The optimal approximation of non-linear operators

U.S.S.R. Comput. MathsMath. Phys. Vol. 18, pp, 236-242 0 Pergamon Press Ltd. 1979. Printed in Great Britain.

0041-5553/78/0601-0236$07.50/O

THE OPTIMAL APPROXIMATION OF NON-LINEAR OPERATORS* A. I. GREBENNIKOV

Moscow (Received 24 December 1976; revised 13 April 1977)

THE OPTIMALITYin respect of order of the methods of interpolating for the approximation

of the values of an arbitrary non-linear

and regularized splines

operator is proved for discrete

initial data. 1. Many applied problems may be formulated the values of some (possibly unbounded

mathematically

as a problem of calculating

operator B, acting from the Hilbert

and non-linear)

space H into the linear normed space I’. The initial data about the elements B is calculated

are as a rule specified as a collection

linear functionals

2, . . . , m,

Zi, i=l,

the approximate

fi=li(U)

We introduce

values

of values

f;, i=l,

UEH,

2, . . . , m,

on which of some

on these elements. Here instead of the exact values f[=l, (u)+gi,

where the ti are errors, are often given.

the operator Au=(Z,(o),...,1,(u)),

acting from H into the m-dimensional

Euclidean space F.

UEM~={UED:IILUII~~R},

We will assume that

O
< +m,

where L is a linear operator acting from H into the Hilbert space G and

D&D=D,IlD,.

We write f= (fir . . 1 fm),

f= VI, .--1 t),

E=(E1, ..-,

Em).

We define the set of exact initial data as the set Nw={f=Au,

and the set of approximate

initial data, as Na, R=(f~F:

Here the number

6>0

UEMR),

r=Au+g,

ucM,,

Ilgk6).

describes the accuracy of definition

The problem of calculating

of the elements

f=NR.

the values of the operator B on these assumptions

is ill-posed [ 1] .

There is considerable interest in methods of solving ill-posed problems which are in some sense optimal. For linear problems the questions of the construction of optimal methods have been fairly fully studied (see, for example, [2,3] ), but non-linear problems have been investigated to a lesser extent. Here we must mention the paper of V. P. Tanana [4] , where for the case of the solution of a non-linear equation of the first kind with an approximate right side the optimality in respect of order of the method of discrepancy and the method of quasi-solutions of V. K. Ivanov [S] was proved. *Zh_ @hid.

Mat. mat. Fiz., 18, 3, 762-766, 1978.

236

237

Short communicdons

The problem considered below differs in the fact that the operator B calculated is regarded as non-linear,

and the initial information

is specified by a linear operator A. This leads to a

position where the element on which the calculation

of the value of the operator B gives the least

error (in respect of order), belongs to a linear space of interpolating and regularizing (in the case of approximate data) splines. In the case of approximate

data this paper investigates the optimal properties of Tikhonov’s

therefore as in [3] , a relative error of approximation

method of regularization, regularization

parameter is chosen. The introduction

enabled us, when considering prior information,

optimization

criteria

problems, not to require the existence of quantitative etc, and to manage with only qualitative

prior

that is, knowledge of the operators A and L. This made it possible to reduce the

problem of optimization

for different initial data to the choice of the regularization

On the other hand the introduction due to the fairly complex structure definition

depending on the

of a parameter in the optimality

such as, for example, the level of error of the input data or the constant

occurring in the method of quasi-solutions information,

(in the case of exact data)

of the traditionally

parameter.

of this relative error enabled us to avoid dif~culties

of the sets of admissible elements, which participate

in the

used error [2] , since in the proposed approach the admissible

elements vary in a linear set D, which facilitates the investigation. Efficient ~go~thms

for the numerical

realization

of the optimal order methods indicated

in this paper are given, for example, in [6]. 2. Let

6=0,

that is, the input information

is specified exactly.

U,(j) ={u=.UR:

We introduce

the set

Au=/)

for some fixed vector f= NR, and let T be an arbitrary operator defined on A [D] , acting in V. The error in calculating the value of the operator B for the data vector fby means of the operator T is described by the quantity

and on the whole set NR by the quantity

Problems of finding optimal operators from the condition all-possible admissible operators constructed

T were formulated

of the minimum

of these errors for

in [3] and such optimal operators were

for a linear B.

For an arbitrary non-linear

operator B the construction

of optimal operators is difficult.

However, operators optimal in respect of order can be indicated,

that is, operators for which the

error differs from the optimal by some constant. Let U(f)={u4: Au=f}, element USED such that

fEA[D].

IlLufllci =

We consider an auxiliary problem:

inf IILuIIc. UGU(i)

find the

238

A. I. tirebennikov

It is assumed that problem (1) has a unique solution (conditions on A [D] , and also on

for example, in [2] ). Consequently, operator

S : Sf=u,,

sufficient

Y,rA[D]

for this are indicated, the linear bounded

is defined.

Theorem I

made the operator To = BS is of optimal order, the following estimates

With the assumptions being valid:

2oonT(B) a 0s CR,To) 2 wm (B) = sup OonT(B, f).

(3)

f=NR Proof:

For any

u=U,(f)

T the following inequalities

and arbitrary

T, f))

a(& ) -

1 2

max{llBu-Tfllv,

{IPJJ-Tfllv

hold:

Wf-Tfllv) (4)

1

+II~f-~urIIv)~

- 2 IIBU-%fllv,

from which the left side of (2) follows. The right side of (2) is obvious. If in the series of inequalities and then with respect to fEivR, (4) we take the exact upper bound first with respect to u=UR(f), we obtain (3). The theorem is proved. 3. For a rather different choice of the error and the supplementary operator to be approximated

conditions

on the

it is possible to indicate an optimal algorithm of approximation

of

the values of the operator B. Let P be an arbitrary operator defined on the space F, acting in D. The error in calculating the value of the operator B for the data vector fis described by the quantity sup IlB(u-Pf)

P, f) =

aB(&

IIT’,

U-=UB(f)

and on the whole set NR by the quantity &, (R, P) = sup GE @, p, f)

.

fENR We

say that the operator

&

isf-optimal if GB(R, PO,f) = inf i;iS @, P, f) = h CRY f),

P

and we say that the operator PO is NR -optimal if

where inf is taken over all-possible operators P. We introduce

the set UB={u=MR:

Au=O}.

(5)

239

Short communications

We write WE(B) = sup IlBullv. UECJR Theorem 2 If the operator B is antisymmetric,

that is,

B(-u)

then the operator

PO=S,

=-Bu

BED,,

is NB -optimal, where &%z)

If, in addition,

(7)

03)

=aB@).

the relation

$

G

holds, then the same operator S is f-optimal,

vui, UZEDB,

IIBui+Buzllv

(9)

where

GJB (R, f) =

sup

IIB(a-@)

(10)

IV-

u=.un(f)

Proof:

element

The properties of interpolating (2uf-U)

splines imply that for any element

also belongs to the set &y).

Also, for any

ud,(f)

u=U,(f) and arbitrary

the f=NR

we have the obvious inequality &(R,

P, f)>max

By relations (7) and (9) for any

{IIB(2ur--u--Pf)IIv,

IIB(u---Pf)Ilv}.

(11)

and any permissible operator P we have the

u~lJ,(f)

inequalities Ilv+liB(u-Pf)

2IIB(u,-U) Il6llB(2r~~-~-Pf)

llv, IIB(u-W)

~2 max {llB(2u,-zz-Pf) From these inequalities

and (11) by the arbitrariness

~B(R,

p,

f) 3

sup

of

IIv

IV).

u= U,(f)

IIB(u-Sf) Ilv.

we obtain the inequality

02)

u=Un(f)

Here equality in (12) is attained for P = S, and this implies that the solution of problem (5) is the operator PO = S, and Eq. (10) follows. For any u=Uv,(f) and arbitrary j=NR by the properties of interpolating splines the element (uf-u) =UR. From this and Theorem 1 of [3] there follow relations valid for operators B satisfying only condition (7):

A. I. Grebennikov

240

liJ,(I?,P)~

sup IIB(u-Uj)IlV. ~ENRUEURU)

WB(R)~ sup

The extreme terms in (13) are identical for P = S,and accordingly, from (13). The theorem is proved.

&=S.

(13) Equation (8) follows

We give an example of non-linear operators satisfying the relations (7) and (9). Let H=Wik) then conditions (7) and (9) are satisfied by the operators la, bl, V=LIa, bl, B*(z)= (J.C$) onlyfor

where (2~I)<0

2”-‘,

u(~)(I)+O

n=O, *i,

and

*2,. . . ,

k=O, 1,. . . ,

~=[a, b].

4. In the case of approximate data, for DO, we consider an approach to the problem of optimizing the calculation of the values of the operator B, based on A. N. Tikhonov’s method of regularization [ 1 ] . For any

UEF, h>O

we define the functional @Au, vl=~IIAu--vll~~+IILullc~,

and consider the auxiliary problem: find an element

U&ED

u=D,

such that

(14)

It is assumed that this problem has a unique solution uh for any hr0 &: &v=u~ED is defined. Thereby for all v=F the linear operator

and

u=F.

We specify the non-empty set NcF. We defme the measure of approximation of the operator B at the point v=N by the quantity OS&; T, 0) = sup {IIBU-T4lv/uAu, UED

VI),

and on the whole set N by the quantity os(h; T) =

sup OB(h; T, 0). USN

Here and below we regard as admissible the operator T defined on the whole set N and acting in K We write OF (h; u)= inf os(h; T, v), T

WF (h) = inf oB(li; T), T

where inf is taken over all admissible operators. l%eorem 3 The operator

TL=BS1.

is of optimal order, and the following relations are satisfied:

241

Short communications

ProoJ: For any u=D, u=N and arbitrary T the following inequalities are satisfied:

which imply the left sides of inequalities (15). The right sides of these inequalities are trivial. The theorem is proved. The chosen errors of approximation of the operator B depend on the parameter A, whose introductjon permitted us to reduce the problem of constructing the optimal appro~mating operator for various initial data to the choice of a suitable value of the parameter X depending on the initial data of the specific problem. 5. As the set N let the set be specified. We denote by i?j, the solution of Na. R. problem (14) corresponding to the parameter il=h (6, r), chosen from the conditions of the discrepancy jMn,--fib,=6 (see [2] )_We write

CO:’ (6, R, f) = inf COB (6, R, T, f), T Theorem

wyT (6, R) = inf og(6, R, T). P

4

The operator

TAta,y)=Ti

is of optimal order, the following relations holding:

2@;==(8, R, f) 2 eM6, R, TA,f) 3 O? (6, R, f,, 2m”B”’(S,fR)>

sup

on (6, R, FL, f) ,, OF (8, R).

f-EN&. R

The proof of this theorem essentially repeats the proof of Theorem 3. 6. We note that the operators To and 7j constructed have the structure To=BS, T1=BSI., where the operators S and fi are independent of the approximate operator B, and are completely determined by the operators A and L. Thereby the optimal properties of these operators are preserved for the class of problems, that is, for ah-possible permissible operators B. When the condition of B-complementarity is satisfied the convergence of the optimal order methods presented can be investigated as in [2]. Translated by J. Berry.

242

Yu. L. Gaponenko REFERENCES

1.

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

2.

MOROZOV, V. A. Regular methods of solving ill-posedproblems (Regulyamye metody resheniya nekorrektno postavlennykh zadach), Izd-vo MGU, Moscow, 1974.

3.

MOROZOV, V. A. and GREBENNIKOV, A. I. The optimal approximation Nauk SSSR, 223,6,1307-1310,1975.

4.

TANANA, V. P. Optimal methods of solving non-linear unstable problems. Dokl. Akad. Nauk SSSR, 200, 5,1035-1037,1975.

5.

IVANOV, V. K. and KOROLYUK, T. I. Error estimates for solutions of i&posed problems. Zh. vj@hisl. Mat. mat. Fiz., 9,1,30-40,1969.

6.

LORAN, P. Zh. Approximation and optimization (Approksimatsiya

US.S.R. Comput. Maths Math. Phys. Vol. 18, pp. 242-245 0 Pergamon Press Ltd. 1979. Printed in Great Britain.

of operators. Dokl. Akad.

i optimizatsiya),

“Mir”, Moscow, 1975.

0041-5553/78/0601-0242$07.50/O

THE METHOD OF CONSISTENT APPROXIMATION FOR SOLVING NON-LINEAR OPERATOR EQUATIONS* Yu . L. GAPONENKO

Moscow (Received 16 July 1976) A NEW REGULARIZATION method is proposed for the approximate solution of non-linear equations. The method is based on the well-known idea of the construction of a compact set. As regularization parameter a mesh step compulsorily consistent with the error of the initial data is used. The method guarantees uniform convergence to the exact solution of the class in the one-dimensional and two-dimensional cases. Operator equations of the first kind constitute an important class of ill-posed problems. The problem of constructing an approximate solution of an ill-posed equation for approximate initial data can be solved by the methods of the theory of regularization (see, for example, [l-7]). In this paper a regularization method for the approximate solution of non-linear equations is presented. 1. Statement of the problem

We consider the operator equation Av=f,

(1)

where A is a continuous operator acting from the space C[O, I] into the space L2 [0, I] . We will assume that Eq. (1) has a unique solution u(z) 43 ‘10, 11 and that for the operator A the prior estimate lIAv-Awll~6l(llv-wl~o,

max(llvllo,

Il~llo));

(2)

is known, here Il.li and ll.lk-~denote the norms in the metrics of L2 [0,l] and C[O, l] respectively, satisfying and Q(t, R) denotes a non-negative continuous function, O