- Email: [email protected]
The speed of convergence of projection methods for linear equations
Short communications
265
REFERENCES 1.
LEDYANKIN, Yu. Ya. On the accuracy of the transmission of information and its control in digital systems. In: Qbernetic techniques (Kibernetich. tekhn.), Vol. 6,29-39, IK Akad. Nauk Ukr SSSR, Kiev, 1970.
2.
MALINOVSKII, B. N., BOYUN, V. P. and KOZLOV, L. G. Algorithms for the solution of systems of linear algebraic equations, oriented to a structural realization. Upravlyayushchiesistemy i mashiny, $79-84, 1977.
3.
SAMARSKII, A. A. Theory ofdifference schemes (Teoriya raznostnykh &hem), “Nauka”, Moscow, 1977.
U.S.S.R. Comput. Maths. Math. Phys. Vol. 19, pp. 265-272 0 Pergamon Press Ltd. 1980. Printed in Great Britain.
THE SPEED OF CONVERGENCE OF PROJECTION METHODS FOR LINEAR EQUATIONS* A. G. ZARUBIN Khabarovsk (Received 3 May 1978; revised 14 Jury 1978)
THE SPEED of convergence of approximate solutions, constructed by the moment method and by the Bubnov-Gal&in method, for a class of linear equations, are discussed. Applications to boundary value problems for differential equations (ordinary and partial) are given.
1. The method of moments In [ 1,2] the question of the speed of convergence of the moment method was investigated. Below, conditions will be indicated for which estimates of the speed of convergence different from the conditions of [ 1,2] can be obtained. Let E and El be separable Banach spaces, E1 being densely imbedded in E. Let the operators A, K belong to the space L(E1, E), where by L(E1, E) we denote the space of linear continuous operators acting from E, into E and defined on E,. We consider the operator equation Au+Ku=u,
(1.1)
where v is a specified element of E. For the approximate solution of (1.1) we will use the method of moments, which consists of the following. Let Eln be a sequence of closed subspaces of E,, each EIn corresponding to a subspace AEIn = E” C E. On E let a sequence of projection operators P, be defined, each of which projects E on to the corresponding E”. We assume that the projectors P, are uniformly bounded and converge strongly to unity.
*Zh. vjkhisl. Mat. mat. Fiz., 19,4, 1048-1053,
1979.
A. G. Zarubin
266
An approximate solution u, E El” of Eq. (1.1) is found as the solution of the equation
(l-2)
Au,tP,Ku,=P,v.
Let the operators A and K have bounded inverses A- 1 and (A t a-1,
defined on all E.
Lemma If the conditions IIA-'(r-p,)zllE~'cp(n)llzll~
VzsE,
cp(n)+O,
n-+m,
(l-3)
are satisfied, where the constants c and 7, 0 Q r < 1, are independent of the choice of the element u, then for sufficiently large n an operator (A + P&)-l exists, defined on all E, and II (A+PnK)-ill~+~t
where the constant cl is independent of n. Boof: We consider the operator ZtKA-iP,=
(ZtKA-‘)
-KA-‘(Z-P,).
The operator I + KA- 1 has a bounded inverse (z+KA-‘)-‘=A(A+K)-‘=I-K(A+K)-I.
The second term satisfies the inequality IIKA-‘(Z-P,)zllaG$‘-7~(n)llzll~,
which follows from (1.4) and (1.3). We choose n such that 1)(Z+KA-‘)-‘III+EIIKA-~(Z-Pn)
IIE+~<~cp(‘--r)(n)lI (ZtKA-‘)-ill~l.
Then it is known (see [6]) that an operator (I+ KA-lP,,)-l
exists, defined on all E, and
II(Z+KA-'P,)-'IIE-E~C~,
where the constant c2 is independent of n. Moreover, it is easy to see that the operator (ZtP,KA-‘)
-‘=I-P,(Z+KA-‘P,)
-lKA-’
is a linear bounded operator from E into E. Then A +P,,K has an inverse operator A-~(z+P,KA-1) which is bounded. The lemma is proved.
-1,
267
Short communications
Note that the conditions of reversibility given above differ from Theorem 1 of [l] . Theorem 1
When the conditions of the lemma are satisfied the approximate solutions u, exist for fairly large n and converge to the exact solution u of Eq. (1.1) in the norm of the space El. Proof: The solvability of Eqs. (1.2) follows from the lemma, and the existence of u follows from the reversibility of A t K. It is easy to see that the identity
(A+P,K) (u,-u)
= (Pn-I) w,
holds, where w = v - Ku. This and the lemma imply he estimate
The convergence of the approximate solutions in the norm of the space El follows from this inequality. Theorem 2 If the conditions of the lemma are satisfied the speed of convergence of the approximate solutions is given by Ilua-ulle~ (q(n)
+cczcsllP,!III~-‘ll~(‘-‘)
(n)) II PIE,
where c3 is the norm of the imbedding operator El into E. ProoJ: The identity u,-u=
(AtP,K)
--i (P,,-I) W.
holds. This and the form of the operator (A + P,K)- l imply that ~~u,--u~(~=II(A-‘-A-‘P,(Z+KA-IF’,)-‘ICA-l)
(I-P,)z&.
From the last equation and condition (1.3) we have I/n,-u[&cp(n) IIwll~+llA-‘P,(Z+KA-‘P,)-‘KA-‘(Zlp,)wllE. Since the operator A has a bounded inverse from E into El, and El is imbedded in E, therefore
Then
where c3 is the norm of the operator of imbedding of El into E. Estimating the second term by means of (1.3) and (1.4), we complete the proof. In [l] an estimate of the speed of convergence of order cp(n)was obtained for conditions more complex for checking.
268
A. G. Zarubin
2. The Bubnov-Gal&kin method Let the Hilbert space H1 be densely imbedded in the Hilbert space H. We consider the linear operators4 B, acting fromHI intoH. We will assume that D(A) =D(B) =H,~H, then +4and B may be regarded as operators acting in the Hilbert space H. Let the operators A and B be convergent in H [3] and make an acute angle [4]. We consider Eq. (1.1) in H. Let A and A t K .haveinverse bounded operators A- 1 and (A +KF1, acting from N into HI. The reversibility of the operator A t K implies the existence and uniqueness of the solution of Eq. (1.1) for any v of H. We assume that the inverse operator B-l is completely continuous. Let X1, . . . , A,, . . . be the eigenvalues of the operator (O
(2.1)
u,=P,v.
Its exact solution U, E H” is called an approximate solution of Eq. (1 .l), found by the Bubnov-Galkkin method. Theorem 3 Let the space HI be compactly imbedded in H and let the inequality Illr’ull~ G cllAull~~ll&
i--T
,
(2.2)
hold for any element u E HI, where the constants c and 0 < r < 1 are independent of the choice of u. Then Eqs. (2.1) are uniquely solvable for fairly large n and the following identity holds for any arbitrarily small positive E: lim ~~~3lu,-4l~ n+m
= 0.
Proof: Since HI is compactly imbedded in H, then (2.2) implies that the operator KA- 1 is completely continuous in H. Then for sufficiently large n Eqs. (2.1) are uniquely solvable [S] .
The solutions u, satisfy the inequality
(Pn-k,+PnKu,z, Bu,) 3 (mllAu,ll-IIKu,ll)
Il~unll.
If we use the inequality (2.2) and that of Young, then from (2.3) we obtain
(2.3)
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Short communications
(P,Au,+P,Kun,
( >
MUnIl
II&J. 1
l, then r/(1-r)
( > 5
(m-4
(1-z) IlkIll
-
El
LA us take e1 = m(k)-
(
5/(1-z)
7
-c
a
hd
IIAUnlI~IIUII+ c
-5. (
(i-~)Ilu,li=ll4l+
m >
Crll~nll.
(2.4)
We rewrite Eqs. (2.1) in the form Au,+P,Ku,=P,v+
(2.5)
(I-P,)Au,.
Since the operators A and B are similar, the operator A- 1B is bounded [6] , and therefore ll~-‘(z-P,)ull~llA-‘BIIIIB-~(z-P,)ull~
hn;ihl-‘BIlllull.
From this by the lemma we obtain the reversibility of the operator A + P,K. Applying to (2.5) the operator (A + P&)-l, we obtain ll~,ll~II~ll+ll(-4+P,K)-‘(I-P.)Au,ll.
(2.6)
We estimate the second term of the inequality (2.6) using the form of the operator (A + P,K)- l. We have
II(A+P,K)-‘(I-Pn)Aunll G (IIA-‘Bllh,-:1+CC2C~llA-‘IllIA-~BIl~-~h~~;+:)
IIAU,,II.
Then (2.4), (2.6), (2.7) imply the inequality
( > $
II~unll~llvIIt
cc~IIuII
+ ~~(ll~-‘~llh,;l,+~~z~~II~-‘IIIIA-’BII’-Th:IS:) This implies the uniform boundedness of the norm IIAu, II.
Let u be the exact solution of Eq. (1.1). Then the identity (A+PnK) (G-U) = (Pn-I)
is satisfied, where w,=v-Ku-Au,
w,,,
IIAunll.
(2.7)
270
A. G. Zarubin
By what was said above the norms 11w, 11H are uniformly bounded. It remains to invert the operator (A + P,K) and reason as in the proof of Theorem 2. Theorem 3 is a generalization of the results obtained in [5] .
3. Applications to boundary value problems Let us consider the differential equation
O
(3.1)
i=l, 2 ,...,m.
(3.2)
with the linear homogeneous boundary conditions m--l
J-l lai,ucr)(-l)+Pi,U(r)(l)
]=I),
We will consider that the function v(t) E L2 [-1,1 ] , and the coefficients pr(t) are continuous. We assume that the equation @)= 0 with the boundary conditions (3.2) has only a trivial solution and problem (3.1), (3.2) is uniquely solvable for any function v(t) of L, [-1, 11. We put (I
Au=zdm),
Ku=
pi(t)u(i)(t). I!c i=o
We denote by W,* [-1, l] a set of functions u(t) absolutely continuously differentiable (m - l)-times (*I(t) belongs to L, [-1, 11. We denote by W 2m [-1, l] the aggregate on [-1, l],andsuchthatu of all the functions of Wz* [-1, l] satisfying the conditions (3.2). This will be a subspace of the space W,*[-1, 1].LetE=L2,EI = W,*. If we denote by Eln theset of all polynomials of degree not higher than n + m, then E” = AE, n is the set of polynomials of degree not higher than n. Let P,, be an operator establishing a connection between each function of L, [-1, l] and a segment of its Fourier series in Legendre polynomials. Then it is known that the following estimate holds: IM-‘(I-+7%)ullL,~C5~-mllUII~,. Since the coefficients pi(t) are continuous, we have IlWl~6
sup max Ipi (t) I IIUIIW~T i
t
Applying multiplicative inequalities, we obtain IIKLuI~~~~c~IIu~~~~~~~~IluilL~~8’m. All the condition of Theorem 2 are satisfied. Accordingly,
271
Short communications
Estimates of more rapid convergence were obtained in [ 1,2] on the assumption that the coefficients p,(t) are continuously differentiable t times. The case where the p&t) are only continuous was not investigated in [ 1, 21. We consider the Dirichlet problem for second-order elliptic equations in a bounded domain St C Rs with smooth boundary Xl: a
au
as_
W(X) --g
I
-z i,j=i
( ’
8
ai ) I2 i=i +
I
(4
$+a(x)u=v(x).
(3.3)
1
UI ap=O.
(3.4)
We will assume that the coefficients arp ai and a are such that probl%m(3.3), (3.4bis uniquely solvable for any function v(x) E L2(s2), where the solution U(X)E WS2(CI),and Wz2(CZ)is a subspace of the space W22(CI).We write I
Au=i,j=t
i-i
Let aii(x) = a&i). We consider the operator BU = - Au. By what was said above the operators A and B are similar on D(A) =D(B)=ti2z(Q)d,z(S2). Moreover, they form an acute angle (see [7]). Also, if we assume that ai and a(x) are continuous, then
All the conditions of Theorem 3 are satisfied if H” is the linear shell of the functions ql, . . . , tpn, and the pi are the eigenfunctions of the Dirichlet problem for the Laplace equation. By Theorem 3 the estimate of the speed of convergence has the form
This estimate is weaker than the result obtained in [3]. But if the operator K has the form Ku = a(x)u, the following estimate holds:
The author thanks M. A. Krasnosel’skii for discussing the results. Translated by J. Berry.
272
F. P. Vasii’ev,M. A. Vorontsov and 0. A. Litvinova REFERENCES
1.
DAUGAVET, I. K. On the method of moments for ordinary differentiaf equations. Sibirskii matem. zh., 6,1, 70-85, 1965.
2.
VAINIKKO, G. M. On the speed of convergence of the method of moments for ordinary differential equations. Sibirskii matem. zh., 9, 1, 21-28, 1968.
3.
MIKHLIN, S. G. Variationalmethods in mathematicalphysics (Variatsionye metody v matematicheskoi mike), “Nauka”, Moscow, 1970.
4.
SOBOLEVSKII, P. E. On equations with operators forming an acute angle. Dokl. Akad. Nauk SSSR, 116,5,X4-X1,1957.
5.
SOBOLEVSKII, P. E. On a comparison principle in the theory of approximate methods. Proceedings of a seminar on functional analysis (Tr. seminara po funkts. analizu), No. 11, 244-25 1, Izd-vo VGU, Voronezh, 1968.
6.
KRACNOSEL’SKII, M. A. et al. The approximate solution of operator equations (Priblizhennoe reshenie operatornykh uravnenii), “Nauka”, Moscow, 1969.
I.
LADYZHENSKAYA, 0. A. and URAL’TSEVA, N. N. Linear and quasilinearelliptic equations (Lineinye i kvazilineinye uravneniya eilipticheskogo tipa), “Nat&a”, Moscow, 1964.
iJ.SS.R. Comput. Maths. Math. Phys. Vol. 19, pp. 212-218 0 Pergamon Press Ltd. 1980. Printed In Great Britain.
0041-5553/0801-0272$07.50/O
ON THE OPTIMAL CONTROL OF THE PROCESS OF THERMAL BLOOMING* F. P. VASIL’EV, M. A. VORONTSOV and 0. A. LITVINOVA
Moscow (Received 3 Jury 1978)
A GRADIENT method of optimizing the solution of the problem of the spread of a beam of light in conditions of thermal blooming is presented. The results of numerical calculations are given.
1. Physical statement of the problem It is well-known [ I] that the thermal blooming of light beams spreading in non-linear media leads to considerable spreading of the beam and the shifting of its energy centre. In many cases distortions of this kind are undesirable. Existing methods of compensating for them are based on a preliminary choice of the initial amplitude and phase profile of the beam, which in a number of cases enables non-linear distortions to be reduced [2]. In this connection it is of considerable interest to find the optimal initial profde of the beam. We will consider a fairly general formulation of the problem of the optimal control of the initial phase front of a beam of light, propagating in a medium without loss in Conditions of unsteady thermal blooming, on the assumption that the medium moves along the x-axis at constant speed.
lZh. vghisl. Mat. mat. Fiz., 19,4, 1053-1058,
1979.
265
REFERENCES 1.
LEDYANKIN, Yu. Ya. On the accuracy of the transmission of information and its control in digital systems. In: Qbernetic techniques (Kibernetich. tekhn.), Vol. 6,29-39, IK Akad. Nauk Ukr SSSR, Kiev, 1970.
2.
MALINOVSKII, B. N., BOYUN, V. P. and KOZLOV, L. G. Algorithms for the solution of systems of linear algebraic equations, oriented to a structural realization. Upravlyayushchiesistemy i mashiny, $79-84, 1977.
3.
SAMARSKII, A. A. Theory ofdifference schemes (Teoriya raznostnykh &hem), “Nauka”, Moscow, 1977.
U.S.S.R. Comput. Maths. Math. Phys. Vol. 19, pp. 265-272 0 Pergamon Press Ltd. 1980. Printed in Great Britain.
THE SPEED OF CONVERGENCE OF PROJECTION METHODS FOR LINEAR EQUATIONS* A. G. ZARUBIN Khabarovsk (Received 3 May 1978; revised 14 Jury 1978)
THE SPEED of convergence of approximate solutions, constructed by the moment method and by the Bubnov-Gal&in method, for a class of linear equations, are discussed. Applications to boundary value problems for differential equations (ordinary and partial) are given.
1. The method of moments In [ 1,2] the question of the speed of convergence of the moment method was investigated. Below, conditions will be indicated for which estimates of the speed of convergence different from the conditions of [ 1,2] can be obtained. Let E and El be separable Banach spaces, E1 being densely imbedded in E. Let the operators A, K belong to the space L(E1, E), where by L(E1, E) we denote the space of linear continuous operators acting from E, into E and defined on E,. We consider the operator equation Au+Ku=u,
(1.1)
where v is a specified element of E. For the approximate solution of (1.1) we will use the method of moments, which consists of the following. Let Eln be a sequence of closed subspaces of E,, each EIn corresponding to a subspace AEIn = E” C E. On E let a sequence of projection operators P, be defined, each of which projects E on to the corresponding E”. We assume that the projectors P, are uniformly bounded and converge strongly to unity.
*Zh. vjkhisl. Mat. mat. Fiz., 19,4, 1048-1053,
1979.
A. G. Zarubin
266
An approximate solution u, E El” of Eq. (1.1) is found as the solution of the equation
(l-2)
Au,tP,Ku,=P,v.
Let the operators A and K have bounded inverses A- 1 and (A t a-1,
defined on all E.
Lemma If the conditions IIA-'(r-p,)zllE~'cp(n)llzll~
VzsE,
cp(n)+O,
n-+m,
(l-3)
are satisfied, where the constants c and 7, 0 Q r < 1, are independent of the choice of the element u, then for sufficiently large n an operator (A + P&)-l exists, defined on all E, and II (A+PnK)-ill~+~t
where the constant cl is independent of n. Boof: We consider the operator ZtKA-iP,=
(ZtKA-‘)
-KA-‘(Z-P,).
The operator I + KA- 1 has a bounded inverse (z+KA-‘)-‘=A(A+K)-‘=I-K(A+K)-I.
The second term satisfies the inequality IIKA-‘(Z-P,)zllaG$‘-7~(n)llzll~,
which follows from (1.4) and (1.3). We choose n such that 1)(Z+KA-‘)-‘III+EIIKA-~(Z-Pn)
IIE+~<~cp(‘--r)(n)lI (ZtKA-‘)-ill~l.
Then it is known (see [6]) that an operator (I+ KA-lP,,)-l
exists, defined on all E, and
II(Z+KA-'P,)-'IIE-E~C~,
where the constant c2 is independent of n. Moreover, it is easy to see that the operator (ZtP,KA-‘)
-‘=I-P,(Z+KA-‘P,)
-lKA-’
is a linear bounded operator from E into E. Then A +P,,K has an inverse operator A-~(z+P,KA-1) which is bounded. The lemma is proved.
-1,
267
Short communications
Note that the conditions of reversibility given above differ from Theorem 1 of [l] . Theorem 1
When the conditions of the lemma are satisfied the approximate solutions u, exist for fairly large n and converge to the exact solution u of Eq. (1.1) in the norm of the space El. Proof: The solvability of Eqs. (1.2) follows from the lemma, and the existence of u follows from the reversibility of A t K. It is easy to see that the identity
(A+P,K) (u,-u)
= (Pn-I) w,
holds, where w = v - Ku. This and the lemma imply he estimate
The convergence of the approximate solutions in the norm of the space El follows from this inequality. Theorem 2 If the conditions of the lemma are satisfied the speed of convergence of the approximate solutions is given by Ilua-ulle~ (q(n)
+cczcsllP,!III~-‘ll~(‘-‘)
(n)) II PIE,
where c3 is the norm of the imbedding operator El into E. ProoJ: The identity u,-u=
(AtP,K)
--i (P,,-I) W.
holds. This and the form of the operator (A + P,K)- l imply that ~~u,--u~(~=II(A-‘-A-‘P,(Z+KA-IF’,)-‘ICA-l)
(I-P,)z&.
From the last equation and condition (1.3) we have I/n,-u[&cp(n) IIwll~+llA-‘P,(Z+KA-‘P,)-‘KA-‘(Zlp,)wllE. Since the operator A has a bounded inverse from E into El, and El is imbedded in E, therefore
Then
where c3 is the norm of the operator of imbedding of El into E. Estimating the second term by means of (1.3) and (1.4), we complete the proof. In [l] an estimate of the speed of convergence of order cp(n)was obtained for conditions more complex for checking.
268
A. G. Zarubin
2. The Bubnov-Gal&kin method Let the Hilbert space H1 be densely imbedded in the Hilbert space H. We consider the linear operators4 B, acting fromHI intoH. We will assume that D(A) =D(B) =H,~H, then +4and B may be regarded as operators acting in the Hilbert space H. Let the operators A and B be convergent in H [3] and make an acute angle [4]. We consider Eq. (1.1) in H. Let A and A t K .haveinverse bounded operators A- 1 and (A +KF1, acting from N into HI. The reversibility of the operator A t K implies the existence and uniqueness of the solution of Eq. (1.1) for any v of H. We assume that the inverse operator B-l is completely continuous. Let X1, . . . , A,, . . . be the eigenvalues of the operator (O
(2.1)
u,=P,v.
Its exact solution U, E H” is called an approximate solution of Eq. (1 .l), found by the Bubnov-Galkkin method. Theorem 3 Let the space HI be compactly imbedded in H and let the inequality Illr’ull~ G cllAull~~ll&
i--T
,
(2.2)
hold for any element u E HI, where the constants c and 0 < r < 1 are independent of the choice of u. Then Eqs. (2.1) are uniquely solvable for fairly large n and the following identity holds for any arbitrarily small positive E: lim ~~~3lu,-4l~ n+m
= 0.
Proof: Since HI is compactly imbedded in H, then (2.2) implies that the operator KA- 1 is completely continuous in H. Then for sufficiently large n Eqs. (2.1) are uniquely solvable [S] .
The solutions u, satisfy the inequality
(Pn-k,+PnKu,z, Bu,) 3 (mllAu,ll-IIKu,ll)
Il~unll.
If we use the inequality (2.2) and that of Young, then from (2.3) we obtain
(2.3)
269
Short communications
(P,Au,+P,Kun,
( >
MUnIl
II&J. 1
l, then r/(1-r)
( > 5
(m-4
(1-z) IlkIll
-
El
LA us take e1 = m(k)-
(
5/(1-z)
7
-c
a
hd
IIAUnlI~IIUII+ c
-5. (
(i-~)Ilu,li=ll4l+
m >
Crll~nll.
(2.4)
We rewrite Eqs. (2.1) in the form Au,+P,Ku,=P,v+
(2.5)
(I-P,)Au,.
Since the operators A and B are similar, the operator A- 1B is bounded [6] , and therefore ll~-‘(z-P,)ull~llA-‘BIIIIB-~(z-P,)ull~
hn;ihl-‘BIlllull.
From this by the lemma we obtain the reversibility of the operator A + P,K. Applying to (2.5) the operator (A + P&)-l, we obtain ll~,ll~II~ll+ll(-4+P,K)-‘(I-P.)Au,ll.
(2.6)
We estimate the second term of the inequality (2.6) using the form of the operator (A + P,K)- l. We have
II(A+P,K)-‘(I-Pn)Aunll G (IIA-‘Bllh,-:1+CC2C~llA-‘IllIA-~BIl~-~h~~;+:)
IIAU,,II.
Then (2.4), (2.6), (2.7) imply the inequality
( > $
II~unll~llvIIt
cc~IIuII
+ ~~(ll~-‘~llh,;l,+~~z~~II~-‘IIIIA-’BII’-Th:IS:) This implies the uniform boundedness of the norm IIAu, II.
Let u be the exact solution of Eq. (1.1). Then the identity (A+PnK) (G-U) = (Pn-I)
is satisfied, where w,=v-Ku-Au,
w,,,
IIAunll.
(2.7)
270
A. G. Zarubin
By what was said above the norms 11w, 11H are uniformly bounded. It remains to invert the operator (A + P,K) and reason as in the proof of Theorem 2. Theorem 3 is a generalization of the results obtained in [5] .
3. Applications to boundary value problems Let us consider the differential equation
O
(3.1)
i=l, 2 ,...,m.
(3.2)
with the linear homogeneous boundary conditions m--l
J-l lai,ucr)(-l)+Pi,U(r)(l)
]=I),
We will consider that the function v(t) E L2 [-1,1 ] , and the coefficients pr(t) are continuous. We assume that the equation @)= 0 with the boundary conditions (3.2) has only a trivial solution and problem (3.1), (3.2) is uniquely solvable for any function v(t) of L, [-1, 11. We put (I
Au=zdm),
Ku=
pi(t)u(i)(t). I!c i=o
We denote by W,* [-1, l] a set of functions u(t) absolutely continuously differentiable (m - l)-times (*I(t) belongs to L, [-1, 11. We denote by W 2m [-1, l] the aggregate on [-1, l],andsuchthatu of all the functions of Wz* [-1, l] satisfying the conditions (3.2). This will be a subspace of the space W,*[-1, 1].LetE=L2,EI = W,*. If we denote by Eln theset of all polynomials of degree not higher than n + m, then E” = AE, n is the set of polynomials of degree not higher than n. Let P,, be an operator establishing a connection between each function of L, [-1, l] and a segment of its Fourier series in Legendre polynomials. Then it is known that the following estimate holds: IM-‘(I-+7%)ullL,~C5~-mllUII~,. Since the coefficients pi(t) are continuous, we have IlWl~6
sup max Ipi (t) I IIUIIW~T i
t
Applying multiplicative inequalities, we obtain IIKLuI~~~~c~IIu~~~~~~~~IluilL~~8’m. All the condition of Theorem 2 are satisfied. Accordingly,
271
Short communications
Estimates of more rapid convergence were obtained in [ 1,2] on the assumption that the coefficients p,(t) are continuously differentiable t times. The case where the p&t) are only continuous was not investigated in [ 1, 21. We consider the Dirichlet problem for second-order elliptic equations in a bounded domain St C Rs with smooth boundary Xl: a
au
as_
W(X) --g
I
-z i,j=i
( ’
8
ai ) I2 i=i +
I
(4
$+a(x)u=v(x).
(3.3)
1
UI ap=O.
(3.4)
We will assume that the coefficients arp ai and a are such that probl%m(3.3), (3.4bis uniquely solvable for any function v(x) E L2(s2), where the solution U(X)E WS2(CI),and Wz2(CZ)is a subspace of the space W22(CI).We write I
Au=i,j=t
i-i
Let aii(x) = a&i). We consider the operator BU = - Au. By what was said above the operators A and B are similar on D(A) =D(B)=ti2z(Q)d,z(S2). Moreover, they form an acute angle (see [7]). Also, if we assume that ai and a(x) are continuous, then
All the conditions of Theorem 3 are satisfied if H” is the linear shell of the functions ql, . . . , tpn, and the pi are the eigenfunctions of the Dirichlet problem for the Laplace equation. By Theorem 3 the estimate of the speed of convergence has the form
This estimate is weaker than the result obtained in [3]. But if the operator K has the form Ku = a(x)u, the following estimate holds:
The author thanks M. A. Krasnosel’skii for discussing the results. Translated by J. Berry.
272
F. P. Vasii’ev,M. A. Vorontsov and 0. A. Litvinova REFERENCES
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LADYZHENSKAYA, 0. A. and URAL’TSEVA, N. N. Linear and quasilinearelliptic equations (Lineinye i kvazilineinye uravneniya eilipticheskogo tipa), “Nat&a”, Moscow, 1964.
iJ.SS.R. Comput. Maths. Math. Phys. Vol. 19, pp. 212-218 0 Pergamon Press Ltd. 1980. Printed In Great Britain.
0041-5553/0801-0272$07.50/O
ON THE OPTIMAL CONTROL OF THE PROCESS OF THERMAL BLOOMING* F. P. VASIL’EV, M. A. VORONTSOV and 0. A. LITVINOVA
Moscow (Received 3 Jury 1978)
A GRADIENT method of optimizing the solution of the problem of the spread of a beam of light in conditions of thermal blooming is presented. The results of numerical calculations are given.
1. Physical statement of the problem It is well-known [ I] that the thermal blooming of light beams spreading in non-linear media leads to considerable spreading of the beam and the shifting of its energy centre. In many cases distortions of this kind are undesirable. Existing methods of compensating for them are based on a preliminary choice of the initial amplitude and phase profile of the beam, which in a number of cases enables non-linear distortions to be reduced [2]. In this connection it is of considerable interest to find the optimal initial profde of the beam. We will consider a fairly general formulation of the problem of the optimal control of the initial phase front of a beam of light, propagating in a medium without loss in Conditions of unsteady thermal blooming, on the assumption that the medium moves along the x-axis at constant speed.
lZh. vghisl. Mat. mat. Fiz., 19,4, 1053-1058,
1979.