The Optimal Order of Convergence for Stable Evaluation ... - CiteSeerX

Electronic Journal of Differential Equations

Vol. 1993(1993), No. 04, pp. 1-10. Published October 14, 1993. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110 or 129.120.3.113

The Optimal Order of Convergence for Stable Evaluation of Dierential Operators  C.W. Groetsch and O. Scherzer

Abstract

An optimal order of convergence result, with respect to the error level in the data, is given for a Tikhonov-like method for approximating values of an unbounded operator. It is also shown that if the choice of parameter in the method is made by the discrepancy principle, then the order of convergence of the resulting method is suboptimal. Finally, a modi ed discrepancy principle leading to an optimal order of convergence is developed.

1 Introduction

Suppose that L : D(L)  H1 ! H2 is a closed densely de ned unbounded linear operator from a Hilbert space H1 into a Hilbert space H2 . The problem of computing values y = Lx, for x 2 D(L), is then ill{posed in the sense that small perturbations in x may lead to data x satisfying kx ? x k  , but x 2= D(L), or, even if x 2 D(L), it may happen that Lx != Lx as  ! 0, since the operator L is unbounded. Morozov has studied a stable method for approximating the value Lx when only approximate data x is available (see [7] for information on Morozov's work). This method takes as an approximation to y = Lx the vector y = Lz , where z minimizes the functional

kz ? x k + kLz k 2

over D(L): This is equivalent to

2

( > 0)

y = L(I + L L)?1 x :

(1.1) (1.2)

 1991 Mathematics Subject Classi cations: Primary 47A58, Secondary 65J70. Key words and phrases: Regularization, unbounded operator, optimal convergence, stable.

c 1993 Southwest Texas State University and University of North Texas Submitted: June 14, 1993. Supported by the Austrian Fonds fur Forderung der wissenschaftlichen Forschung, project P7869-PHY, the Christian Doppler Society, Austria (O.S.), and NATO project CRG-930044 (C.W.G.)

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Morozov shows that if  = () ! 0 as  ! 0, in such a way that p ! 0, then y ! Lx as  ! 0. He also develops an a posteriori method, the discrepancy principle, for choosing the parameter , depending on the data x , that leads to a stable convergent approximation scheme for Lx. As a simple concrete example of this type of approximation, consider dierentiation in L2(R). That is, the operator L is de ned on H 1 (R), the Sobolev space of functions possessing a weak derivative in L2(R), by Lx = x0 . For a given data function x 2 L2 (R) satisfying kx ? x k  , the stabilized approximate derivative (1.2) is easily seen (using Fourier transform analysis) to be given by Z 1  y (s) =  (s ? t)x (t) dt where the kernel  is given by

?1

p

(t)  (t) = ? sign 2 exp(?jtj= ):

Another concrete example of this stable evaluation method is provided by the Dirichlet to Neumann map. Consider for simplicity the unit disk D and unit circle @D. For a given function g on @D we denote by u the function which is harmonic in D and takes boundary values g. The operator L is then de ned @u . To be more speci c, L is the closed operator de ned on the dense by Lg = @n subspace ( ) X D(L) = g 2 L2 (@D) : jnj2 jg^(n)j2 < 1 of L (@D) by

n2Z

2

where

(Lg)(ei ) =

X

n2Z

g^(n) = 21

jnjg^(n) exp(in)

Z 2 0

g(t)e?int dt :

The stable approximation (1.2) for Lx, given approximate data x , is then

y (ei ) =

X

n2Z





n ^ 1 + n2 x (n) exp(in):

Our aim is to provide an order of convergence result for fy g and to show that this order of convergence is essentially best possible. Our approach, which is inspired by a work of Lardy [6] on generalized inverses of unbounded operators, is based on spectral analysis of certain bounded operators associated with L (see also [5] where other consequences of this approach are investigated). We also determine the best possible order of convergence when the discrepancy principle

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is used to determine . This order of convergence turns out to be suboptimal. For results of a similar nature pertaining to Tikhonov regularization for solving rst kind equations involving bounded operators, see [4, Chapter 3]. Finally, we propose a modi cation of the discrepancy principle for approximating values of an unbounded operator that leads to an optimal convergence rate.

2 Order of Convergence To establish the order of convergence of (1.2) it will be convenient to reformulate (1.2) as y = LL [I + (1 ? )L ]?1 x (2.1) where L = (I + L L)?1 and LL are known to be bounded everywhere de ned linear operators and L is self{adjoint with spectrum (L )  [0; 1] (see, e.g. [8, p.307]). Because x in (2.1) is operated upon by a product of bounded operators, we see that for xed  > 0, y depends continuously on x , that is, the approximations fy g are stable. The representation (2.1) has certain advantages in that the dependence of y on bounded operators (L and LL ), which are independent of , is explicit. To further simplify the presentation, we introduce the functions We then have

T (t) = ( + (1 ? )t)?1 ;  > 0; t 2 [0; 1]: y = LL T (L )x :

The approximation with no data error will be denoted by y :

y = LL T(L )x:

Theorem 2.1 If x 2? D2
(LLL) and  = () satis es 23 ! C > 0 as  ! 0, then ky ? Lxk = O  3 : Proof: Let w = (I + LL)Lx: Then Lx = L^ w, where L^ = (I + LL)? . 1

Since

LL = L(I + L L)?1 = (I + LL)?1 L = L^ L and Lx = (I + LL)?1 w = L^ w, we obtain from (2.1) ?




y ? Lx = L L ? [I + (1 ? )L ] [I + (1 ? )L ]?1 x = L(L ? I )[I + (1 ? )L ]?1 x = (L^ ? I )[I + (1 ? )L^ ]?1 Lx = (L^ ? I )T (L^ )L^ w:

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Since kT(L^ )L^ k  1, we nd that

ky ? Lxk = O():

Also,

(2.2)

ky ? y k = (L LL T (L )(x ? x); L T (L )(x ? x)) = ((I ? L )T2 (L )(x ? x); L T(L )(x ? x))  kI ? L k  since kT (L )k   : Therefore, 2

(2.3)

1

  ky ? y k = O p :

We then have

ky ? Lxk = O() + O





p = O  23 ; ?




since 2 ! C > 0: 2  L) on the This theorem shows that under the regularity condition x 2 D ( LL  exact data the order of convergence O  32 is attainable by the approximation (2.1) using approximate data with error level . In the next section we show that this order is best possible, except for the trivial case when x 2 N (L), i.e., when Lx = 0. 3

3 Optimality

?




We begin by showing that any improvement in the order O  32 entails a certain convergence rate for the parameter . ?
Lemma 3.1 ?If 2x
2= N (L) and ky ? Lxk = o  32 for all x satisfying kx ? x k  , then  = o  3 : Proof: Let x = x ? u, where u is a unit vector and let e = y ? Lx. Then ?
[I + (1 ? )L^ ]e = [I + (1 ? )L^ ] LL (I + (1 ? )L )?1 x ? Lx ? [I + (1 ? )L^ ]LL (I + (1 ? )L )?1 u = (L^ ? I )Lx ? LL u: ?




Since kek = o  32 , by assumption, and since ?
kLL uk  kLLk = o  32 ;

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C.W. Groetsch and O. Scherzer

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we nd that

 k(L^ ? I )Lxk ! 0; as  ! 0:  23   But x 2= N (L) = N ((L^ ? I )L) and hence  = o  32 : We now

2

show that for a wide class of operators the order of convergence

O  32 can not be improved. We will consider the important class of operators L L which have a divergent sequence of eigenvalues. Such is the case if L is the derivative operator, when ?L L is the Laplacian operator, or more generally whenever L is a dierential operator for which L is compact. 



Theorem 3.1 If LL has eigenvalues n ! 1 and ky ? Lxk = o  23 for all x with kx ? x k  , then x 2 N (L). Proof: If x 2= N (L), then  = o? 23
, by Lemma 3.1. Let e = y ? Lx, then

kek = ky ? Lxk + 2(y ? Lx; y ? y ) + ky ? y k 2 and by hypothesis ky ? 34Lxk ! 0 as  ! 0 (since x = x satis es kx ? x k  ). 2

2

2

Therefore we must have 2(y ? Lx; y ? y ) + ky ? y k2 ! 0 as  ! 0: 4

3

(3.1)

Suppose that fun g are orthonormal eigenvectors of LL associated with fn g. Then fung are eigenvectors of L associated with the eigenvalues n = 1+1n and n ! 0 as n ! 1. Now let x = x + un. Then
? ky ? y k2 = 2 L (I + (1 ? )L )?1 un; L LL (I + (1 ? )L )?1 un = 2 2n n ( + (1 ? )n )?2 = 2 n (1 ? n )( + (1 ? )n )?2 : 3

Therefore, if  = n = n2 , then n ! 0 as n ! 1 and

kyn ? y k = (1 ?  )  + 1 ?  n 4 2 3 3 2

n

n

!?2

! 1 as n ! 1:

(3.2)

Finally, we have

j(y ? Lx; yn ? y )j  ky ? Lxk kyn ? y k ! 0: 4

n3

2

n3

2

n3

This, along with (3.2), contradicts (3.1) and hence x 2 N (L).

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4 The Discrepancy Principle We may write the approximation y to Lx as y = Lz where z = L T (L )x : (4.1) Morozov [7, p.125] has shown that if kx k >  (i.e., the signal{to{noise ratio is greater than one), then there is a unique  = () > 0 such that kz () ? x k = : (4.2) Moreover, he showed that y () ! Lx as  ! 0. We now provide an order of convergence for this method and show that, in general, it can not be improved. p Theorem 4.1 If x 2 D(LL) and x 2= N (L), then ky () ? Lxk = O( ). Proof: First note that [I + (1 ? )L ](z ? x ) = (L ? I )x : Moreover, note that (cf. (2.1)) kI + (1 ? )L k  1: Therefore, if  is chosen by (4.2), then k(L ? I )x k  kz ? x k =  and hence k(L ? I )x k  () : Since x 2= N (L), we have x 2= N (L ? I ) and hence

0 < k(L ? I )xk  lim inf  : !0 ( )

We therefore have

 = O(): (4.3) Since minimizes (1.1) over D(L) and () satis es (4.2), it follows that 2 + ()kLz () k2 = kz () ? x k2 + ()kLz () k2  kx ? x k2 + ()kLxk2  2 + ()kLxk2 and hence kLz ()k  kLxk. We then have z

ky  ? Lxk = ky  k ? 2(y  ; Lx) + kLxk  2(Lx ? y  ; Lx) = 2(x ? z  ; L Lx)  4kLLxk ( )

2

( )

2

( )

( )

2

( )

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C.W. Groetsch and O. Scherzer

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p

and hence ky () ? Lxk = O( ): 2 It turns out that if the parameter is chosen by the discrepancy method (4.2), then the order of convergence derived in Theorem 4.1 can not be improved in general. To see this, suppose that L has a sequence of eigenvalues n ! 0 and thatfun g is a corresponding sequence of orthonormal eigenvectors. Furthermore, let n = 1+1n ,x = u1 , and xn = u1 + n un : An easy calculation then gives

kyn ? Lxk  ( + (1?n ) ) n n : (4.4) n Now set n = nn 2 , then n ! 0 as n ! 1. We will show that if  satis es p (4.3), then kyn ? Lxk = o( n ) is not possible. Indeed, if this were the case, 2

(1+

2

2

2

)

then by (4.4) we have

 + (1 ? ) n ?2 =  2  ( + (1 ? ) )?2 ! 0 n n n n n n and hence n + (1 ? ) nn ! 1. But if  is chosen by (4.2), then by (4.3), is bounded and hence nn ! 1. But nn = 1n + 1 ! 1; a contradiction. 



 n

In the next section we show how the discrepancy principle can be modi ed to recover the optimal order of convergence.

5 Optimal Discrepancy Methods Engl and Gfrerer [1],[2],[3] have developed discrepancy principles of optimal order for approximating solutions of bounded linear operator equations of the rst kind by Tikhonov regularization. In this section we investigate similar principles for approximating values of unbounded linear operators. We begin by considering the function () = 2 kz ? x k: By using a spectral representation of the operator L T (L ) which de nes z via (4.1), it is easy to see that the function  ! () is continuous, strictly increasing and satis es lim () = 0 and lim !1 () = 1 !0+

(we assume that x 62 N (L), for otherwise the approximations are trivial). Therefore, there is a unique  = () > 0 satisfying

kz ? x k =  : 2

2

(5.1)

We will show that the modi ed discrepancy principle (5.1) leads, under suitable conditions, to an optimal order of convergence for the approximations y to Lx.

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Theorem 5.1 Suppose2 x 2 D(L L) and x 2= N (L). If  = () is chosen by condition (5.1), then 3  ! kL Lxk > 0 as  ! 0. Proof: To simplify notation we set  = () in the proof. First we show that  ! 0 as  ! 0. Since [I + (1 ? )L ](z ? x ) = (L ? I )x and kL k  1; ( )

we have

2 k(L ? I )x k  kz ? x k =  2 : (5.2) Also, (L ? I )x ! (L ? I )x 6= 0 as  ! 0 since L x = x implies L Lx = 0, i.e., x 2 N (L). Therefore, from (5.2), we nd that  ! 0 as  ! 0. Next we show that  ! 0 as  ! 0. In fact, kz ? z k = kL T (L )(x ? x )k  

and

x ? z = x ? L T(L )x ! 0 as  ! 0 since N (L ) = f0g and tT (t) ! 1 as  ! 0 for each t 6= 0. Therefore kx ? z k ! 0 as  ! 0. We then have 2 = kz  ? x k  kz  ? xk +  ! 0 as  ! 0   2 and hence  ! 0 as  ! 0: We can now show that LLz ! LLx as  ! 0. Indeed,

and

LLz ? L Lx = (L T (L ) ? I )L Lx ! 0 as  ! 0

(5.3)

LL(z ? z ) = (I ? L )T (L )(x ? x);

therefore,

kLL(z ? z )k  k(I ? L )k  : (5.4) Since kT(L )k   . But, since  ! 0; we nd from (5.3) and (5.4) that LLz ! L Lx as  ! 0: 1

Finally, we have

and hence, by (5.1)

x ? z = L Lz 2 = kL Lz  k ! kLLxk as  ! 0:  3

From Theorem 2.1 and 5.1 we immediately obtain

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 L), x 2= N (L) and  = () is chosen by (5.1), Corollary 5.1 If x 2 ?D(2LL
 then ky  ? Lxk = O  3 : The Corollary requires the \smoothness" condition x 2 D(LL L) in order ( )

to guarantee the optimal convergence rate, but it is possible to obtain a \quasi{ optimal" rate without any additional smoothness assumptions on the data x. It follows from the proof of Theorem 2.1 (speci cally, from (2.3)), that 1 ky ? Lxk2  ky ? Lxk2 + C 2 : (5.5)  2   Let m(x; ) be the in mum, over  > 0, of the right hand side of (5.5). It is possible, following ideas of Engl and Gfrerer [2], to choose a parameter  = () such that ky () ? Lxk2 has the same order as m(x; ) which we call the quasioptimal rate. In fact, minimizing the right hand side of (5.5) leads to a condition of the form ?
f (; x) := [(I ? L )T (L )]3 x; x = C2 : If we denote the spectral resolution of the identity generated by the operator L by fE :  2 [0; 1]g, then Z 1 (1 ? ) 3 dkE z k2 : f (; z ) =  (1 ? ) +  0 >From this it follows that for any z 62 N (L), f (
; z ) is a monotonically increasing continuous function satisfying 2 lim f (; z ) = 0 and lim !1 f (; z ) = kPz k ; !0 where P is the orthogonal projector of H1 onto N (L)? . Therefore, for any  > 0 and x 62 N (L) and any positive constant which is dominated by the signal-to-noise ratio of data x , that is, satisfying 0 < < kPx k=; there is a unique choice of the parameter  = () satisfying f ((); x ) = ( )2 : It can be shown, but we will not provide the details, that, this a posteriori choice of the parameter always leads to the quasi-optimal rate ky ?Lxk2 = O(m(x; )), without any additional smoothness assumptions on the data x.

References [1] H.W. Engl, Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates, J. Optimiz. Theory Appl. 49(1987), 209{215. MR 88b:49045.

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[2] H.W. Engl and H. Gfrerer, A posteriori parameter choice for general regularization methods for solving linear ill{posed problems, Appl. Numer. Math. 4(1988), 395{417. MR 89i:65060. [3] H. Gfrerer, An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill{posed problems leading to optimal convergence rates, Math. Comp. 49(1987), 507{522. MR 88k:65049. [4] C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, London, 1984. [5] C.W. Groetsch, Spectral methods for linear inverse problems with unbounded operators, J. Approx. Th. 70(1992), 16-28. [6] L.J. Lardy, A series representation of the generalized inverse of a closed linear operator,Att. Accad. Naz. Lincei Cl. Sci. Mat. Natur., Ser VIII, 58(1975), 152{157. MR 48# 13540. [7] V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer{ Verlag, New York, 1984. [8] F. Riesz and B. Sz{Nagy, Functional Analysis, Ungar, New York, 1955. Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025 USA E-mail: groetschucbeh.san.uc.edu

Institut fur Mathematik Universitat Linz A{4040 Linz Austria scherzerindmath.uni-linz.ac.at