Math. Nadir. 96, 27-31 (1080)

On the Kryanev-Lardy Method for Ill-Posed Problems

By C. W. GROETSclH of Ohio (U.S.A.)

(Eingegangen am 30.3.1979)

1. Introduction

KRYANEV'S method for solving the ill-posed operator equation

( 1 ) AU = f ,

where A is a linear operator on a real &BERT Bpace, consists of choosing a bounded, positive definite operator B and forming the sequence of iterates defined by

(2) XO = 0, AX,, + Bxn = Bx,-, + f ,

KRAYANEV [3] established the convergence of th0 method under the assumption that A is a hounded positive semi-defnite operator and equation (1) has a unique solution. The author [a] proved the convergence of a related method in the case when A is a densely defined closed linear operator, again under the assumption that for a given f equation (1) has it unique solution. Our aim in this note is to investigate the convergence of the method to a generalized solution of (1) when A is a closed unbounded operator and the existence of a unique solution is n6t assumed.

2. Results

Suppose that H , and H2 are real HILBEET spaces (the inner product in each space will be designated by (., .)) and that D(A) is a dense subspace of H,. Let A : D(A) --f H2

,he a closed linear operator. We shall investigate an iterative method for approximating A+/ , where A+ is the MOORE-PENROSE generalized inverse of A. We recall that A+ is the closed linear operator defined in the dense subspace

D(A+) = R(A) 0 R(A)'

of H , by A+/ = u, where u is the solution of minimal norm of the equation

( 3 ) Ax = &/, and Q is the orthogonal projection of H2 onto R(A). We note that this definition of the generalized inverse is the same as that for a bounded operator (see e.g. [l]) and that for f D(A+) the set of solutions of (3) is convex, nonempty and closed (since A is closed). Therefore the vector A+f is uniquely defined.

We shall suppose that R : H , --f H , is a bounded, self-adjoint operator satisfying

(4)

-

(Br, x ) 2 c2 II4*

28 Groetsch, On the Kryanev-Lardy Method

for all x E H , and some c + 0. We may define an equivalent inner product on H , by

[G Y1 = (Bx, Y), and we shall denote the HILBBRT space which consists of H I with the new inner product [ m , -1 by 2,. The norm in H , will be denoted II.lla, that

1141; = (BZ, 2) = [z, 4. Since the norms II.II and 11 - are equivalent, the subspace D(A) is also dense in 3EpI. The adjoint of A considered as an operator on 8, will be designated by A’. That is,

A’ : D(A‘) --f H , satisfies

(A%, Y) = [x, A’yl

for all y E D(A’) = {y E H , : (Ax, y) = [x, 21, some z E H , and all x E D(A)I. The adjoint of A considered as an operator on H , will be designated by the custonrary symbol, A*. Note that these two adjoints are related in a simple way. Namely, D(A’) = D(A*) and A* = BA‘. By [6 , page 3071 the operator

(I, + A’A)-l: if, -+ 2 1 ,

where I, is the identity operator on S1, ’isa bounded, self-adjoint operator on H, satisfying

( 5 ) Il(11 + A’A)-lllB 5 1 .

. Lemma 1. Given / E H,, there is a unique u1 E D(A) such thut

+ (A%, A 4 = (1, Aw) /or all v E D(A) . Moreover, if / E D(A+), then U, = u - Wu, where W = ( I , + A’A)-1 and u = A+/ .

Proof. Since A is closed, D(A) is a HILBJJRT space under the inner product (., .) defined by

(5, Y) = [x, Y1 + (Ax, AY). The linear functional 4 defined on D(A) by 4(er) = (f, Av) is clearly continuous with respect to the norm induced by the inner product (., .). Therefore, by the RIESZ Theorem, there is a unique u1 E D(A) with

(1, Av) = +(w) = ( U l , v) for all w E D(A) , which was to be shown.

v E D(A) and Wu E D(A*A), we have If f E D(A+) and u = A+/, then Au = Qf, and since (f, Av) = (Qf, Aw) for all

(B(u - Wu), v) + (A(u - WU), Av)

= (Bu, V) - ( (B + A*A) WU, v) f (Qf, Av) = (Bu, w) - (B(I , + A’A) wu, v) + (f, Aw) = (f, Aw).

= (Bu, W) - (BWU, V) - (AWU, Av) + (Au, Aw)

But then, by the first pert of the Lemma, u, = u - Wu, which was to be proved.

Qroetsch, On the Kryanev-Lardy ilIethod 29

We will study the sequence of iterates defined by

(6) u, = wu,-I + u1, n = 1 ) 2,3, ... where u1 is given by Lenima 1 . Kote that this is equivalent to the requirement that

(Bun, v) + (Aura, Av) == (Bun-1) v) + (f, At')

for all 21 t D ( A ) , which establishes the connection with KRYANEV'S method. This can be established exactly as in the proof of 1,eniiria 1 by considering the linear functional

+(w) (Bun-1, v) + (1, Av)

on the HILBEHT space D ( A ) . In the case B = I and f t U ( A ' ) , the iiiethod (6) redores to that investigated by h R D Y [4].

Lenirna 2. I f f E D ( A + ) and u = .4+f, then 11 - un = Wnu. Proof. By (6) and Leninia 1 , we h a w

u - u, = 11 - u1 - wu,-I = u - (u - W u ) - Wu,,-I = W ( u - u,-l).

Therefore, u - ZL, = W'u, n = 1, 1, 3, ....

henceforth denote the operator A*A by A. We niay now provide an error bound for the method (6). For convenience we will

Theorem 1. Suppose R(2) & K ( A ' A ) mid Qf = A B z for soine i E I$,, then for some

y t ff,,

Proof. Since R(1) & R(A'A) , we have Az = A'Ay, for some y E H I . Let ( E , ] be the resoltition of the identity in XI induced by the self-adjoint operator A'A. Since Qf = Axz and Az t N ( A ) I , we have Az = A + / . Therefore by 1,eninia 4,

m

0

It then follows that m

We note that if we make the weaker assuniption that f E D(A+) (i.e. Q f c R ( A ) ) , rather than the stronger assumption that Qf c R(AA) , then the method still converges. For in this case we have by leninia 2,

m

11u - U J ~ = J (1 + ~ 1 - n d [ ~ , u , u] -+ o as n + oo. 0

30 Groetsch, On the Krymev-Lardy Method

However, if f 4 D(A+), then the sequence {u,} diverges and in fact has no weakly con- vergent subsequence. For if the subsequence (u,J converges weakly to y , then since W is bounded and therefore weakly continuous, we have by (6)

(7) y - w y =u,.

Since u1 E D ( A ) and W y E D ( 2 ) c D(A) , we find that y E D(A) . Also, by Lenima 1 and (7)

(1, Av) = ( B y - B W y , v) + ( A y - A W Y , Aw)

= (BY, .) - ((B + 4 WY, V) + (AY, = (Ay , Av), for all v E D(A) .

Therefore, f - A y E B(A)I , that is, f E D(A+). We summarize these results in the follow- ing :

Theorem 2. If f E D(A+) , then u, + A + / . However, if f D(A+) , then (u,} has no

Since bounded sets in &BERT space are weakly compact, we obtain immediately

Corollary. If f 8: D(A+), then llunll + bo.

Finally, we investigate the method under the assumption that the exact data f is unavailable, but an approximation 1 satisfying 111 - jll 5 S is on hand. The first approxi- mation 6, (corresponding to the corrupted data 1) then satisfies

weakly convergent subsequence.

the following :

(fi,, w) = (BQ,, v) + (AQ,, Av) = ( j , Aw)

(u, - 61, w) = (f - r', Aw)

(u, - Q,, u, - 21) = (f - j , A(u, - 61)).

IIAxl12 5 (Bx, x) + (Ax, Ax) = 11x11:

for all w D(A) . By Lemma 1 we then have

for all li E D(A) . Setting v = u1 - Q,, this gives

(8)

If we designate the norm on D(A) induced by the inner product (., .) by 11. [ I 1 , then we have

for all x E D ( A ) . Therefore by (8), it follows that

Ibl - Q l l t 4 Ilf - I l l IIA(u1 - ~ l ) l l a I; SllUl - Qllll

llu, - QIIIB 5 11% - ~111, 5 6 .

6" = W6,-, + Q,,

and hence

Later approximations using the data 1 satisfy by (6)

and therefore ?k-1

u, - 6, = 2 Wk(u, - C,), n = 1,2, 3, .... k=O

Groetsch, On the Kryanev-Lardy Method 31

But IIWllB 5 1, by (5 ) , and therefore

IIun - .iinll* 5 n llul - .iilHB 5 n8.

It then follows that

11% - UIIB 5 IlGn - ~ ~ I I B + IIun - ~ I I B I nd + IIun - ~ I B - But jlu, - uIJB --f 0 as n --+ 00 i f f E D(A+). Therefore, given E > 0 there is a B ( E ) and n, such that

]/fine - ullB < E for 0 < 6 5 d ( ~ ) ,

that is, the method is a regularizing alogrithm in the sense of TIKHONOV (see [ S ] ) if f E D(A+).

References [ 13 C. W. GROETSCH, Generalized Inverses of Linear Operators: Representation and Approxi-

[2] -, Sequential regularization of ill-posed problems involving unbounded operators, Comm.

[3] A. V. KRYANEV, An iterative method for solving incorrectly posed problems, U.S.S.R. Compu-

[4] L. J. LARDY, A series representation of the generalized inverse of a closed linear operation

[5] F. RIESZ and B. SL-NAQY, Functional Analysis (translated by Leo F. Boron), Ungar, New

[6] A. N. TIKHONOV and V. T. ARSENIN, Solutions of 111-Posed Problems (translated from the

mation, Dekker, New York 1977.

Math. Univ. Carol. 18, 489-498 (1977).

tational Math. and Math. Phys. 14, 24-33 (1974).

Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur. Ser. VIII, 68, 152-157 (1975).

York 1955.

Russian), Wiley, New York 1977.

Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221 U.S.A.