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A General Framework for Regularized Evaluation of Unstable Operators
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
203, 451]463 Ž1996.
0390
A General Framework for Regularized Evaluation of Unstable Operators* Charles W. Groetsch Department of Mathematical Sciences, Uni¨ ersity of Cincinnati, Cincinnati, Ohio 45221-0025
and Martin Hanke Institut fur ¨ Praktische Mathematik, Uni¨ ersitat ¨ Karlsruhe, D-76128 Karlsruhe, Germany Submitted by Joseph A. Ball Received January 26, 1995
A general approach is developed for stable evaluation of unbounded operators when the data are corrupted by error. The essence of the approach is to recast the direct problem of evaluating the unstable operator as an inverse problem involving an associated bounded operator. Classical general regularization methods for inverse problems are then elucidated within the new setting, and new order optimal parameter choice strategies for iterative and noniterative regularization methods are developed. Finally, a regularizing stopping rule for a conjugate gradient acceleration of Lardy’s method is provided. Q 1996 Academic Press, Inc.
1. INTRODUCTION In the past two decades the notion of regularization methods has been developed as a tool to approximate the solution x of an ill-posed in¨ erse problem Kx s g , where K is a bounded linear operator which lacks a continuous Žgeneralized. inverse. Integral equations of the first kind can be considered as *Supported by NATO ŽCRG930044.. 451 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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prototype problems of this sort, and there is a variety of applications in the natural sciences which lead to ill-posed inverse problems, cf., e.g., w2, 6x. However, there are also direct problems Lx s y,
Ž 1.1.
where y is to be evaluated, that are ill-posed because L is an unbounded operator; numerical differentiation is a well-known example for this kind of ill-posed problem. Often, regularization methods for a stable evaluation of Ž1.1. can be derived by recasting the direct problem as an equivalent inverse problem. For example, the problem of numerical differentiation can be rewritten as a first kind integral equation with a Green’s function kernel. For some unbounded operators, however, e.g., for integro-differential operators, such transformations fail because the operator L has nonclosed range. As an example consider the elliptic differential equation ydiv Ž a=u . s f
Ž 1.2.
in a domain V ; R n with boundary data on V .
usq
Ž 1.3.
Assuming that the conductivity a g H 2 Ž V . is bounded from below by a positive constant and V has sufficiently smooth boundary, problem Ž1.2., Ž1.3. has a unique solution u s F Ž a. g H 2 Ž V . l H01 Ž V .. A typical inverse problem is the determination of the conductivity a from the potential field u. Note that this is a nonlinear problem. A first step in approaching this problem may therefore be linearization. The Frechet derivative of F is ´ given by F9 Ž a . h s A Ž a .
y1
D Ž a . h,
with A Ž a . w s ydiv Ž a=w .
and
D Ž a . h s div Ž h=F Ž a . . .
Under certain assumptions on a and F Ž a., the operators AŽ a. and DŽ a. are closed injective linear operators on subspaces of H 2 Ž V ., resp. H 1 Ž V ., with appropriate boundary conditions, cf. Ito and Kunisch w10x. Consequently, the solution of the linearized problem is h s F9 Ž a .
y1
u ˜ s D Ž a.
y1
A Ž a . u, ˜
EVALUATION OF UNSTABLE OPERATORS
453
i.e., h is given as the solution of a direct problem involving the linear operator L s DŽ a.y1AŽ a.. Note that the spectrum of L clusters at zero and at infinity in general. In other words, L is an unbounded operator with nonclosed range. A completely different and considerably easier example has been investigated in w7x. The Žnonlinear. heat transfer law at the interface of a given solid may be determined numerically by evaluating the integro-differential operator t
H0 w˜ 9 Ž t . k Ž t y t . dt ,
t G 0,
Ž 1.4.
given the measured surface temperature history w ˜ Ž t .. Here, the kernal Ž . function k in 1.4 depends on the geometry of the problem and, in general, has a mild singularity at the origin. In particular, for a semi-infinite solid the operator t
Ž Kx . Ž t . s H x Ž t . k Ž t y t . dt 0
is the familiar Abel integral operator, and commutes with the differentiation operator Dw s w 9. In general, however, the operators K and D do not commute, and the spectrum of the operator KD occurring in Ž1.4. will again cluster at the origin and at infinity. As a means to regularize direct problems of the form Ž1.1. given approximate data ˜ x Morozov w13x suggested minimizing the quadratic functional 5zy˜ x 5 2 q a 5 Lz 5 2
Ž 1.5.
over z g D Ž L. for some small a ) 0. Denoting the minimizer of Ž1.5. by ˜ z he proposed to take Lz˜ as approximation of y in Ž1.1.. This approach was subsequently studied in w8x where Žamong several other results . an a posteriori parameter choice for a has been introduced that provides quasi-optimal approximations Lz of y with respect to the data error in ˜ x. In w5x a class of regularization methods for evaluating Ž1.1. was suggested which parallels the spectral theoretic approach taken earlier for the regularization of inverse problems, cf. e.g., w4, 12, 2x. Although all results in w5, 8x seem to have their natural analog in the context of inverse problems, there has been lacking a rule of how to carry over these results in a general way. It is the aim of this paper to provide such a rule, and we will demonstrate the potential applicability of this scheme by providing error estimates for the parameter choice rule suggested in w8x for Morozov’s
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method. Note that no proof for its quasi-optimality was given in w8x because of many tedious technical details. Another regularization method that fits into the general scheme considered in w5x is the iterative method of Lardy w11x which is defined by a sequence of variational problems. Here, the regularization parameter is the iteration index and the stopping rule thus takes the role of regularization. In the final section we develop some kind of discrepancy principle for stopping the iteration, and again, provide error estimates for the corresponding iterates. Unfortunately, far too many variational problems have to be solved in a practical implementation of Lardy’s scheme. We will therefore describe an accelerated version of this method based on the well-known conjugate gradient method, and again prove the suitability of the discrepancy principle for the termination of this iteration. Once more it has to be emphasized that all these results have been established earlier in the context of inverse problems, and their extension to direct problems follows comparatively easily from a connection that will be established in the following section.
2. THE BASIC TRANSFORMATION Let X and Y be Hilbert spaces, and L be a closed linear operator L : D Ž L. ; X ª Y . It shall be assumed throughout that D Ž L. is a dense subset of X , and RŽ L. is dense in Y . Following w5x define
ˇ s Ž I q L*L . L
y1
,
ˆ s Ž I q LL* . L
y1
.
Ž 2.1.
Both operators in Ž2.1. are bounded and selfadjoint with spectra in w0, 1x, cf., e.g., w1x. Assume that we want to determine y s Lx from approximations x d of x which are close in the sense that 5 x y xd 5 F d
Ž 2.2.
EVALUATION OF UNSTABLE OPERATORS
455
for some small d ) 0. To this end let Ta : a ) 04 be a family of bounded real-valued functions with Ta Ž l . ª
1
l
< lTa Ž l . < F c,
,
pointwise on Ž 0, 1 as a ª 0,
Ž 2.3.
uniformly for all a ) 0 and l g w 0, 1 x .
We define approximations of Lx by choosing
ˇ aŽ L ˇ. x d , xad s LT
yad s Lxad .
Ž 2.4.
One can think of xad as being a mollification of the perturbed data x d. As shown in w5x, xad belongs to D Ž L., and yad ª Lx d
if x d g D Ž L . .
Otherwise 5 yad 5 ª ` as a ª 0, and a proper choice of a s a Ž d . will provide a regularized approximation of Lx. Instead of the continuous Žregularization. parameter a ) 0 one can also consider approximations indexed by nonnegative integers n g N 0 . In this case we shall speak of iterati¨ e regularization, and consider the limit n ª ` rather than a ª 0 as above. Two particular examples from the literature fit into this general scheme. The minimizer xad Žand the corresponding approximation yad s Lxad . of Ž1.5. as suggested by Morozov w13x is obtained for Ta Ž l . s
1
a Ž 1 y l. q l
.
Ž 2.5.
Lardy’s iterative scheme w11x, which will be described in more detail in Section 4, is obtained from the polynomials ny1
Tn Ž l . s
Ý Ž 1 y l. j .
js0
We will now derive a relation between the above approach of constructing general regularization methods for the evaluation of unbounded operators with the more classical setting concerning the regularization of inverse problems. To this end recall that the selfadjoint positive definite operators ˇ L, ˆ L*L, and LL* all admit arbitrary positive powers, cf., e.g., w1x. L, Concerning these operators we state the following lemma.
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ˆn . and LEMMA 1. For any real n ) 0 we ha¨ e D ŽŽ LL*. n . s RŽ L n n ˇ D ŽŽ L*L. . s RŽ L .. Proof. For a natural number n it is obvious that D ŽŽ LL*. n . s D ŽŽ I q ˆn . by the definition Ž2.1. of L ˆ LL*. n .. Since the latter subset equals RŽ L the first assertion is obviously fulfilled by n g N. For real n the claim therefore follows by interpolation, cf., e.g., w1x. The second statement is proved in the same way.
ˇ1r2 . ; DŽ L.. hence, we can In particular, Lemma 1 implies that RŽ L Ž . define new true and perturbed ‘‘data’’ ˇ1r2 x, w s LL
ˇ1r2 x d . w d s LL
Ž 2.6.
ˇ1r2 has the following properties. The operator LL ˇ1r2 : X ª Y is a bounded operator with norm 5 LL ˇ1r2 5 F 1. LEMMA 2. LL Moreo¨ er, the following identity holds: ˇ1r2 . * Ž LL ˇ1r2 . s I y L. ˆ Ž LL
Ž 2.7.
ˇ1r2 x s L ˆ1r2 Lx for Proof. We first recall from w5, Lemma 2.2x that LL Ž . Ž . Ž . x g D L . Thus, for a fixed x g D L and any z g D L we have ˇ1r2 . *LL ˇ1r2 x, z : s ² LL ˇ1r2 x, LL ˇ1r2 z : ² Ž LL ˆ1r2 Lx, L ˆ1r2 Lz : s ² L*LLx, ˆ s ²L z:. Now, obviously,
ˆ s L*LL ˇ s Ž I q L*L y I . L ˇsIyL ˇ L*LL on D Ž L. showing that
ˇ1r2 . *LL ˇ1r2 x, z : s ² Ž I y L ˇ . x, z : ² Ž LL for all z in the dense subset D Ž L. of X . Thus Ž2.7. has been established ˇ is a for x g D Ž L., and it extends to the whole space X since I y L Ž . bounded operator on X . From 2.7 follows further that
ˇ1r2 . * Ž LL ˇ1r2 . 5 s 5 I y L ˇ 5 F 1, ˇ1r2 5 2 s 5 Ž LL 5 LL and the proof is complete.
ˇ1r2 5 s 1 if and only if L is an A spectral analysis easily reveals that 5 LL unbounded operator, i.e., if and only if the direct problem Ž1.1. is ill-posed.
EVALUATION OF UNSTABLE OPERATORS
457
As we will establish below in Proposition 3 the desired quantity y s Lx is a solution of the ‘‘in¨ erse problem’’
ˆ1r2 y s w. L
Ž 2.8.
ˆ clusters at the origin, and If L is unbounded then the spectrum of L Ž . hence, the solution of 2.8 is ill-posed. Concerning the perturbed data w d we conclude from Lemma 2 and Ž2.2. that ˇ1r2 Ž x y x d . 5 F 5 x y x d 5 d . 5 w y w d 5 s 5 LL
Ž 2.9.
ˇ1r2 x with x g X . Then w g RŽ L ˆ1r2 . if and PROPOSITION 3. Let w s LL Ž . Ž . only if x g D L . If x g D L then y s Lx is the unique solution of Ž2.8.. Proof. As noted above, if x g D Ž L. then
ˇ1r2 x s L ˆ1r2 Lx g R Ž L ˆ1r2 . , w s LL ˆ1r2 y s w. Since L ˆ1r2 is injective this solution and y s Lx is a solution of L 1r2 ˆ . then L ˆ1r2 w g RŽ L ˆ ., which equals is unique. Vice versa, if w g RŽ L D Ž LL*. by Lemma 1. This means that ˆ1r2 w g D Ž L . . L*L
Ž 2.10.
On the other hand we have
ˆ1r2 w s L*L ˆ1r2 LL ˇ1r2 x s L*LLx ˇ sŽIyL ˇ . x s x y Lx, ˇ L*L and hence, Lemma 1 and Ž2.10. imply
ˆ1r2 w q Lx ˇ g D Ž L. . x s L*L This completes the proof. The general theory of regularization methods Žas developed, for example, in w4, 12x. for the solution of a not necessarily selfadjoint equation Ž2.8. suggests approximations of the form
ˆ. L ˆ1r2 w d s Ta Ž L ˆ. L ˆ1r2 LL ˇ1r2 x d , yad s Ta Ž L where the functions Ta should fulfill precisely the same requirements as above, namely Ž2.3.. In other words, the classical theory applied to the inverse problem Ž2.8. results in the same class of approximations as the theory developed in w5x for the direct problem, cf. Ž2.4.. Note that the ˆn . readily translate to the direct classical smoothness assumptions y g RŽ L setting via Lemma 1.
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With these prerequisites it is easy to identify the results in w5, Sect. 3x concerning the regularizing properties of the approximations Ž2.4. for the direct problem as corollaries of the results on inverse problems as given, e.g., in w4, Chap. 3x. It is only the saturation result w5, Theorem 3.5x concerning Morozov’s scheme Ž1.5. which cannot be obtained in this straightforward manner. In the following sections we will use this new insight into the approximations Ž2.4. to obtain some important further results.
3. A POSTERIORI PARAMETER CHOICE RULES Following Engle and Gfrerer w3x, or Raus w14x, the best possible worst case error of the regularization method Ž2.4. for a given function x g X is defined as
c˜ Ž d . [ sup inf 5 Lx y yad 5 : a ) 0 4 : 5 x y x d 5 F d 4 . On the other hand, if a Ž d . denotes a parameter choice strategy depending on the noise level d in Ž2.2. then the worst case error of this parameter choice rule is given as
c Ž d . [ sup 5 Lx y yadŽ d . 5 : 5 x y x d 5 F d 4 . A parameter choice strategy is called quasi-optimal if
c Ž d . F cc˜ Ž d . for some c ) 0. Consider once again Morozov’s method Ž1.5. of computing approximations yad of Lx. In w8x, the following method for choosing the regularization parameter a has been suggested. Parameter Choice Rule for Morozo¨ ’s Methods. Choose a s a Ž d . as the solution of the equation
ˇ . Ta Ž L ˇ. ² xd, a Ž I y L
3
x d :1r2 s td .
Ž 3.1.
As stated in w8x the nonlinear equation Ž3.1. has a unique solution, and one could follow the line of argument in w3x to prove the quasi-optimality of this method. Here we will use the results of the foregoing section to prove a slightly weaker result which allows the following error bounds. THEOREM 4. Let x g D Ž L., and t ) 1 be fixed. For gi¨ en data x d subject to Ž2.2. determine a Ž d . from Ž3.1.. Then yadŽ d . ª Lx as d ª 0. Moreo¨ er, if
459
EVALUATION OF UNSTABLE OPERATORS
x g D ŽŽ LL*. n L. for some 0 - n F 1 then 5 Lx y yadŽ d . 5 s O Ž d 2 n rŽ2 nq1. . ,
d ª 0.
Ž 3.2.
Proof. The functions Ta of Ž2.5. satisfy Assumptions 2.1 and 2.10 in w3x with g Ž a . s 1ra ,
a ˜ Ž l. s l .
L s 2,
Actually these assumptions are only fulfilled for l g Ž0, 1. but this causes ˆ has no eigenspace corresponding to l s 1 since no problems because L ˆ . is a dense subset of Y . Engl and Gfrerer now suggest to choose the RŽ L regularization parameter a s a Ž d . for the inverse problem Ž2.8. as the solution of the equation w3, Ž2.17.x, which here takes the particular form 2
3
ˇ . Ta Ž L ˇ . w d : s gd 2 , ²wd , 2 a 3 Ž I y L
Ž 3.3.
where g G L s 2. Note that the projector Q onto the closure of the range ˆ is the identity operator by virtue of Lemma 1 since RŽ L. is assumed of L to be dense in Y . Using Ž2.6. and the identity Ž2.7., we recognize Ž3.3. as being identical to the parameter choice strategy Ž3.1.. As is shown in w3, Theorem 2.11x this parameter choice strategy is quasi-optimal with respect to perturbations in w of order d . Although this does not imply quasi-optimality with respect to perturbations in x of order d , it suffices to conclude the assertions of the theorem. To this end we note first that we have < Ta Ž l . < F 1ra ,
< ln Ž 1 y lTa Ž l . . < F cn a n , 0 - n F 1.
From this and w4, Theorem 2.3.3x we conclude that yad ª Lx provided
a ª 0,
d 2ra ª 0.
Note that 5 w y w d 5 F d by virtue of Ž2.9.. On the other hand, if x g DŽŽ LL*. n L. for some 0 - n / 1 then the choice a ; d 1rŽ2 nq1. leads to the error estimate Ž3.2., cf., e.g. w12, Satz 3.4.3x. Now the assertions of the theorem follow from the quasi-optimality of the parameter choice strategy Ž3.1. with respect to the perturbations in w.
4. LARDY’S METHOD AND CONJUGATE GRADIENTS Lardy’s iteration method, cf. w11x, consists in the following procedure: given an initial approximation x 0 s x 0d g D Ž L. of x d , compute d d d ˇ d x nq 1 s xn q LŽ x y xn . ,
d d ynq1 s Lx nq1 ,
n g N0 .
Ž 4.1.
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ˇ is contained in DŽ L. by Lemma 1, and hence, Recall that the range of L this iterative scheme is well-defined. A practical implementation will probably utilize the following weak ˇ and y s Lz satisfy formulation: given any x g X , the quantities z s Lx ² x, ¨ : q ² Lx, L¨ : s ² z, ¨ :
for all ¨ g D Ž L . .
Ž 4.2.
It is obvious from Ž4.1. that the approximations ynd of Lardy’s method are nothing else than the iterates of the classical Landweber iteration applied to Ž2.8.. In this context the discrepancy principle is a useful stopping rule, i.e., parameter choice strategy for n s nŽ d . g N 0 . This ˆ1r2 ynd 5, which requires the evaluation of the so-called discrepancy 5 w d y L can be done by virtue of Ž2.7. as
ˇ1r2 Ž x d y x nd . 5 2 ˆ1r2 ynd 5 2 s 5 LL 5wd y L ˇ1r2 . * Ž LL ˇ1r2 . Ž x d y x nd . : s ² x d y x nd , Ž LL ˇ . Ž x d y x nd . : . s ² x d y x nd , Ž I y L
Ž 4.3.
Thus, the discrepancy principle for the Landweber iteration yields the following stopping rule for Lardy’s method. Stopping Rule for Lardy’s Method. Choose n s nŽ d . as the smallest nonnegative integer for which d ² x d y x nd , x d y x nq1 : F td .
Ž 4.4.
THEOREM 5. If t ) 1 is fixed and x g D Ž L. then the stopping rule Ž4.4. determines a unique Ž finite. stopping index nŽ d ., and ynŽd d . ª Lx as d ª 0. If x g D ŽŽ LL*. n L. for some n ) 0, and if 5 x y x d 5 F d then 5 Lx y ynŽd d . 5 s O Ž d 2 n rŽ2 nq1. . ,
d ª 0.
Proof. It is clear from the definition Ž4.1. of the iterates that Ž4.3. coincides with the left-hand side of Ž4.4.. Thus the assertions of the theorem follow immediately from Vainikko’s results w15x on the discrepancy principle for the Landweber iteration. We emphasize that each iteration in Lardy’s method requires the solution of a linear equation as in Ž4.2.. As compared to the Landweber iteration for inverse ill-pose problems, Lardy’s method should therefore not be considered an explicit but rather an implicit iteration method. In fact, each iteration will require approximately the same amount of work as the solution of Morozov’s minimization problem for one particular choice
EVALUATION OF UNSTABLE OPERATORS
461
of a . In other words, Lardy’s iteration will only be competitive to Morozov’s method if it converges rapidly, which will rarely be the case Žsimilar to the Landweber iteration for inverse problems.. We therefore recommend a combination of Lardy’s method with an acceleration scheme of conjugate gradient type, i.e., the use of the much faster conjugate gradient method for solving the normal equations corresponding to Ž2.8., i.e.,
ˆ sL ˆ1r2 w s LLx. ˇ Ly
Ž 4.5.
We refer to w9x for an extensive treatment of the conjugate gradient method for the regularization of inverse problems. For this particular problem the resulting algorithm Ži.e., w9, Algorithm 2.2x. looks as follows: v
v
v
ˇŽ x d y x 0d . and r 0d s Lz 0d as in (4.2) Compute z 0d s L d s z 0d and s s r 0d for n s 0, 1, . . . ,
ˇ and Ls ˆ s LLd ˇ as in (4.2) }compute Ld d d ˆ: } a s ² rn , rn :r² s, Ls d d }x nq 1 s x n q a d, d d ynq 1 s yn q a s, d ˇ z nq 1 s z nd y a Ld, d d ˆ rnq 1 s rn y a Ls 2 d } b s 5 rnq 1 5 r5 rnd 5 2 d }d s z kq 1 q b d, d s s r kq 1 q b s end for.
ˆ1r2 w d y Ly ˆ nd corresponding In this algorithm rnd is always the residual L to Ž4.5.. In addition, the algorithm determines ˇ Ž x d y x nd . , z nd s L
n g N0 .
Ž 4.6.
This can easily be seen by induction. Similarly, besides the usual search direction s the quantity d is updated so as to relate to s via Ld s s. This allows the evaluation of the following stopping rule for this method. Stopping Rule for the Conjugate Gradient Method. Choose n s nŽ d . as the smallest nonnegative integer for which 5 x d y x nd 5 2 y ² x d y x nd , z n : F Ž td . 2 . We have the following results for this stopping rule.
Ž 4.7.
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THEOREM 6. If t ) 1 is fixed and x g D Ž L. then the stopping rule Ž4.7. determines a unique Ž finite. stopping index nŽ d ., and ynŽd d . ª Lx as d ª 0. If x g D ŽŽ LL*. n L. for some n ) 0, and if 5 x y x d 5 F d then 5 Lx y ynŽd d . 5 s O Ž d 2 n rŽ2 nq1. . ,
d ª 0.
Ž 4.8.
Proof. To prove this result we have to show that the left-hand side expression in Ž4.8. coincides with the square of the discrepancy 5 w d y ˆ1r2 ynd 5. Comparing Ž4.7. and Ž4.3. we observe that this is the case if and L only if
ˇ Ž x d y x nd . : . ² x d y x nd , z n : s ² x d y x nd , L Because of Ž4.6. this is indeed the case. In other words, the stopping rule Ž4.7. is the usual discrepancy principle for the conjugate gradient method applied to the normal equation of Ž2.8., and hence, the assertions follow from the analysis of the conjugate gradient method for inverse problems, compare w9, Sect. 3.3x.
REFERENCES 1. R. Dautray and J.-L. Lions, ‘‘Mathematical Analysis and Numerical Methods for Science and Technology,’’ Vol. 3, Springer-Verlag, BerlinrHeidelbergrNew York, 1990. 2. H. W. Engl, Regularization methods for the stable solution of inverse problems, Sur¨ eys Math. Indust. 3 Ž1993., 71]143. 3. H. W. Engl and H. Gfrerer, A posteriori parameter choice for generalized regularization methods for solving linear ill-posed problems, Appl. Numer. Math. 4 Ž1988., 395]417. 4. C. W. Groetsch, ‘‘The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind,’’ Pitman, BostonrLondonrMelbourne, 1984. 5. C. W. Groetsch, Spectral methods for linear inverse problems with unbounded operators, J. Approx. Theory 70 Ž1992., 16]28. 6. C. W. Groetsch, ‘‘Inverse Problems in the Mathematical Sciences,’’ Vieweg, BraunschweigrWiesbaden, 1993. 7. C. W. Groetsch, A numerical method for some inverse problems in nonlinear heat transfer, Comput. Math. Appl. 1 Ž1994., 49]59. 8. C. W. Groetsch and O. Scherzer, Optimal order of convergence for stable evaluation of differential operators, Electron. J. Differential Equations, 1993r04. 9. M. Hanke, ‘‘Conjugate Gradient Type Methods for Ill-Posed Problems,’’ Longman, Harlow, 1995. 10. K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems, J. Math. Anal. Appl. 188 Ž1994., 1040]1066. 11. L. J. Lardy, A series representation of the generalized inverse of a closed linear operator, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ž 8 . 58 Ž1975., 152]157.
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12. A. K. Louis, ‘‘Inverse and schlecht gestellte Probleme,’’ Teubner, Stuttgart, 1989. 13. V. A. Morozov, ‘‘Methods for Solving Incorrectly Posed Problems,’’ Springer-Verlag, New YorkrBerlinrHeidelberg, 1984. 14. T. Raus, The principle of the residual in the solution of ill-posed problems with ¨ Toimetised 715 Ž1985., 12]20. wIn Russianx nonselfadjoint operator, Tartu Riikl. Ul. 15. G. M. Vainikko, Error estimates of the successive approximation method for ill-posed problems, Automat. Remote Control 40 Ž1980., 356]363.
203, 451]463 Ž1996.
0390
A General Framework for Regularized Evaluation of Unstable Operators* Charles W. Groetsch Department of Mathematical Sciences, Uni¨ ersity of Cincinnati, Cincinnati, Ohio 45221-0025
and Martin Hanke Institut fur ¨ Praktische Mathematik, Uni¨ ersitat ¨ Karlsruhe, D-76128 Karlsruhe, Germany Submitted by Joseph A. Ball Received January 26, 1995
A general approach is developed for stable evaluation of unbounded operators when the data are corrupted by error. The essence of the approach is to recast the direct problem of evaluating the unstable operator as an inverse problem involving an associated bounded operator. Classical general regularization methods for inverse problems are then elucidated within the new setting, and new order optimal parameter choice strategies for iterative and noniterative regularization methods are developed. Finally, a regularizing stopping rule for a conjugate gradient acceleration of Lardy’s method is provided. Q 1996 Academic Press, Inc.
1. INTRODUCTION In the past two decades the notion of regularization methods has been developed as a tool to approximate the solution x of an ill-posed in¨ erse problem Kx s g , where K is a bounded linear operator which lacks a continuous Žgeneralized. inverse. Integral equations of the first kind can be considered as *Supported by NATO ŽCRG930044.. 451 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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prototype problems of this sort, and there is a variety of applications in the natural sciences which lead to ill-posed inverse problems, cf., e.g., w2, 6x. However, there are also direct problems Lx s y,
Ž 1.1.
where y is to be evaluated, that are ill-posed because L is an unbounded operator; numerical differentiation is a well-known example for this kind of ill-posed problem. Often, regularization methods for a stable evaluation of Ž1.1. can be derived by recasting the direct problem as an equivalent inverse problem. For example, the problem of numerical differentiation can be rewritten as a first kind integral equation with a Green’s function kernel. For some unbounded operators, however, e.g., for integro-differential operators, such transformations fail because the operator L has nonclosed range. As an example consider the elliptic differential equation ydiv Ž a=u . s f
Ž 1.2.
in a domain V ; R n with boundary data on V .
usq
Ž 1.3.
Assuming that the conductivity a g H 2 Ž V . is bounded from below by a positive constant and V has sufficiently smooth boundary, problem Ž1.2., Ž1.3. has a unique solution u s F Ž a. g H 2 Ž V . l H01 Ž V .. A typical inverse problem is the determination of the conductivity a from the potential field u. Note that this is a nonlinear problem. A first step in approaching this problem may therefore be linearization. The Frechet derivative of F is ´ given by F9 Ž a . h s A Ž a .
y1
D Ž a . h,
with A Ž a . w s ydiv Ž a=w .
and
D Ž a . h s div Ž h=F Ž a . . .
Under certain assumptions on a and F Ž a., the operators AŽ a. and DŽ a. are closed injective linear operators on subspaces of H 2 Ž V ., resp. H 1 Ž V ., with appropriate boundary conditions, cf. Ito and Kunisch w10x. Consequently, the solution of the linearized problem is h s F9 Ž a .
y1
u ˜ s D Ž a.
y1
A Ž a . u, ˜
EVALUATION OF UNSTABLE OPERATORS
453
i.e., h is given as the solution of a direct problem involving the linear operator L s DŽ a.y1AŽ a.. Note that the spectrum of L clusters at zero and at infinity in general. In other words, L is an unbounded operator with nonclosed range. A completely different and considerably easier example has been investigated in w7x. The Žnonlinear. heat transfer law at the interface of a given solid may be determined numerically by evaluating the integro-differential operator t
H0 w˜ 9 Ž t . k Ž t y t . dt ,
t G 0,
Ž 1.4.
given the measured surface temperature history w ˜ Ž t .. Here, the kernal Ž . function k in 1.4 depends on the geometry of the problem and, in general, has a mild singularity at the origin. In particular, for a semi-infinite solid the operator t
Ž Kx . Ž t . s H x Ž t . k Ž t y t . dt 0
is the familiar Abel integral operator, and commutes with the differentiation operator Dw s w 9. In general, however, the operators K and D do not commute, and the spectrum of the operator KD occurring in Ž1.4. will again cluster at the origin and at infinity. As a means to regularize direct problems of the form Ž1.1. given approximate data ˜ x Morozov w13x suggested minimizing the quadratic functional 5zy˜ x 5 2 q a 5 Lz 5 2
Ž 1.5.
over z g D Ž L. for some small a ) 0. Denoting the minimizer of Ž1.5. by ˜ z he proposed to take Lz˜ as approximation of y in Ž1.1.. This approach was subsequently studied in w8x where Žamong several other results . an a posteriori parameter choice for a has been introduced that provides quasi-optimal approximations Lz of y with respect to the data error in ˜ x. In w5x a class of regularization methods for evaluating Ž1.1. was suggested which parallels the spectral theoretic approach taken earlier for the regularization of inverse problems, cf. e.g., w4, 12, 2x. Although all results in w5, 8x seem to have their natural analog in the context of inverse problems, there has been lacking a rule of how to carry over these results in a general way. It is the aim of this paper to provide such a rule, and we will demonstrate the potential applicability of this scheme by providing error estimates for the parameter choice rule suggested in w8x for Morozov’s
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GROETSCH AND HANKE
method. Note that no proof for its quasi-optimality was given in w8x because of many tedious technical details. Another regularization method that fits into the general scheme considered in w5x is the iterative method of Lardy w11x which is defined by a sequence of variational problems. Here, the regularization parameter is the iteration index and the stopping rule thus takes the role of regularization. In the final section we develop some kind of discrepancy principle for stopping the iteration, and again, provide error estimates for the corresponding iterates. Unfortunately, far too many variational problems have to be solved in a practical implementation of Lardy’s scheme. We will therefore describe an accelerated version of this method based on the well-known conjugate gradient method, and again prove the suitability of the discrepancy principle for the termination of this iteration. Once more it has to be emphasized that all these results have been established earlier in the context of inverse problems, and their extension to direct problems follows comparatively easily from a connection that will be established in the following section.
2. THE BASIC TRANSFORMATION Let X and Y be Hilbert spaces, and L be a closed linear operator L : D Ž L. ; X ª Y . It shall be assumed throughout that D Ž L. is a dense subset of X , and RŽ L. is dense in Y . Following w5x define
ˇ s Ž I q L*L . L
y1
,
ˆ s Ž I q LL* . L
y1
.
Ž 2.1.
Both operators in Ž2.1. are bounded and selfadjoint with spectra in w0, 1x, cf., e.g., w1x. Assume that we want to determine y s Lx from approximations x d of x which are close in the sense that 5 x y xd 5 F d
Ž 2.2.
EVALUATION OF UNSTABLE OPERATORS
455
for some small d ) 0. To this end let Ta : a ) 04 be a family of bounded real-valued functions with Ta Ž l . ª
1
l
< lTa Ž l . < F c,
,
pointwise on Ž 0, 1 as a ª 0,
Ž 2.3.
uniformly for all a ) 0 and l g w 0, 1 x .
We define approximations of Lx by choosing
ˇ aŽ L ˇ. x d , xad s LT
yad s Lxad .
Ž 2.4.
One can think of xad as being a mollification of the perturbed data x d. As shown in w5x, xad belongs to D Ž L., and yad ª Lx d
if x d g D Ž L . .
Otherwise 5 yad 5 ª ` as a ª 0, and a proper choice of a s a Ž d . will provide a regularized approximation of Lx. Instead of the continuous Žregularization. parameter a ) 0 one can also consider approximations indexed by nonnegative integers n g N 0 . In this case we shall speak of iterati¨ e regularization, and consider the limit n ª ` rather than a ª 0 as above. Two particular examples from the literature fit into this general scheme. The minimizer xad Žand the corresponding approximation yad s Lxad . of Ž1.5. as suggested by Morozov w13x is obtained for Ta Ž l . s
1
a Ž 1 y l. q l
.
Ž 2.5.
Lardy’s iterative scheme w11x, which will be described in more detail in Section 4, is obtained from the polynomials ny1
Tn Ž l . s
Ý Ž 1 y l. j .
js0
We will now derive a relation between the above approach of constructing general regularization methods for the evaluation of unbounded operators with the more classical setting concerning the regularization of inverse problems. To this end recall that the selfadjoint positive definite operators ˇ L, ˆ L*L, and LL* all admit arbitrary positive powers, cf., e.g., w1x. L, Concerning these operators we state the following lemma.
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GROETSCH AND HANKE
ˆn . and LEMMA 1. For any real n ) 0 we ha¨ e D ŽŽ LL*. n . s RŽ L n n ˇ D ŽŽ L*L. . s RŽ L .. Proof. For a natural number n it is obvious that D ŽŽ LL*. n . s D ŽŽ I q ˆn . by the definition Ž2.1. of L ˆ LL*. n .. Since the latter subset equals RŽ L the first assertion is obviously fulfilled by n g N. For real n the claim therefore follows by interpolation, cf., e.g., w1x. The second statement is proved in the same way.
ˇ1r2 . ; DŽ L.. hence, we can In particular, Lemma 1 implies that RŽ L Ž . define new true and perturbed ‘‘data’’ ˇ1r2 x, w s LL
ˇ1r2 x d . w d s LL
Ž 2.6.
ˇ1r2 has the following properties. The operator LL ˇ1r2 : X ª Y is a bounded operator with norm 5 LL ˇ1r2 5 F 1. LEMMA 2. LL Moreo¨ er, the following identity holds: ˇ1r2 . * Ž LL ˇ1r2 . s I y L. ˆ Ž LL
Ž 2.7.
ˇ1r2 x s L ˆ1r2 Lx for Proof. We first recall from w5, Lemma 2.2x that LL Ž . Ž . Ž . x g D L . Thus, for a fixed x g D L and any z g D L we have ˇ1r2 . *LL ˇ1r2 x, z : s ² LL ˇ1r2 x, LL ˇ1r2 z : ² Ž LL ˆ1r2 Lx, L ˆ1r2 Lz : s ² L*LLx, ˆ s ²L z:. Now, obviously,
ˆ s L*LL ˇ s Ž I q L*L y I . L ˇsIyL ˇ L*LL on D Ž L. showing that
ˇ1r2 . *LL ˇ1r2 x, z : s ² Ž I y L ˇ . x, z : ² Ž LL for all z in the dense subset D Ž L. of X . Thus Ž2.7. has been established ˇ is a for x g D Ž L., and it extends to the whole space X since I y L Ž . bounded operator on X . From 2.7 follows further that
ˇ1r2 . * Ž LL ˇ1r2 . 5 s 5 I y L ˇ 5 F 1, ˇ1r2 5 2 s 5 Ž LL 5 LL and the proof is complete.
ˇ1r2 5 s 1 if and only if L is an A spectral analysis easily reveals that 5 LL unbounded operator, i.e., if and only if the direct problem Ž1.1. is ill-posed.
EVALUATION OF UNSTABLE OPERATORS
457
As we will establish below in Proposition 3 the desired quantity y s Lx is a solution of the ‘‘in¨ erse problem’’
ˆ1r2 y s w. L
Ž 2.8.
ˆ clusters at the origin, and If L is unbounded then the spectrum of L Ž . hence, the solution of 2.8 is ill-posed. Concerning the perturbed data w d we conclude from Lemma 2 and Ž2.2. that ˇ1r2 Ž x y x d . 5 F 5 x y x d 5 d . 5 w y w d 5 s 5 LL
Ž 2.9.
ˇ1r2 x with x g X . Then w g RŽ L ˆ1r2 . if and PROPOSITION 3. Let w s LL Ž . Ž . only if x g D L . If x g D L then y s Lx is the unique solution of Ž2.8.. Proof. As noted above, if x g D Ž L. then
ˇ1r2 x s L ˆ1r2 Lx g R Ž L ˆ1r2 . , w s LL ˆ1r2 y s w. Since L ˆ1r2 is injective this solution and y s Lx is a solution of L 1r2 ˆ . then L ˆ1r2 w g RŽ L ˆ ., which equals is unique. Vice versa, if w g RŽ L D Ž LL*. by Lemma 1. This means that ˆ1r2 w g D Ž L . . L*L
Ž 2.10.
On the other hand we have
ˆ1r2 w s L*L ˆ1r2 LL ˇ1r2 x s L*LLx ˇ sŽIyL ˇ . x s x y Lx, ˇ L*L and hence, Lemma 1 and Ž2.10. imply
ˆ1r2 w q Lx ˇ g D Ž L. . x s L*L This completes the proof. The general theory of regularization methods Žas developed, for example, in w4, 12x. for the solution of a not necessarily selfadjoint equation Ž2.8. suggests approximations of the form
ˆ. L ˆ1r2 w d s Ta Ž L ˆ. L ˆ1r2 LL ˇ1r2 x d , yad s Ta Ž L where the functions Ta should fulfill precisely the same requirements as above, namely Ž2.3.. In other words, the classical theory applied to the inverse problem Ž2.8. results in the same class of approximations as the theory developed in w5x for the direct problem, cf. Ž2.4.. Note that the ˆn . readily translate to the direct classical smoothness assumptions y g RŽ L setting via Lemma 1.
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GROETSCH AND HANKE
With these prerequisites it is easy to identify the results in w5, Sect. 3x concerning the regularizing properties of the approximations Ž2.4. for the direct problem as corollaries of the results on inverse problems as given, e.g., in w4, Chap. 3x. It is only the saturation result w5, Theorem 3.5x concerning Morozov’s scheme Ž1.5. which cannot be obtained in this straightforward manner. In the following sections we will use this new insight into the approximations Ž2.4. to obtain some important further results.
3. A POSTERIORI PARAMETER CHOICE RULES Following Engle and Gfrerer w3x, or Raus w14x, the best possible worst case error of the regularization method Ž2.4. for a given function x g X is defined as
c˜ Ž d . [ sup inf 5 Lx y yad 5 : a ) 0 4 : 5 x y x d 5 F d 4 . On the other hand, if a Ž d . denotes a parameter choice strategy depending on the noise level d in Ž2.2. then the worst case error of this parameter choice rule is given as
c Ž d . [ sup 5 Lx y yadŽ d . 5 : 5 x y x d 5 F d 4 . A parameter choice strategy is called quasi-optimal if
c Ž d . F cc˜ Ž d . for some c ) 0. Consider once again Morozov’s method Ž1.5. of computing approximations yad of Lx. In w8x, the following method for choosing the regularization parameter a has been suggested. Parameter Choice Rule for Morozo¨ ’s Methods. Choose a s a Ž d . as the solution of the equation
ˇ . Ta Ž L ˇ. ² xd, a Ž I y L
3
x d :1r2 s td .
Ž 3.1.
As stated in w8x the nonlinear equation Ž3.1. has a unique solution, and one could follow the line of argument in w3x to prove the quasi-optimality of this method. Here we will use the results of the foregoing section to prove a slightly weaker result which allows the following error bounds. THEOREM 4. Let x g D Ž L., and t ) 1 be fixed. For gi¨ en data x d subject to Ž2.2. determine a Ž d . from Ž3.1.. Then yadŽ d . ª Lx as d ª 0. Moreo¨ er, if
459
EVALUATION OF UNSTABLE OPERATORS
x g D ŽŽ LL*. n L. for some 0 - n F 1 then 5 Lx y yadŽ d . 5 s O Ž d 2 n rŽ2 nq1. . ,
d ª 0.
Ž 3.2.
Proof. The functions Ta of Ž2.5. satisfy Assumptions 2.1 and 2.10 in w3x with g Ž a . s 1ra ,
a ˜ Ž l. s l .
L s 2,
Actually these assumptions are only fulfilled for l g Ž0, 1. but this causes ˆ has no eigenspace corresponding to l s 1 since no problems because L ˆ . is a dense subset of Y . Engl and Gfrerer now suggest to choose the RŽ L regularization parameter a s a Ž d . for the inverse problem Ž2.8. as the solution of the equation w3, Ž2.17.x, which here takes the particular form 2
3
ˇ . Ta Ž L ˇ . w d : s gd 2 , ²wd , 2 a 3 Ž I y L
Ž 3.3.
where g G L s 2. Note that the projector Q onto the closure of the range ˆ is the identity operator by virtue of Lemma 1 since RŽ L. is assumed of L to be dense in Y . Using Ž2.6. and the identity Ž2.7., we recognize Ž3.3. as being identical to the parameter choice strategy Ž3.1.. As is shown in w3, Theorem 2.11x this parameter choice strategy is quasi-optimal with respect to perturbations in w of order d . Although this does not imply quasi-optimality with respect to perturbations in x of order d , it suffices to conclude the assertions of the theorem. To this end we note first that we have < Ta Ž l . < F 1ra ,
< ln Ž 1 y lTa Ž l . . < F cn a n , 0 - n F 1.
From this and w4, Theorem 2.3.3x we conclude that yad ª Lx provided
a ª 0,
d 2ra ª 0.
Note that 5 w y w d 5 F d by virtue of Ž2.9.. On the other hand, if x g DŽŽ LL*. n L. for some 0 - n / 1 then the choice a ; d 1rŽ2 nq1. leads to the error estimate Ž3.2., cf., e.g. w12, Satz 3.4.3x. Now the assertions of the theorem follow from the quasi-optimality of the parameter choice strategy Ž3.1. with respect to the perturbations in w.
4. LARDY’S METHOD AND CONJUGATE GRADIENTS Lardy’s iteration method, cf. w11x, consists in the following procedure: given an initial approximation x 0 s x 0d g D Ž L. of x d , compute d d d ˇ d x nq 1 s xn q LŽ x y xn . ,
d d ynq1 s Lx nq1 ,
n g N0 .
Ž 4.1.
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GROETSCH AND HANKE
ˇ is contained in DŽ L. by Lemma 1, and hence, Recall that the range of L this iterative scheme is well-defined. A practical implementation will probably utilize the following weak ˇ and y s Lz satisfy formulation: given any x g X , the quantities z s Lx ² x, ¨ : q ² Lx, L¨ : s ² z, ¨ :
for all ¨ g D Ž L . .
Ž 4.2.
It is obvious from Ž4.1. that the approximations ynd of Lardy’s method are nothing else than the iterates of the classical Landweber iteration applied to Ž2.8.. In this context the discrepancy principle is a useful stopping rule, i.e., parameter choice strategy for n s nŽ d . g N 0 . This ˆ1r2 ynd 5, which requires the evaluation of the so-called discrepancy 5 w d y L can be done by virtue of Ž2.7. as
ˇ1r2 Ž x d y x nd . 5 2 ˆ1r2 ynd 5 2 s 5 LL 5wd y L ˇ1r2 . * Ž LL ˇ1r2 . Ž x d y x nd . : s ² x d y x nd , Ž LL ˇ . Ž x d y x nd . : . s ² x d y x nd , Ž I y L
Ž 4.3.
Thus, the discrepancy principle for the Landweber iteration yields the following stopping rule for Lardy’s method. Stopping Rule for Lardy’s Method. Choose n s nŽ d . as the smallest nonnegative integer for which d ² x d y x nd , x d y x nq1 : F td .
Ž 4.4.
THEOREM 5. If t ) 1 is fixed and x g D Ž L. then the stopping rule Ž4.4. determines a unique Ž finite. stopping index nŽ d ., and ynŽd d . ª Lx as d ª 0. If x g D ŽŽ LL*. n L. for some n ) 0, and if 5 x y x d 5 F d then 5 Lx y ynŽd d . 5 s O Ž d 2 n rŽ2 nq1. . ,
d ª 0.
Proof. It is clear from the definition Ž4.1. of the iterates that Ž4.3. coincides with the left-hand side of Ž4.4.. Thus the assertions of the theorem follow immediately from Vainikko’s results w15x on the discrepancy principle for the Landweber iteration. We emphasize that each iteration in Lardy’s method requires the solution of a linear equation as in Ž4.2.. As compared to the Landweber iteration for inverse ill-pose problems, Lardy’s method should therefore not be considered an explicit but rather an implicit iteration method. In fact, each iteration will require approximately the same amount of work as the solution of Morozov’s minimization problem for one particular choice
EVALUATION OF UNSTABLE OPERATORS
461
of a . In other words, Lardy’s iteration will only be competitive to Morozov’s method if it converges rapidly, which will rarely be the case Žsimilar to the Landweber iteration for inverse problems.. We therefore recommend a combination of Lardy’s method with an acceleration scheme of conjugate gradient type, i.e., the use of the much faster conjugate gradient method for solving the normal equations corresponding to Ž2.8., i.e.,
ˆ sL ˆ1r2 w s LLx. ˇ Ly
Ž 4.5.
We refer to w9x for an extensive treatment of the conjugate gradient method for the regularization of inverse problems. For this particular problem the resulting algorithm Ži.e., w9, Algorithm 2.2x. looks as follows: v
v
v
ˇŽ x d y x 0d . and r 0d s Lz 0d as in (4.2) Compute z 0d s L d s z 0d and s s r 0d for n s 0, 1, . . . ,
ˇ and Ls ˆ s LLd ˇ as in (4.2) }compute Ld d d ˆ: } a s ² rn , rn :r² s, Ls d d }x nq 1 s x n q a d, d d ynq 1 s yn q a s, d ˇ z nq 1 s z nd y a Ld, d d ˆ rnq 1 s rn y a Ls 2 d } b s 5 rnq 1 5 r5 rnd 5 2 d }d s z kq 1 q b d, d s s r kq 1 q b s end for.
ˆ1r2 w d y Ly ˆ nd corresponding In this algorithm rnd is always the residual L to Ž4.5.. In addition, the algorithm determines ˇ Ž x d y x nd . , z nd s L
n g N0 .
Ž 4.6.
This can easily be seen by induction. Similarly, besides the usual search direction s the quantity d is updated so as to relate to s via Ld s s. This allows the evaluation of the following stopping rule for this method. Stopping Rule for the Conjugate Gradient Method. Choose n s nŽ d . as the smallest nonnegative integer for which 5 x d y x nd 5 2 y ² x d y x nd , z n : F Ž td . 2 . We have the following results for this stopping rule.
Ž 4.7.
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GROETSCH AND HANKE
THEOREM 6. If t ) 1 is fixed and x g D Ž L. then the stopping rule Ž4.7. determines a unique Ž finite. stopping index nŽ d ., and ynŽd d . ª Lx as d ª 0. If x g D ŽŽ LL*. n L. for some n ) 0, and if 5 x y x d 5 F d then 5 Lx y ynŽd d . 5 s O Ž d 2 n rŽ2 nq1. . ,
d ª 0.
Ž 4.8.
Proof. To prove this result we have to show that the left-hand side expression in Ž4.8. coincides with the square of the discrepancy 5 w d y ˆ1r2 ynd 5. Comparing Ž4.7. and Ž4.3. we observe that this is the case if and L only if
ˇ Ž x d y x nd . : . ² x d y x nd , z n : s ² x d y x nd , L Because of Ž4.6. this is indeed the case. In other words, the stopping rule Ž4.7. is the usual discrepancy principle for the conjugate gradient method applied to the normal equation of Ž2.8., and hence, the assertions follow from the analysis of the conjugate gradient method for inverse problems, compare w9, Sect. 3.3x.
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12. A. K. Louis, ‘‘Inverse and schlecht gestellte Probleme,’’ Teubner, Stuttgart, 1989. 13. V. A. Morozov, ‘‘Methods for Solving Incorrectly Posed Problems,’’ Springer-Verlag, New YorkrBerlinrHeidelberg, 1984. 14. T. Raus, The principle of the residual in the solution of ill-posed problems with ¨ Toimetised 715 Ž1985., 12]20. wIn Russianx nonselfadjoint operator, Tartu Riikl. Ul. 15. G. M. Vainikko, Error estimates of the successive approximation method for ill-posed problems, Automat. Remote Control 40 Ž1980., 356]363.