly convergent minimizing sequence for a continuous convex functional, Zh. vjchisl. Mat. mat. Fiz., 9, No. 2,286-299, 1969.

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BAKUSHINSKII, A. B., Some problems in theory of regularizing algorithms, in: Collection of papers of the VTs MGU, No. 12,56-79,1969.

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VASIN, V. V., On the stable computation 13, No. 6,1383-1389,1973.

of the derivative in space Cc- 00, + m), Zh. vjjchisl.Mat. mat. Fiz.,