An algorithm for reconstructing functions

U.S.S.R.

Comput.Maths.Math.Phys.,

Vo1.27,No.3,pp.B%88,1987

Printed in Great Britain

0041-5553/87 $10.00+0.00 01988 Pergamon Press plc

SHORT COMMUNICATIONS AN ALGORITHM FOR RECONSTRUCTINGFUNCTIONS* N.G. USHAKOV

An algorithm is proposed for reconstructing functions that are given with random errors. The algorithm is based on the regularizationmethod and allows a sequence of approximations to be constructed that converges with probability 1 to the function being reconstructed for extremely weak limitations on the noise. In this paper the problem of the reconstruction (filtration)of functions that are given with random errors is considered. The proposed method, based on regularization,essentially differs from methods such as, for example, the method of moving averages,and, for sufficiently wide assumptions related to the process that realizes the error, allows a sequenceoffunctions to be obtained that converges with probability 1 to the function being reconstructed. Suppose a real continuous function is given in the interval [a,'~]. Suppose that it is studied on some subset A of this interval with random error E(a),u=A i.e. the function is f(c)=fW+E(a), ceA. It is required that the function f(z)is reconstructed from the function f(a). To solve this problem quite rigid limitations are usually placed on the family of random quantities E(a) (for example mutual dependence). However, in practical situations these conditions cannot be satisfied. The algorithm proposed below allows the function f(z)to be effectively reconstructed for significantly wider assumptions relating to the process &(a).It is true that a supplementary condition of the boundedness of E(a)has to be introduced but for practical situations this condition is not too rigid. The algorithm is based on the theorems proved below. Suppose a family of random quantities E(a),aeA satisfy the following conditions. 1. For some b>O and for any a=A, 2. For any a>0 there exists e>O such that for any a*A' the inequalities

aeA

and for any finite subset A'cA,

P(E(a)<-b+WE(r), yeA')>s and p(e(a)ab-W&(r),r=A')>e

hold (thesearetheconditionalprobabilitiesrelatingtothe s-algebra generated by the family of random quantities E(r),yeA'). Conditions 1 and 2 are satisfied, in particular, in the case when E(a) are mutually dependent, differently distributed, random quantities and b is an exact upper bound for the error. At the same time the class of random functions that satisfy 1 and 2 significantly broader than the set of sets of independent, differently distributed, random quantities. In fact, suppose &l(a), aeA, i=1,2,...,n is a set of random quantities that satisfy the following condition: P(l&r(a)lO, i=l,2,...,n, P(v-r)- 1. z

t-1

For every a put E(a)==&.(a). It is easy to see that E(a)is dependent: II P(E(ad=&

Eb)~Et)-

I2

P(E,(a,)EE~,El(al)-EaIv-l)P(v=i)-

ti P(e,(a36EI)P(El(ar)~Ez)P(v--i)~ '-I ,-, * n

c

=

P(E,(ad -%)P(v-9

P'(E'(ai) = EI)P(Eb)

P(Ej(a~)=EdP(v=i)1-1

= Ed

*Zh.vychisl.Mat.mat.Fiz.,27,5,771-776,1987

85

86

for any Bore1 sets El and E 2 that satisfy conditions

P(l&Wl~ *IFurthermore,

1 and 2:

P(l&,(a)la; blv-i)P(v-r)P(IC~@)lc *)P(v-f)P(v=d)- 1. i-1 I-1 1-a

we fix an aribitrary C-0 and put es=min (P&(a)<-b+6), P&(a)sb-6)).

e=min (ei,...,e,). From the definition of Let &i(c) it follows that all the ec are positive, a,...,aN that are pairwise different (N is arbitrary) and any Bore1 sets EI,...,EN then For any P(E(a)6-b+blC(al)EEi,...,E(aN)~~~)p(&(or)<-b+G,E(ar)EEI,...r&(CLN)~EN) _ P(E(~~)EE~,...,~(~N)~~N) II

[E --I Wj@r)~ El,. . . , &~(a~) = &P(v==l) 1 l--I n Edmd~W++ F,P(E((a)6-*+8)P(&((al)EE,,..., d-1 II E&,..., [ZP(CjW P(&i(u)<-b+g,Er(a~)EEl,.... i-i n

[Z

&J(%dEEN)P(v-j)

]-‘h.

,--r

The

second inequality of condition 2 is proved in the same way. the error distribution at each The example quoted has a simple physical interpretation: is determined by the value of some random quantity, one and the same for all points (for example, the error distribution of a signal that is recorded on some apparatus can depend on the conditions in which detection takes place: temperature, humidity and so on). Note.

The requirement Ef(c)=f(c)

is not imposed. Theorem 1. Suppose A is a denumerable, everywhere compact subset of the interval Ia,*I /E(Z), z=[a,*l, is a and the random quantities &(a),~~%4 satisfy conditions 1 and 2. Then if continuous function such that for any a=A the inequality f(c)-bcft(e)Gf(c)+*, holds,

then

Proof. an arbitrary such that and

P(/~(z)=f(.z))=i. Suppose that for any fixed CGA the event (fo(a)=f(a)) has probability 1. aa=A. Let% denote the class of all functions 9E(X) that are continuous

19C(ao)-f(a+)I>i/n.00

is chosen

f(c)-*~9~(c)=G(o)+* to be sufficientlysmallso

that for

/z-%165

(I) the relation

if(~)--f(ao)l~4/(4~) is satisfied (this can be done by virtue of the continuity of the function f(s)).Let denote the class of all continuous functions 9pl(z)that satisfy (1) and such that where

~(6,9e)

is the continuity

9(1/m,9ptNU(4n), of the function

module

Fix in I

(2) a.,-

9c(z) i.e.

9(&9t)= mar maxlcpk(~+Q-cpb(dI.

IV,<6 I

We shall show that P(a.n%,,)=o. In fact, if then

vE(z)wonlf).,,

for la-aol
(the trivial

case

a,n~,,,=o

I%(a)-cprw A=min(8, l/m).

From

(2),

is not included

for consideration)

l
(3) and the definition

(3) of the class

9.

we have

~9r(~)-f(~)l=I9~(~)-9~(~O)+9c(~O)-f(~O)+f(~O)--f(a)I~I9E(aO)-1(ap)I--~9r(a)-9r(ao)~-~f(ao)-f(or)~>i/(2") for all aeA, /a-ao(
we shall assume

that

rpt(ao)>f(ao)(for the case

9po(c) -f (W =-up) (4) i=i, Z,...,ai+a,, a=A, la-aoIcA. An arbitrary sequence a,.a~,... is chosen sueh that Ia,--aolGA, iPi. From the condition cpt(+)=snIl%, it follows that cpk(m)
from which we obtain

87

W%flZLdcP(

ij

(5)

b)).

h(adcf(ad+

L-L

Furthermore P f(w)+ b
I

EW,fcf
>

*e-s(n)>0

(6)

and

i fW+

b>f(a<)+---,l-=2,...,N X )

P f(ar)+b=-fCaa)+ i (

I

P f(Q)+ (

f(at)+bWa,)+d,*-3,...,N

b=-f@d+$

x...

1 X P f(an+r)+ b>f (aN+d+ ( 4n for any N=,i,2,....From this and

(6) it follows

i f(at)+b>f(ad + 4n

P

Letting

that <(i-e)N.

I'+-- we obtain _

P

From

(4) and

(7) it follows

(

$[

fW+

b>f(at) + ‘))= 4n

0.

that

P(i~,kt(a~)~~Lai)+b))6P(,~{f(a,)t b>f(ac)+c})=O. from which,

taking

into account

(5) we finally obtain P(a,.n%ln)=o.

Furthermore, P(ft(ad+f(ad)=P(

i a.)-P(

n-L

Taking into account the continuity (ft(z)=f(z) v+~Ia, bl)=(ft(a) =f(a) v-4 Vfd@ff(z)

Vz=[.,

Fj ij

n-1 ml-*

W.n%d)

4~~Pw,nD.,.)=o. I-, In-,

of the functions ft(z) and f(z)(from which the inequality follows) we obtain bl)=P(f,(a)+f(a) va=A)==P(

U (ftb)+fW)) a.x.4

= 0

The theorem is proved. Suppose Q[f] is an arbitrary stabilizing functional on a set of functions that are continuous in the interval [a,b] and uniformly bounded at some point of the interval i.e. a non-negative functional such that the subset If: Q[f]
Suppose AI, AZ,... is an increasing Theorem 2. such that the union

sequence

of finite subsets of the interval

satisfy conditions and the random variables e(a).a=A is dense in the interval [a,b] 2. Then if fl(&,z), f2(&z)... is a sequence of continuous functions such that f(+b~fdE,

a)Gf(a)+b,

minimizes the stabilizing functional and f,,(E. 2) continuous on [a,b] and satisfy (8) then

P(limf.(E,fl==f(f))=“‘OD Proof.

n=l, 2,...,

aeA., Q[f]

on the set of all functions i.

Put 8,-&(r):

f’(a)-b
1 and

a=A., n=i, 2,. ..I,

(8) that are

It is easy to see that ePn+jcLP,, fn(l. z)=PP.nF, n=l, 2,... set ; In %_I with probability 1 consists of the element f(z)

Furthermore by virtue of Theorem 1 the

(It is obvious that

with probability 1 consists of the element f(z)(it is obvious that f(z)EFP). Thus the sequence of imbedded compact sets P,n? n=L 2,...,with probability 1 has as its intersection a set consisting of the single element f(s). From this, taking into account that for any n the function f*(&, +) belongs to the set 8,W, we obtain that the sequence f.(L z), n=l, 2, . . . . converges to f(z) with probability 1. The theorem is proved. For the stabilizing functional Q[f] it is convenient to take a functional of the following type:

b) Wfl-marlf'(z)l. c) sa[f-Varf(z) (Var is the full variation of the function f(z)). (In cases a) and b) it is assumed that a solution is sought in the class of all differentiable functions). Numerically the problem of the reconstruction of f(z) reduces to minimizing a function of simple type in an n-dimensional cube. If a priori information on the behaviour of the function f(z)exists, the domain in which the minimization must be carried out can in essence be reduced, for example in the case when f(z)is monotonic. Note that convergence to an exact solution is ensured for an error fixed at an arbitrary level. We will move on to a description of the reconstructing algorithm. Suppose f(z) is a function continuous in the interval [a,bl,I,<...
Suppose the conditions of Theorem 1 are satisfied with inequalities (1) Theorem 3. replaced by the inequality f(a)-b-c
where c>O.

Then

P(luPlh(z)-f(+)(>c)-0. z Suppose the conditions of Theorem 2 are satisfied with inequalities (8) Theorem 4. replaced by the inequality W&2,..., f(a)-b-cGfn(L c)G(a)+b+c, WA., where c>O. Then P(limsupsuplf.(E,~)-I(r)I>e)-0. ")__ # The proofs of Theorems3and 4 are in principle analogous to the proofs of Theorems 1 and 2; they are therefore not given. If even an approximate value of the exact upper error bound is not known, it is possible, by considering it as an unknown parameter, to use a statistical evaluation. In this case the quantity I/c+r-f8l 2.- mar *<<
As ?a+- 6, converges in probability to b. REFERENCES

1. TIKRONOV A.N. and ARSENIN V.YA., Methods for solving ill-posed problems, Nauka, MOSCOW, 1979. Translated by S.R.