The method of successive approximations for linear equations in hilbert space

THE METHOD OF SUCCESSIVE APPROXIMATIONS FOR LINEAR EQUATIONS IN HILBERT SPACE* V. N. STRAKHOV

Moscow (Received 10 March 1972)

A method of successive approximations is constructed for linear equations of the form f = (E - S) cp with an arbitrary non-increasing operator S acting in the Hilbert space H. 1. The purpose of this note is to show that the use of the theorem due to Nagy on the (minimal) unitary dilatation of a compression in Hilbert space makes it possible to considerably improve the accuracy of the theory of the solution of linear equations by the method of successive approximations. Let H be a (real or complex) Hilbert space, s E L(H, H) the compression, l\Sll< 1. We consider the problem of solving the operator equation

cp= ST +

fo,

foER(E-S).

(1)

The classical result is that if p(s) < 1 (P(S) is the spectral radius of the operator S), the method of simple iteration ‘PO E

H,

(pn =

SQh-i + fo,

n = 1, 2, 3, . . . ,

(2)

converges to the (unique) solution of this equation [l] . This result was strengthened by Krasnosel’skii [2], by showing that the approximations cp,, in (2) converge to one of the solutions of Eq. (I), if S is a self-conjugate operator, and h = -1 is not an eigenvalue of the operator S. In the case where S is an arbitrary compression the question of the construction of a convergent method of successive approximations remained open. 2. We put T=-

E+S 2

and transform Eq. (1) to the following equivalent form: *Zh. vjkhisl. Mat. mat. Fiz., 13,4, 1041-1044, 1973.

269

270

K N. Strakhov

rp= Tv+ff.

(3)

It is obvious that T E L(H, If) and is a compression. Theorem 1.

Let Eq. (3) have a solution for a givenf(possibly not unique). Then for any CPO E H the approximations cpn= Tqn-i + f,

n = 1, 2, . . . ,

(4)

converge to one of the solutions of this equation. Proof: Let cp be some solution of Eq. (3) (which will in fact be improved below). Then it is known that

By the theorem of the unitary dilatation of compressions in Hilbert space [3,4], a Hilbert space K can be found for whichH will be a subspace and in which we can define a unitary operator U such that for all n = 1, 2, . . . we have S” = PP.

(6)

Here P is the operator of orthogonal projection from K into H. It follows from Eq. (6) that x,(S) = Pn,(U), where I-C,is an arbitrary polynomial of degree n. In particular,

Butting CT- 90)= $n,

$n E K,

(8)

Short communications

271

we obtain from the Eq. (5)

Let E (0) , o < 0 < 2n,be an expansion of unity generated by the unitary operator U acting in K. It follows from Eq. (8) that

(10)

=

s

(cos O/2)2n d (E (0) (cp- cpo), cp- cpa).

Let the point A.= 1 not belong to the discrete spectrum of the operator T. In this case the point h = 1 either does not belong to the discrete spectrum of the operator U, or it + 0 as n --+a3 and by Eq. (9) we have It9 - (p,,ll -+ 0 as belongs to it. In the first case IIIJJ~II n -+ 00. But if the point ,A= 1 belongs to the discrete spectrum of the operator U, the subspace in K generated by the eigenvectors of the operator U, has in H a zero projection, and therefore 119- rp,,ll= IIP$nll-+ 0 as n + 00. Therefore, if Eq. (3) has a unique solution, the successive approximations defined by Eq. (4) converge to this solution. Now let the point h = 1 belong to the discrete spectrum of the operator T. We will denote by HT the subspace in H generated by the eigenvectors of the operator T, and by PT the operator of orthogonal projection onto this subspace. In the case considered we choose

where cp’ is a solution orthogonal to HT. With this choice of cp, although the point h = 1 will belong to the discrete spectrum of the operator U, we have as before IIT- (pnll= IIP+,II-+ 0 as n--t m. The theorem is proved. 3. It is obvious from expressions (9) and (10) that the rate of convergence to zero of 11~- rp,li may be arbitrarily slow. However, if the solution cp belongs to a narrower set M c R(E - T) , a definite rate of convergence may exist.

K N. Strakhov

272

We will give a typical result. Theorem 2. Let

(11)

M=M,=R((E--T)“), where m is some positive integer.

If r~- ‘pOE M,,,, a constant C,(cp - cpo), can be found such that for all n = 1,2, . . . we have

(12)

fioof: If cp- ‘POE M,, we can always find an element x E H such that X.

rp-cpcpo=(E-Z’)mx=

By the second relation of (7) we have /E-U\

m

xn= (2)

cp- cpn= Pxn,

X.

(13)

Determining _&Y(B) as above, we find from this 2x llx,,l!2= (Is

cos~(e/2))“(cos~(e/2)

but sup

cP(l--ua)m==

OGXZGI

( *)*(i%)m~ l-

Therefore, from Eqs. (13) and (14) we obtain llrp - cPnl12<

(

1 - --=)n(

&)mllx~ll,

which is what it was required to establish (C,(cp - go) = IIXllz). The theorem is proved.

(14)

273

Short communications Remark. If the operator (E- T) has fractional powers (E - T)“, where m < 0 is a real number, the results (11) and (12) remain true in this case also, but the proof is more delicate.

4. Although the rate of convergence of the norms of successive approximations (4) depends on the properties of the unknown solution and may be arbitrarily small, there is always some lower limit of the rate of convergence for the norms of the errors of these approximations. The error Y,, of the approximation 2, r=

f-

(E-qcp,

=

CF~is defined by the relation

(E-T)(cp-rpcp,)

(~--)~"(q--Pa).

=

(15)

Theorem 3.

For any cp and cpo m

c

-=cikp - wiV < +

(n + l)Ilv,llz

Oov

n=o

consequently llvnll =

0

(-!-). n

n-em

Proof: From Eqs. (15) and (7) we have Ilv,ll = l&,11, where

yn=

(F)

(T)

“w-(pa)

and consequently, m z

0 (n + 1) IIY~II~

<

n=o

But

c n-o

(n +

~)llYnl12.

2n llynllZ =

s 0

(1--~s2(8/2))(cos2(~/2))nd(E(8)(cp-~o),cp-~o),

from which it is easy to find that

eo

$I!CP - ‘pot12 < c

(n + 1) llynl12

n=o

This implies all the statements of the theorem.

-c

Ilv - ‘p0112.

(16)

K N. Strukhov

274

Note, It is obvious from the proof that in the general case relation (16) determines the limiting rate of convergence (which cannot be increased). 5. In conclusion we mention the following. Practically all the results presented in this note refer to the “worst” case 11~11 = 1; everything is much better for the case llSl1< LBut in our case the problem of solving Eq. (2) or the equivalent Eq. (3) must be regarded as incorrectly posed, since R(E - 2’)cannot coincide with H. The results of this note may be used to construct regularized algorithms [5-71, and in particular to select sets of a uniform re~la~tion [8]. Translatedby J. Berry REFERENCES M. A. and VAINIKKO, G. M. Approximate solutiod of operator equations reshenie operatornykh uravnenii), “Nauka”, Moscow, 1969.

1.

KRASNOSEL’S~I, (Pribbzhennoe

2.

KRASNOSEL’SKII, M. A. The solution by the method of successive approximations with self-conjugate operators. Usp. mat, Nauk., 15,3, 161-165, 1961.

3.

Sz.-NAGY, B. and FOYASH, Ch. Harmonic analysisof operators in Hilbert space (Garmonicheskii analiz operatorov v ~~ert~orn prostranstve), “Mir”, Moscow, 1970.

4.

HALMOS, P. The HiIbert space in problems (Gil’bertovo prostranstvo v zadachakh), “Mif’, Moscow, 1970.

5.

LAVRENT’EV, M. M. Some incorrect problems of mathematical physics (Onekototykh nekorrektnykh zadachakh matematicheskoi fmiki), Izd.vo SO Akad. Nauk SSSR, Novosibirsk, 1972.

6.

TIKHONOV, A. N. The solution of incorrectly posed problems and a method of regularization, Dokl. Akad. Nauk SSSR, 151, 3, 501-504, 1963.

7.

TIKHONOV, A. N. On incorrectly posed problems. In: CoNection of papers of the VTs MGU. 8, 3-33, Izd-vo MGU, Moscow, 1967.

8.

IVANOV, V. K. The uniform re~l~ation 546-X58,1966.

of equations

of unstable problems. Sibirskii matem zh., 7,3,