- Email: [email protected]
A remark on a class of regularizing algorithms
A REMARK ON A CLASS OF REGULARIZING ALGORITHMS” A. B . BAKUSHINSKII
Moscow (Received 22 May 1972) RESULTS on the convergence of a family of regularizing algorithms for the solution of an operator equation with an irreversible operator, are made more precise. This family was studied previously in [ 11. In the present note we refine some results on a class of regularizing algorithms for the solution of an operator equation of the first kind. This class was studied in [l] . Necessary and sufficient convergence conditions for regularizing algorithms of this form will be given below. We also refine the convergence co&&ion for a perturbation of the operator, of each specific regularizing algorithm of the family considered. 1. We will use a particular case of the general definition of a regularizing algorithm introduced by A. N. Tikhonov [2], namely, the family of linear operators T,, depending on the positive parameter CY, is called a regularizing algorithm with respect to the bounded operator A acting from the Hilbert space Hr into HZ, if 1) IITall<-J,
a>O,
2) Iim T,Av==v,for any VEH~, where vI is the part of v situated in the orthogonal a+0 complement to ker A. The operators T, act from HZ into Hr . We denote by s (A’A) the spectrum of the operator A *A and by El the spectral function of A *A. Let cp(L,a) be a real function measurable with respect to El for any a>O, and cp(0, a) =O. Also, let
We generate a family of operators T, by the formula Ta=\l)(A’A, a) A’. *Zh. vjkhisl. Mat. mat. Fiz., l-3,6, 1596-1598, 1973.
278
(1)
2’79
Short communications
It is of interest to clarify what are the necessary and sufficient conditions which must be imposed on the function cp7in order that the family (1) will possess the properties l), 2). Sufficient conditions were given previously in [l] . The question of necessary conditions was discussed in [3], where a solution was obtained in the case of a discrete spectrum of A *A. It seems to us that the following statement corresponds more precisely to the question posed. Theorem I
For the family (1) to possess the properties 1) and 2), it is necessary and sufficient that the following conditions be satisfied:
vraisup(cp(h,a)]< cqr %a
=,
(3)
lim]g(h, a)- I]= 0 O-DO
(4)
almost everywhere with respect to any measure (&uL, UJ. &oof: Necessity. Condition (2) is necessary for the property l), since l& is the norm of the operator (1) (see [ 1J ). Also A
T,Av
=
s
(5)
rp(4 a) d&v,
0
where A is the upper bound of the spectrum of A *A. Since by hypothesis TaA is a bounded operator,
But if (3) is not satisfied, it would easily be obtained from (5) that sup llZ’,All=m, then by the Banach-Steinhaus theorem the property (2) could not bearsatisfied. Consequently, it is necessary that (3) be satisfied
but
A. B. Bakushinskii
280
We notice that the functions (&u, V)and(E XVI, uL) are identical everywhere, except perhaps for the point h=O; moreover, the function (&uL, VA)is continuous for h=O. We have A
IITc4~u-u1112=
s
[cp(h,a)-
(6)
Ij2d(E*uI,UJ.
0
If Eq. (4) were not satisfied, the left side of Eq. (6) would obviously not tend to zero as a-+0, which would contradict the property 2). Consequently, Eq. (4) is necessary for 2). The sufficiency of the conditions of the theorem is verified in the same way as in [l] with the use in Eq. (6) of Lebesgue’s theorem on passing to the limit under the integral sign. 2. In the formation of T, by Eq. (l), let the perturbed operator x be used in place of the exact operator A. Let the equation As=f be solvable and x be its normal solution orthogonal to ker A (ker A*A)). Let the function cp(A, a) be defined (with respect to A) everywhere on A is the upper bound of the spectrum of d’A, and be continuous.
[0,
A],
where
The following theorem holds, strengthening the statement of section 3 of [l] . T&eorem 2 If the modified conditions (2) - (4) are satisfied (vrai sup is replaced by sup in (2) and (3), and in (4) the convergence is uniform for all h= (c, m) n [0,A] for any arbitrarily small c>O ), the relation Iim a-0,
holds, if
&8+0.
Ilrp(ii’iq coa’f-sll-
0,
(7)
b-0
Here 6-M-AI.
Indeed, lI$(iP”A, a)P’f-zll
~)PKP-*(*if,
a)A’As-AI+ a)A.A;zll&&YMl+II~
(A’&
a)A*Az-zll.
281 To prove Eq. (7) it is merely necessary to establish that
The expression under the limit sign on the left side of Eq. (8) can be written in the form *
J (p,(a, 4 -
1) 2d(&x, 2).
(9)
0
Since slker A’A by hypothesis, the function (I&Z,I) which increases monotonically as 3\. increases, is continuous with respect to 1\ for h=O and (_&x,Z) =O. We fm 00. Let A+ be a point of the positive semi-axis which is not a point of the discrete spectrum of A *A and is such that (J&+z, Z)O because of the denumerability of the discrete spectrum and the continuity of (l&x, x) for A=O. Estimating (9), we obtain A
A
J
J
(9 (h, 4 - 1)2d (J.&x,x) =c (cPha)-
+
1)2~b%~,X)
(10)
h+
0
MI (EAJ, x) - V&+x,x)1+ M b%x, x) + M(&+x, 2).
Here M=
sup (q(h, u)-I)~. a,A=ro,Al
This quantity is finite by (3). By Rellich’s theorem ([4], p. 396) lim 1(I&.+x,5) - (&*z, 5) I= 0
for xEHl,
a+0
lim @OX,x) = 0, a+0
x E Hi,
x 1 ker A’A.
Moreover, because of the assumption of uniform convergence in Eq. (4) on the left side of (10) the integral between the limits A+ and A tends to zero as a-0 and 6+0. Finally
282
A. B. Bakushinskii
x I ker A’A.
In view of the arbitrariness of e this implies Eq. (8). Theorem 2 is proved. We finally mention that the values of &were calculated in [l] for some specific regularizing algorithms. 7’ranslated by J. Berry REFERENCES 1.
BAKUSHINSKII, A. B., A general method of constructing regularizing algorithms for a linear ill-posed equation in Hilbert space. Zh. vpchisl.Mat. mat. Fiz., 7, 3,672-677, 1967.
2.
TIKHONOV, A. N., The regularization of ill-posed problems. Dokl. Akad. Nauk SSSR, 153,1, 49-52, 1963.
3.
KHUDAK, Yu. I., On the convergence of a family of regularizing algorithms. Zh. vpchisl. Mat. mat. Fiz., 12, 2,497-502, 1972.
4.
RIESZ, F. and NAGY, B. Sz., Lectures on functional analysis(Lektsii po funktslonal’nomu analizu), Izd-vo in, lit., Moscow, 1954.
Moscow (Received 22 May 1972) RESULTS on the convergence of a family of regularizing algorithms for the solution of an operator equation with an irreversible operator, are made more precise. This family was studied previously in [ 11. In the present note we refine some results on a class of regularizing algorithms for the solution of an operator equation of the first kind. This class was studied in [l] . Necessary and sufficient convergence conditions for regularizing algorithms of this form will be given below. We also refine the convergence co&&ion for a perturbation of the operator, of each specific regularizing algorithm of the family considered. 1. We will use a particular case of the general definition of a regularizing algorithm introduced by A. N. Tikhonov [2], namely, the family of linear operators T,, depending on the positive parameter CY, is called a regularizing algorithm with respect to the bounded operator A acting from the Hilbert space Hr into HZ, if 1) IITall<-J,
a>O,
2) Iim T,Av==v,for any VEH~, where vI is the part of v situated in the orthogonal a+0 complement to ker A. The operators T, act from HZ into Hr . We denote by s (A’A) the spectrum of the operator A *A and by El the spectral function of A *A. Let cp(L,a) be a real function measurable with respect to El for any a>O, and cp(0, a) =O. Also, let
We generate a family of operators T, by the formula Ta=\l)(A’A, a) A’. *Zh. vjkhisl. Mat. mat. Fiz., l-3,6, 1596-1598, 1973.
278
(1)
2’79
Short communications
It is of interest to clarify what are the necessary and sufficient conditions which must be imposed on the function cp7in order that the family (1) will possess the properties l), 2). Sufficient conditions were given previously in [l] . The question of necessary conditions was discussed in [3], where a solution was obtained in the case of a discrete spectrum of A *A. It seems to us that the following statement corresponds more precisely to the question posed. Theorem I
For the family (1) to possess the properties 1) and 2), it is necessary and sufficient that the following conditions be satisfied:
vraisup(cp(h,a)]< cqr %a
=,
(3)
lim]g(h, a)- I]= 0 O-DO
(4)
almost everywhere with respect to any measure (&uL, UJ. &oof: Necessity. Condition (2) is necessary for the property l), since l& is the norm of the operator (1) (see [ 1J ). Also A
T,Av
=
s
(5)
rp(4 a) d&v,
0
where A is the upper bound of the spectrum of A *A. Since by hypothesis TaA is a bounded operator,
But if (3) is not satisfied, it would easily be obtained from (5) that sup llZ’,All=m, then by the Banach-Steinhaus theorem the property (2) could not bearsatisfied. Consequently, it is necessary that (3) be satisfied
but
A. B. Bakushinskii
280
We notice that the functions (&u, V)and(E XVI, uL) are identical everywhere, except perhaps for the point h=O; moreover, the function (&uL, VA)is continuous for h=O. We have A
IITc4~u-u1112=
s
[cp(h,a)-
(6)
Ij2d(E*uI,UJ.
0
If Eq. (4) were not satisfied, the left side of Eq. (6) would obviously not tend to zero as a-+0, which would contradict the property 2). Consequently, Eq. (4) is necessary for 2). The sufficiency of the conditions of the theorem is verified in the same way as in [l] with the use in Eq. (6) of Lebesgue’s theorem on passing to the limit under the integral sign. 2. In the formation of T, by Eq. (l), let the perturbed operator x be used in place of the exact operator A. Let the equation As=f be solvable and x be its normal solution orthogonal to ker A (ker A*A)). Let the function cp(A, a) be defined (with respect to A) everywhere on A is the upper bound of the spectrum of d’A, and be continuous.
[0,
A],
where
The following theorem holds, strengthening the statement of section 3 of [l] . T&eorem 2 If the modified conditions (2) - (4) are satisfied (vrai sup is replaced by sup in (2) and (3), and in (4) the convergence is uniform for all h= (c, m) n [0,A] for any arbitrarily small c>O ), the relation Iim a-0,
holds, if
&8+0.
Ilrp(ii’iq coa’f-sll-
0,
(7)
b-0
Here 6-M-AI.
Indeed, lI$(iP”A, a)P’f-zll
~)PKP-*(*if,
a)A’As-AI+ a)A.A;zll&&YMl+II~
(A’&
a)A*Az-zll.
281 To prove Eq. (7) it is merely necessary to establish that
The expression under the limit sign on the left side of Eq. (8) can be written in the form *
J (p,(a, 4 -
1) 2d(&x, 2).
(9)
0
Since slker A’A by hypothesis, the function (I&Z,I) which increases monotonically as 3\. increases, is continuous with respect to 1\ for h=O and (_&x,Z) =O. We fm 00. Let A+ be a point of the positive semi-axis which is not a point of the discrete spectrum of A *A and is such that (J&+z, Z)
A
J
J
(9 (h, 4 - 1)2d (J.&x,x) =c (cPha)-
+
1)2~b%~,X)
(10)
h+
0
MI (EAJ, x) - V&+x,x)1+ M b%x, x) + M(&+x, 2).
Here M=
sup (q(h, u)-I)~. a,A=ro,Al
This quantity is finite by (3). By Rellich’s theorem ([4], p. 396) lim 1(I&.+x,5) - (&*z, 5) I= 0
for xEHl,
a+0
lim @OX,x) = 0, a+0
x E Hi,
x 1 ker A’A.
Moreover, because of the assumption of uniform convergence in Eq. (4) on the left side of (10) the integral between the limits A+ and A tends to zero as a-0 and 6+0. Finally
282
A. B. Bakushinskii
x I ker A’A.
In view of the arbitrariness of e this implies Eq. (8). Theorem 2 is proved. We finally mention that the values of &were calculated in [l] for some specific regularizing algorithms. 7’ranslated by J. Berry REFERENCES 1.
BAKUSHINSKII, A. B., A general method of constructing regularizing algorithms for a linear ill-posed equation in Hilbert space. Zh. vpchisl.Mat. mat. Fiz., 7, 3,672-677, 1967.
2.
TIKHONOV, A. N., The regularization of ill-posed problems. Dokl. Akad. Nauk SSSR, 153,1, 49-52, 1963.
3.
KHUDAK, Yu. I., On the convergence of a family of regularizing algorithms. Zh. vpchisl. Mat. mat. Fiz., 12, 2,497-502, 1972.
4.
RIESZ, F. and NAGY, B. Sz., Lectures on functional analysis(Lektsii po funktslonal’nomu analizu), Izd-vo in, lit., Moscow, 1954.