On the stability of the discrepancy method for solving operator equations of the first kind with perturbed operators

141

14

KRAWCZYK R., Newton-Algcrithmec zur Bestimmung von Nullstellcn mit Fchlcrschranken, Computing, 4, 187-29?, 1969. HANSEN E. and SENGUPTA S., Bounding solutions of systems of equations using interval analysis, BIT, 21, 2, 203-211, 1981. MOORE R. , A test for existence of sclutions tc non-linear systems, SIAM J. Numer. Anal., 14, 611-615, 1977.

15 16

Translated

:~U.S.S.R.Comput.Harhs.~~th.Fhys.,vo1.29,~-.5,pp.lal-147,19B9 Printed in Great Britain

by

Z.L.

0041-5553/89 $10.00+0.00 01991 Pergamon Press plc

ON THE STABILITY OF THE DISCREPANCY METHOD FOR SOLVING OPERATOR EQUATIONS OF THE FIRST KIND WITH PERTURBED OPERATORS" N.V. KINH operator equations The stability of the discrepancy method for solving of the firs: kind with approximate operators is established assuming weak continuity of the operators in locally convex topological spaces. The eqcivalenc e cf certain. generalized variational problems in these spaces is alsc shcwr, The 8-stabiilty of the discrepaccy method for solving operator equations of the first kind with approximative operators being continuous (or se!?:-continuous) or closed (or preclosed), has been investigated by many authors (see i;-101:. In this paper, we investigate prcblens, ir, which given operators are weaKly co:tinuous (see Section 11, and we shall also investigate the equivalence sf varlaticna: prcklems called 1, 2 (see Section, 2). 1.

1.

in

which

vector fying

Definition of B-stability the operator equatisr. We consider

X is

a separated

lczaily

space correspz-ding to the fc;:owing cor.Ziticn:

car.-ex

the

cf

:>e

vec:cr

metr:c

g being

a fixed

real-valued

function.

g(O) -0.

A acting

The operator For and

c-4,. ~0)

which to

the

is

operator

Y, and moreover

from

X intc

not

defined

Ak acting (A,. y,)

exactiy

defir.ed

but

is

the

number

T is

fixed

fcr

;~:a:::;

dis;lace-ent,

co:Ivex f,rther,

satis-

l!

such

only

that

cz X, of

is

defined

following

weak;y

the

continuous.

solutions

of

approximately

on X,

is

weakly

Eq.

(1)

is

nonempty

ty

(A,. yt). in continuous, ye belongs

condlrions:

P(1/1. YG)Gb

Y-h. 620.

q,(h. 6)-O

as h. 6-O.

and

q,(k, 6) is a nonnegative real-valued of operators defined by

*~Zh.UyChiSi!.Hat.mot.Fiz.

a metriza:le

y‘i.= [O. 11.

Y and defined the

d,(Ahr A,!Gh. r’h+b
ir. [C,

X‘r consisting

from X into satisfies

!’ is

invar;a:le

and g(i.)-g(l-i.)Cl

Y are that

k:r,d:

Yi.=[O.11. Yy=>..

increasing

g(lr=l

we assume

(A<. y,).

sl;ace,

P be:ng

p(i.y. Oi
firs:

fznctioz

,29,10,1458-1465,1989

cf

the

variables

h,

6.

and dM is

a quasimetric

142

dw(A, Ao)- sup n is a stable functional, following cond:tions:

i.e.

it

‘($;:J) ,

is a functional

n(z)>0

e0,

Ycx,

defined

on dom R, satisfying

the

(2)

YzEdom R,

XJdom W0,

(3)

R,-(r=domR:R(t)
(4)

is 8 compact set in X yc>O First let us put M O&b)

=

&,e,=

if E x :P L&r, ye) f M,,(h, s),qdom L2,

p1

(h, 6)),

Nov we introduce Definition

the following definition of B-stability. 1. The discrepancy method is called B-stability for Eq. (1) if the net of h, 6-O (see 191). sets X, (h, 6) B-converges to .&ndomR as 2. The foliowing lemmas are used to establish B-stability. Lemm 1. If the net of sets (kfO.oelr) &converges to the set M then: (see [91). a) each of ltssubnets converges to M; and A’=,!! then the net (m,. a=X) B-converges to N. b) if g+m.<.l!,. 0=x If the net of sets Lemna 2. (see [91j. {M,. oeI). M,cX. being a separated topologiof the net each subnet {If) cal space, satisfies the condition: {I$). zo=.~f,* holds at least one of the properties:

where LT.)’ is the set (W.. 0=X) B-converges Lem

3.

(see

(x’) =dom I!

sequence

of liniting tc [9!J.

pcints

of the

kc1

set

and

M=X‘.

then

the

net

#. 7T.e stable

functicnai

R

is

lcwrr

seri-ccnrin;ous

i.e.

eaci

suck. that lim I,===. n-m

ondl1nlnnf G(1,)=b
and R(r) G w:tk centre at :/ in the mctrizable locally convex space (y. P) definec in Ii), is s ciosed convex set in Y. Proof. We will first s.how that B(y rl is ccnvex. Since the metric p is invariable for displacement, it is sufficient to show that the ciosed sphere Bi@,r) of radius r with centre 0 is a closed convex set. Let arbitrary elements Y,. Y:EBt(7. r). hence then

there

Lenm

pi!< We

@)Cr.

p,y:.

O)cr

have U)Cp(i.y,+(l-i.)yr.

(l-i.)Yl)+ O)C Cg(%)p(y,. O)+g(l-i.)p(y2,O)Cr(g(~-)-

p(i.y,-

tp(

(I-i.)y:.

(l-L)yr.

O)Cp(Ly,, o)+p((l-A)YI,

tg(l-A))Gr hy,+ (1->.)y+R(O. r). Thus the sphere B(O,r) is convex. the sphere B(O,r, is closed. Lemna 6. If x is a separated locally convex vector space then the weak convergence the nat in X is unique. Proof. We assume that there exists a net {f.)=X, that converges to either of z,, +,=X. z,+z:. Since z,-&+o and x is a separated localiy convex space, there exists a linear COntlnuOUS funCtlOna1 defined on x such that I(z,-z~)+@. Therefore we have f@J+f(&): consequently Clearly,

of

143

may converge to at most a point This is not true, because the net (f(r,)) We have proved the lenma. Thus z,=z,. then the set X.,(h, 6) LenaM 6. If AkV h r 0, is weakly continuous Q, XJ~dom Q+S, Since By property (3) cf the functional PrOOf.

exists

zr. z,=Xp*‘dom R. such

an element

that

Q(Z,)
p(A.r,. y,Jip(Az‘. A&)fp(Ad~. (R’(ro)h~b
and Q(r)

w=X,.

is

lover

mh,GQ(.r:) If m,,-R(r,). If m,,
then

2, belongs

then,

there

exists

R(z.)GI!(zr)

nonempty.

for

such

10, we obtain

yb)C

Therefore z&J,, (h.0. bounded, there exists

m,.b = Clearly,

is

to X.

inf Q(z), ~ww= 0

T>

there

belonging

inf P(I). ~E~,QL.b,

to X.,(8, c,. a sequence

(zI)~DllGh,(,

Yn and R(z.)-mrs

such

that: (5)

as n-m.

exists that z.C=Q, Yn. c=G (zp). Since the set R. is compact, there Frorr (5,, it follows En,. a subsequence of i2n) such that converges to and without loss of generality, we may assume that the sequence (2,) itself converges to f. By Lemma 4, the sphere B(y, IJ,(/~, 6)) is closed convex in Y, therefore, it is alsc weakly closed ill]. We obtain A,.-‘[B(yr.~.(h.6!)l-M,;,,,, since

.KT,h,,, A h r 0, is weakly CO%t:T:JC’JS, SLC!. 2 Se: is weakly closed h’ Or. the other hand, we have and therefcre, G-1, I>, f & .b < .vc,,, .bi Fro- (5) and Lemma 3, it follows that JEJom!1 Thus

=D,

in X. r=Al,,

L

l(+_e,.

(61

By !5) we obtain (7)

mAs- lim inl Q(G) - lim R (I,) >R (I), .-. . ..I

it follows that R(r;=m,,,. X=X., 0.. L]. Thus the set X,,,,,, +g. Hence the lemma. Now we establish S-stability. Theorem 1. Under the hypothesis of Lemma 6, the sequence X.,p.o B-converges to when h, 6-O. X,fidom R Proof. By Lemma 6, .X,rl,+b,*~. Arbitrarily choosing z,,EX.,~~,~,, we establish the net {s,, h, 6-O). Similar to proof such that of Lemma 6, there exists an element z@X,ndomQ consequently to=D.,,n,e,t’b, h; +-a., c-R (20). From (61, (71, We have proved

Since Qo is compact, there exists a subnet of the net such {II, I h, e-01 verges to and without loss of generality, it can be assumed that the r@sn,. converges to Z. Now we show that 5 belongs to X$dom R. We have

that net

it

con-

itself

~(A,~,~,~~~~p(A~~.,.~~~,;)tp(~~~.,~~~)+p(y~.y~)c

(8)

~n’(z,,)h+-~.(h,6)+6C~‘h+e,(h,6)+6~2~,(h.6). The right-hand

side

of

(6’

converges

to zero

as

h, 6-0,

hence

AG,,*~o

as

h, 6-0,

con-

144

scqucntly

We obtain

At+,, -w yo.

G,,::i, From Lemma 5 and (91,

It

follows

A&:

yo.

(9)

that AS-y,.

f=dom R ; consequently On the other hand Thus we obtain fExo. By Lemma 2, the net {X.l,h. L,, h, 6-O) B-converges to X,ndom R as proved the theorem. 2. Equivalence of variational problems 1, 2 In this section, we shall establish the equivalence of variational retain the hypothesis of X, Y, (A*, pc), R, as in preceding section. Variational DrOblem 1: r-min Variational

Droblem

:~(AG,

X*,-argmin {Q(t) : p(A&, #JCR’(r)h+6)*0). Variational problems 1, 2 are calied identical

hence

ye& Put

If assume

We

y.=B., the

and A is continuous,

solvable

and the contrary,

sequence which

VA,

they

are

if

its

then

set

these

consisting

6>0

solvable

and these

sets

Yh, 6>0.

The following lemma is obvious. and T.-r as Lenmu 7. If O-Q.+,Cr. Yn, a sphere of radius r with centre at ys in Y, then n

We

if

equlva:ent

X,,-X,r

Lenvna 8.

1, 2.

gJCR’(z)h+d).

(Q(r) : p(A,,.r, ~,)Cr’h+b)}+I

Xhe=arg (r-min

Proof.

problems

have

2:

If X is an &-space and Y is a Banach space Remark 1. variational problems are equivalent (see [2]). First we introduce the following deflnltion. Definition 2. Variational problem 1 (or 2) is called of solutions is nonempty, i.e.

are

We

(Q(z) : p(A.+r, I/~)Gr’h+6).

min {R(I)

(or

rrXJldom Q h, 6-O.

lyni means,

n-a: B,=B,,,

and Vn

&-{yeY: and

P(g,Yo)
weakly converges to Y~Y there exists a no such that

is

then y=B. Vn. yfB,.

VnTn,. consequently

&‘-B.-y‘

B.‘={L.EY:p(L..O)~r.). The set g,’ is closed, convex Y (Lemma 41, further, it is also a balanced set (see and then - v=B.’ ?.B.‘cB.‘, OChC I. Therefore, [ll]), indeed, clearly, if U-B.’ it is balanced. The set g,’ is absolutely closed convex and Therefore by Hany-y&B. Vn>n,. Banach’s theorem (see IllI), there exists a linear continuous functional defined on X such that: !(y-yO>i

and

VvrB,’

ll(v)l41

consequently !(Y-Y~)~~~~(Y.-Y~)

Vn>n,,

I(Y) rl+l(Yo)

>f(v.)

Vn>n,. It

follows

theorem

that

g.gY.

contrary

to hypothesis.

Thus

y=B. \‘,I.

We have proved

the

I The following Theorem 2.

If

theorem asserts the solvability of variational problem 2. the stable functional R satisfies the condition: each {z.)cdomR,

&-I

145

liminf Q(q)-b<= “.-.

and

Proof. If

Put

imply M*a=(zEX:p(A~,

sp=XJIdOm

Da-M,&

further

such

problem

and

~a)
I/J
~.ED+.~+B.

A&,)+p(Acr,.

2 is

Du-MJdom

solvable.

0.

r,&,,,

it

fc:lows

If

n&-Q(zr)

then

I

If

m6RM

then

there

y,)GR’(r,)h%

Pur inf Q(r). -DII

mrbFrom

variational

then

Q

P(AG, Thus

then

Q(r)=+

that

belongs

G
to

exists

X,,=z such

(I.) =DM

a sequence

O
that: (10)

Yn,

lim Q(G) =nitip. m_ D

ges

to

From

(lC),

Since

R,

it is :t

zEQ,.

can

of generality.

loss

Fror

the

On the

that

&.=R.

there

exists

follows

compa:t, Therefcre

it

a subsequence the

From

(lo)-(12:

We

have

implies

to

XI

Since to

Q, f.

converges

is

compact,

As in ? as

to

and

to

p(y.ya)Gr,.

{znlsuchthatitoonvar-

I as

IER,.

to

without

n-m. that

y,=A,r.-A,.z

the

there

proof

cf

of

that

reM,,

tbe

set

ci

?:ecrei-

2,

tte

X .rZ

net

EC!. c=Oiz

exists

Therefcre

zel!,,‘don,

R=D.,

approximate nef

sclutions

cf

equaticn

(1).

B-converges

{X,6, h, 6-O)

to

F’ii. t,>?. tr.c

(I.8

ii. b-0).

Ctvicusly.

Z:E ,Y .dc,n. !!

j

a scbnet i,

Theorerr

!1 21

theore-.

called

I

T,-R’(I.,)h+d).

fc;lows the

estatllsr.

Therefcre

z,=D,t t’h. 6:Oo.

verges

sequence converges

A,. It fc?lows

c;eratcr

proved

SOL is alsc Remark 2. The set TheoreT 3. Under the typcrhese XJldom R as h, 6-O. Proof. By Thecrea 2, we have I,~ beiong

weakly

Lemmas 5, E, it

and

rEX,,+E.

Let

the

itself

we have

hand,

y.,=B,,=(y=)’

consequently

of

sequence

converges

t f the

ccntir,Gitp

weak other

Yr,. c-nix‘).

that

be assurred

(11)

cf

we can

the

assume

net

{I+,(. h. 6-O)

that

the

net

s:cr. itself,

that

it

con-

{I:.:. i.. C-O)

h, 6-O : --r a.c h b-c!

We shall From

show

that

fPX.‘domS!

Q(r,,)Go(r%).

it

fc;lows

tnat lim inf R (r,J h.L-O

From

(13),

On the

(14),

in

other

'i3,

addition

tc

lower

GR(z,). Of the

semi-continuity

+p(A,r.,.

of

side

9.

it

follows

O)GO'(z

yt!-.O!yt.

,jh-Q.cz,,)h-

(15)

converges

to zerc

as

h,

01

6-O.

Thus

ADZ.f---1/‘ as )r. E-O.

From (131, &E-go

proved

;

(161,

in addition

consequently

By Lemma 2, the theorem.

the

that

(151

I I. A,&!)-

+bc2[R'(r!h-6]~-l[o'~.r:)h-

right-hand

functional

we have

hand,

p(A,r.,. yocpr.4

the

(16,

5~X:‘dom

net

to

weak

(16) of

continuity

the

operator

4~.

it

follows

that

Q

{A’,,. h. 6-O)

B-converges

to

X,Odom R

as

h,

b-0.

We have

146

The following

theorem asserts the Under the hypothesis Variational problem 2 is Proof. (Theorem 31, B-converges to X,:dom R Therefore it is sufficient to show that We have ~+,:‘-R(z) Y~eXha. For variational problem 1, there Theorem

4.

kg is empty. Indeed, we assume the to this set. Therefore we have

equivalence of variational problems 1, 2. of Theorem 2, these problems are equivalent. solvable (Theorem 2), further, the net (X,,,h,640) of course, it also converges to X, (Lemma 1). X~I-X~,YA, 6. are

the

following

cases:

(T-min(R(r):p(&,

p&7%+6))

this

means there

contrary;

On the

other

hand,

ZU’ belonging

we have

This is nc,t true. Thus this T>T*r(, then the set

set

We as.sume tne ccntrary, we alsc cbta:r. a similar

is a:sc, empty, Therefore set. We have

yl)
i.e. there exists an element relation to (17).

{T:P@G. y,)~R’(z,~‘)h+6)=(r:p(Ah~. Clearly equal to

inf R(z). _%

is empty.

.4gr(7-min(R(r):p(A,t,

z&e’ belonging

to this

yajG(71r0)%+6).

(18)

on the set defined on the left-hand side of the minimum values of R(t) side of (18) contains the R(rht’) and the set define d on the right-hand

118)

z&X,+.

7=O:r.:‘)
Thus we have 7==7h:0. Let us artitrariiy

take

This &!EX.‘,.

is not Clearly,

true,

Conversely, R(r,+‘) =7nr‘ an:

let

us arbitrarily

choose

consequently

such z,,( alsc

fi,,‘-.irg{7=min(n(l):p(A,I,

this

beiongs

set

is empty,

to the

following

I&?’ belonging

to the

(201,

in addition

if set: (19)

ye)G~‘h+6)). set

(19);

hence,

we have

(20)

P(A.Z”,‘.Y,)G (Thrcj’h-6 from

set

(17)

7-Z!(11~‘)~7~d~-inf{R(t):p(Aht,yd)<~’(~)~+~)-

elements

an element

exists

the

rnc’=Dha.

conseque-tly

are

then

y~)
pu"Ihr'.

If

o<7c7u”.

if

tc 7*cp= min R(rj, *=D1C

it

follows that I.~‘EX,~. Thus XI~‘-X~~. From the above cases, it follows that X,a-X,t Vh. 6. Thus these problems are equivalent. Therefore, the net (X~lb.h. 6-O) We have proved the theorem. ges to X&dom 9. I would like to thank N.M. Chuong for suggesting this problem.

also

conver-

REFERENCES 1. 2. 3. 4.

Tha stability of the discrepancy method when solving ill-posed problems, TANANAV.P., Izv. VUZ. Matematika, 9, 75-60, 1974. TANANAV.P., A projection-iterative algorithm for solving ill-posed problems with an approximately specified operator, Zh. vychisl. Mat. mat. Fiz., 17, 1, 15-23, 1977. A projection-iterative algorithm for operator equations of the first TANANAV.P., Dokl. Akad. Nauk SSSR, 224, 5, 1028-1029, 1975. kind with a perturbed operator, LISKOVETS O.A., The solutlon of equations of the first kind with a closed operator, Dlf.. Urav. 19, 13, 1666-la?:, 1974.

147

9.

The solution of ill-posed problems with a closed operator, Dokl. LISKOVETS O.A., Akad. Nati SSSA, 219, 5, 1069-1071, 1974. The convergence of an approximate method of solving operator equations MOROZOV V.A., of the first kind, Zh. vychisl. Mat. mat. Fiz., 13, 1, 3-17, 1973. An approximate discrepancy method in non-reflexive spaces, Zh. VINCKUPOV V.A., vychisl. Mat. mat. Fiz., 12, 1, 207-212, 1972. A regularizing algorithm for ill-posed LEONOV A.S. and YAGOLA A.G., GONCHARSKII A.V., problems with an approximately specified operator, Zh. vychlsl. Mat. mat. Fiz., 12, 6, 1592-1594, 1972. Variational Methods of Solving Unstable Problems, Nauka 1 Tekhnika, LISKOVETS O.A.,

10.

Minsk, 1981. IVANOV V.K.,

5. 6. 7. 8.

and

its

VASIN V.V. and TANANA V.P.,

Appiications,

ROBERTSONA.P.

11.

Press,

Nauka,

Moscow,

The

Topological

and ROBERTS% W.,

Theory

of

vector

spaces.

Non-Linear

Problems

Cambridge

Univ.,

1964.

LI.S.S.R.Comput.Ma?hs.Math.Phys.,Vcl.2C,rJc.5,pp.147-157,1989 Printed

Ill-Posed

1976.

in

Great

0041-5553189 $10.00+0.00 C 1991 Pergamon Press plc

Eritain

MEAh SQUARE APPROXIMATION OF A RECTANGULAR MATRIX BY MATRICES OF LOWER RANK!' V.A. DAL’SAVETan2 F .V. YAKCVLEV The probiem defined in the title basis of its connection with the

of this paper is examined, on the decompcsitior, problem of Singular In particular, it is proved that the wellof the matrix in question. known group relaxation methcd is convergent given any initial aPpr:xirr,ation, and its rate of convergence is shown tc depend on the relation the correspcnding sir.galar numbers of the matrix. between

Introduction

1.

Let FiM, N] be an artitrarg The probiem of mean sq,uare

1141 / most

r

(the

set

of

all

rectang_;ar approximation

matrices

such

ilM,

matrix and r L 1 a fixed integer. cf F;ti,flj by matrices Z:K; R: ty F,) is as ?cllc*s. ?I':is dencted

r

c,(z~:-~~F-z112-

where product

Z is (RI = r. of twc

a member

of

Our problem matrices of

F,

if

and

or,ly

may therefore lower rank:

if

I.r,

it

be

I m:r.! I#:, ranr: at

(F[i,j]-Z[i.I])‘-min.

).YJ.W A matrix

r ci

be

expressed

reinterpreted

can

as

C,(X. Y) :-IIF-XY’II’

as

Z[M. .v]-X:&f.

follows:

R]x)“[R,

apprcximate

.v].

F by a

+ min.

11.2;

y, Here is

H, the

is

the

sum of

we have yet matrices of

set at

of most

another rank 1:

pairs

of

r matrices

matrices of

interpreta?ion

(X, rank

of

the

YI each

wit?.

r

columns.

Since

the

product

XYT

1,

originai

problem:

approximate

F by a sum of

7 &(T,...., T,):=UF-&Ii’,-I All Incidentally,

these

interpretations (1.2) is a discrete

of

min . (7, 1ImLti <‘I

the problem are useful version o f the problem

be utilized in the sequel. approximating a function of twc

and will of

(1.3)