Where does smoothness count the most for Fredholm equations of the second kind with noisy information?

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Journal of Complexity 19 (2003) 758–798

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Where does smoothness count the most for Fredholm equations of the second kind with noisy information? Arthur G. Werschulza,b,
,1 a b

Department of Computer and Information Sciences, Fordham University, New York, NY 10023, USA Department of Computer Science, Columbia University, 1214 Amsterdam Avenue, MC0401, New York, NY 10027, USA Received 24 October 2002; accepted 22 January 2003

Abstract We study the complexity of Fredholm problems ðI  Tk Þu ¼ f of the second kind on I d ¼ ½0; 1d ; where Tk is an integral operator with kernel k: Previous work on the complexity of this problem has assumed either that we had complete information about k or that k and f had the same smoothness. In addition, most of this work has assumed that the information about k and f was exact. In this paper, we assume that k and f have different smoothness; more precisely, we assume that f AW r;p ðI d Þ with r4d=p and that kAW s;N ðI 2d Þ with s40: In addition, we assume that our information about k and f is contaminated by noise. We find that the nth minimal error is Yðnm þ dÞ; where m ¼ minfr=d; s=ð2dÞg and d is a bound on the noise. We prove that a noisy modified finite element method has nearly minimal error. This algorithm can be efficiently implemented using multigrid techniques. We thus find tight bounds on the e-complexity for this problem. These bounds depend on the cost cðdÞ of calculating a d-noisy information value. As an example, if the cost of a d-noisy evaluation is proportional to dt ; then the e-complexity is roughly ð1=eÞtþ1=m : r 2003 Elsevier Science (USA). All rights reserved. Keywords: Fredholm equation; Complexity; Optimal algorithms; Multigrid methods




Corresponding author. Department of Computer Science, Columbia University, 1214 Amsterdam Avenue, MC0401, New York, NY 10027, USA. Fax: +212-666-0140. E-mail address: [email protected]. URL: http://www.cs.columbia.edu/~agw. 1 This research was supported in part by the National Science Foundation under Grant CCR-99-87858, as well as by a Fordham University Faculty Fellowship. 0885-064X/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0885-064X(03)00030-X

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1. Introduction We are interested in the worst case complexity of solving Fredholm problems of the second kind ðI  Tk Þu ¼ f

ð1Þ

on the unit cube I d ¼ ½0; 1d ; where Z kð; yÞvðyÞ dy 8vALp ðI d Þ Tk v ¼ Id

for a continuous kernel function k : I d I d -R: Here, pA½1; N; and error is measured in the Lp ðI d Þ-norm. Previous work on this problem has either assumed that we have had complete information about k; or that k and f have had the same smoothness, see, e.g., [5,6,8,10,14,15, Section 6.3], and the references contained therein. What happens when we weaken these assumptions? There are two issues to deal with. First, we want to know where smoothness counts the most for Fredholm problems, as we did in [16] for two-point boundary value problems. That is, we would like to know which is more important—the smoothness of the kernel or of the right-hand side—in determining the complexity. In addition, note that (with the exception of [8]) the references listed above have all assumed that the available information is exact. But in practice, information evaluations are often contaminated by noise [11]. Hence we wish to know how noisy information affects the complexity, as well as which algorithms are optimal when the information is noisy. In this paper, we study the worst case complexity of Fredholm problems under the following assumptions: (1) The right-hand side f belongs to the unit ball of W r;p ðI d Þ; with r4d=p: (2) The kernel k belongs to a ball of W s;N ðI 2d Þ; and I  Tk is an invertible operator on Lp ðI d Þ: (3) Only noisy standard information is available. That is, for any x; yAI d ; we can only calculate f ðxÞ or kðx; yÞ with error at most d; where dA½0; 1 is a known noise level. We are able to determine rn ðdÞ; the nth minimal radius of d-noisy information, i.e., the minimal error when we use n evaluations with a noise level of d: We find that2 rn ðdÞ^nm þ d with a proportionality factor independent of n and d; where nr s o m ¼ min ; : ð2Þ d 2d Moreover, we describe an algorithm using n evaluations with noise level d that is a nearly-minimal error algorithm. This algorithm is a modified finite element method 2

In this paper, we use %; k; and ^ to denote O-, O-, and Y-relations.

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(mfem) using (nonadaptive) noisy information. The modification consists of replacing the kernel k and the right-hand side f that would appear in the ‘‘pure’’ finite element method by their piecewise-polynomial interpolants. Hence this algorithm uses noisy standard information, rather than continuous linear information. We shall refer to this algorithm as the ‘‘noisy mfem.’’ This is, of course, a bit of a misnomer, since the algorithm is not noisy (only the information is noisy); but ‘‘noisy mfem’’ is more succinct than ‘‘mfem using noisy information.’’ We also analyze the cost of the noisy mfem. Let cðdÞ denote the cost of evaluating a function with a noise level d: Then the information cost of this algorithm is cðdÞn: Let us now discuss the combinatory cost of the noisy mfem. This algorithm requires the solution of an n n linear system ðA  BÞu ¼ f: Here, A is the Gram matrix of the finite element space, B depends on the kernel k and f depends on the right-hand side f : If we were considering only a single fixed kernel k; then we could precompute the LU-decomposition of the nonsingular matrix A  B; since this is independent of any particular f : We could then ignore the cost of this precomputation, considering it as a fixed overhead, since it need only be done once. Even so, the combinatory cost of our algorithm would be Yðn2 Þ; since the factors of the LU-decomposition of A  B are dense n n triangular matrices. Of course, things are much worse for our problem, since both the right-hand sides f and the kernels k are varying. Clearly, the factorization of A  B is no longer independent of the problem element being considered, and so we would not be able to ignore the Oðn3 Þ-cost of this factorization. Hence, we see that the combinatory cost of the noisy mfem would overwhelm the information cost as n grows large. We can overcome this difficulty by using a two-grid implementation of the noisy mfem. This algorithm has the same order of error as the original noisy mfem, and its combinatory cost is OðnÞ: Hence, we can calculate the two-grid approximation using YðnÞ arithmetic operations, which is optimal. Using these results, we can determine tight bounds on the e-complexity of the Fredholm problem. There exist positive constants C1 ; C2 ; and C3 ; independent of e; such that the problem complexity is bounded from below by ( & 1=m ’) 1 compðeÞX inf cðdÞ 0odoC1 e C1 e  d and from above by compðeÞpC2

( inf

0odoC3 e

& cðdÞ

1 C3 e  d

1=m ’) :

These upper bounds are attained by two-grid implementations of the noisy modified fem, with d chosen to minimize the right-hand sides of the upper bound. As a specific example, suppose that cðdÞ ¼ dt for some tX0: We find that  tþ1=m 1 compðeÞ^ : e Thus we have found sharp bounds on the e-complexity.

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How much do we lose when we go from exact information to noisy information? Suppose once again that cðdÞ ¼ dt for some tX0: Since exact information is merely noisy information with t ¼ 0; we see that the complexity for exact information is proportional to cð1=eÞ1=m ; where c is the cost of one function evaluation. For the sake 0 of comparison, let us write the complexity for noisy information as ð1=eÞ1=m ; where m0 ¼ m 

1 : 1 þ tm

Note that since the information is noisy, we have t40; and so m0 om: Hence we see that the complexity of our problem using noisy information of smoothness ðr; sÞ is the same as the complexity using exact information of lesser smoothness ðr0 ; s0 Þ; where r0 ¼ r=ð1 þ tmÞ and s0 ¼ s=ð1 þ tmÞ: We now outline the rest of this paper. In Section 2, we precisely describe the problem to be solved. In Section 3, we prove a lower bound on the minimal error using noisy information. It is easy to find a matching upper bound using the general approach of interpolatory algorithms. However, this approach does not address the issue of combinatory cost. Since the problem is nonlinear, it is unclear whether there exists an interpolatory algorithm with (roughly) linear combinatory cost. The remainder of this paper deals with showing that such an algorithm exists, and is given as a two-grid implementation of a noisy modified finite element method (noisy mfem). In Section 4, we define some useful finite element spaces, which are used in Section 5 to define the noisy mfem. In Section 6, we establish an error bound for the noisy mfem. In Section 7, we show that the noisy mfem is a minimal error algorithm. In Section 8, we describe the two-grid implementation of the noisy mfem, showing that its error is essentially the same as the noisy mfem itself, and that its combinatory cost is essentially optimal. Finally, in Section 9, we determine the e-complexity of the noisy Fredholm problem.

2. Problem description In this section, we precisely describe the class of Fredholm problems whose solutions we wish to approximate. For an ordered ring R; we shall let Rþ and Rþþ respectively denote the nonnegative and positive elements of R: Hence (for example), Zþ denotes the set of natural numbers (nonnegative integers), whereas Zþþ denotes the set of strictly positive integers. For a normed linear space X; we let BX denote the unit ball of X: We assume that the reader is familiar with the standard concepts and notations involving Sobolev norms and spaces, as found in, e.g., [3]. We are given dAZþþ and pA½1; N; as well as real numbers r and s satisfying r4d=p and s40: Hence, the Sobolev space W r;p ðI d Þ is embedded in the space CðI d Þ of continuous functions, and W s;N ðI 2d Þ is embedded in CðI 2d Þ; by the Sobolev embedding theorem.

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For kAW s;N ðI 2d Þ; define Tk : Lp ðI d Þ-Lp ðI d Þ as Z ðTk vÞðxÞ ¼ kðx; yÞvðyÞ dy 8xAI d : Id

The operator Tk is compact, see, e.g., [4, p. 518], and hence I  Tk is an invertible operator on Lp ðI d Þ iff 1 is not an eigenvalue of Tk : We are now ready to describe our class of problem elements. We first describe the class of kernels k: Let c1 40 and c2 41 be given. Then we let K ¼ Kc1 ;c2 denote the class of all functions kAW s;N ðI 2d Þ such that jjkjjW s;N ðI 2d Þ pc1 and jjðI  Tk Þ1 jjLin½Lp ðI d Þ pc2 : Here, jj  jjLin½X is the usual operator norm. The class of right-hand sides will be BW r;p ðI d Þ: Finally, we let F ¼ BW r;p ðI d Þ K be our class of problem elements. We are now ready to define our solution operator S : F -Lp ðI d Þ as Sð½ f ; kÞ ¼ ðI  Tk Þ1 f

8½ f ; kAF :

Hence u ¼ Sð½ f ; kÞ is the solution of (1) for ½ f ; kAF : We wish to calculate approximate solutions to this problem, using noisy standard information. To be specific, we will be using uniformly sup-norm-bounded noise. Our notation and terminology is essentially that of [11], although we sometimes use modifications found in [12]. Let dA½0; 1 be a noise level. For ½ f ; kAF ; we calculate d-noisy information z ¼ ½z1 ; y; znðzÞ  about ½ f ; k: Here, for each index iAf1; y; nðzÞg; either jzi  f ðxi Þjpd

for xi AI d ;

or jzi  kðxi ; yi Þjpd

for ðxi ; yi ÞAI 2d :

The choice of whether to evaluate k or f at the ith sample point, as well as the choice of the ith sample point itself, may be determined either nonadaptively or adaptively. Moreover, the information is allowed to be of varying cardinality. For ½ f ; kAF ; we let Nd ð½ f ; kÞ denote the set of all such d-noisy information z about ½ f ; k; and we let [ Z¼ Nd ð½ f ; kÞ ½ f ;kAF

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denote the set of all possible noisy information values. Then an algorithm using the noisy information Nd is a mapping f : Z-Lp ðI d Þ: Remark 1. Note that the permissible information consists of function values of f and k: One could allow the evaluation of derivatives as well. We restrict ourselves to function values alone, since this simplifies the exposition. There is no loss of generality in doing this, since the results of this paper also hold if derivative evaluations are allowed. We want to solve the Fredholm problem in the worst case setting. This means that the cardinality of information Nd is given as card Nd ¼ sup nðzÞ zAZ

and the error of an algorithm f using Nd is given as eðf; Nd Þ ¼ sup

sup

½ f ;kAF zANd ð½ f ;kÞ

jjSð½ f ; kÞ  fðzÞjjLp ðI d Þ :

As usual, we will need to know the minimal error achievable by algorithms using specific information, as well as by algorithms using information of specified cardinality. Let nAZþ and dA½0; 1: If Nd is d-noisy information of cardinality at most n; then rðNd Þ ¼

inf

f using Nd

eðf; Nd Þ:

is the radius of information, i.e., the minimal error among all algorithms using given information Nd : An algorithm f
using Nd is said to be an optimal error algorithm3 if eðf
; Nd Þ^rðNd Þ; the proportionality constant being independent of n and d: The nth minimal radius, rn ðdÞ ¼ inffrðNd Þ: card Nd pn g; is the minimal error among all algorithms using d-noisy information of cardinality at most n: Noisy information Nn;d of cardinality n such that rðNn;d Þ^rn ðdÞ; the proportionality factor being independent of both n and d; is said to be nth optimal information. An optimal error algorithm using nth optimal information is said to be an nth minimal error algorithm. Next, we describe our model of computation. We will use the model found in [11, Section 2.9]. (However, note that in the present paper, the accuracy d is the same for all noisy observations, whereas d may differ from one observation to another in [11].) Here are the most important features of this model:

3

In this paper, we ignore constant multiplicative factors in our definitions of optimality. The more fastidious may use the term ‘‘quasi-optimal’’ if they desire.

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(1) For any xAI d and any f AW r;p ðI d Þ; the cost of calculating a d-noisy value of f ðxÞ is cðdÞ: (2) For any ðx; yÞAI 2d and any kAK; the cost of calculating a d-noisy value of kðx; yÞ is cðdÞ: (3) Real arithmetic operations and comparisons are done exactly, with unit cost. Here, the cost function c : Rþ -Rþþ is nonincreasing. For any noisy information Nd and any algorithm f using Nd ; we shall let costðf; Nd Þ denote the worst case cost of computing fðzÞðxÞ for zAZ and xAI d : We can decompose this as follows. Let costinfo ðNd Þ ¼ supfcost of computing zg zAZ

denote the worst case information cost. Note that if Nd is information of cardinality n; then costinfo ðNd ÞXcðdÞn: Here, equality holds for nonadaptive information, but strict inequality can hold for adaptive information, since we must be concerned with the cost of choosing each new adaptive sample point. We also let costcomb ðf; Nd Þ ¼ sup sup fcost of computing fðzÞðxÞ; given zAZg zAZ xAI d

denote the worst case combinatory cost. Then costðf; Nd Þpcostinfo ðNd Þ þ costcomb ðf; Nd Þ: Now that we have defined the error and cost of an algorithm, we can finally define the complexity of our problem. We shall say that compðeÞ ¼ inffcostðf; Nd Þ: Nd and f such that eðf; Nd Þpe g is the e-complexity of our problem. An algorithm f using noisy information Nd for which eðf; Nd Þpe

and

costðf; Nd Þ^compðeÞ;

the proportionality factor being independent of both d and e; is said to be an optimal algorithm.

3. Lower bounds In this section, we prove a lower bound on the nth minimal error using d-noisy information. Theorem 2. Recall from (2) that nr s o m ¼ min ; : d 2d

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There is a constant M0 ; independent of n and d; such that rn ðdÞXM0 ðnm þ dÞ for all nAZþ and dA½0; 1: Proof. We first claim that rn ðdÞknr=d þ d:

ð3Þ

Indeed, since T0 ¼ 0; we find that Sð½ f ; 0Þ ¼ f for all f AW r;p ðI d Þ: Thus app; the problem of approximating functions from BW r;p ðI d Þ in the Lp ðI d Þ-norm, is a special instance of our problem, and so rn ðdÞXrn ðd; appÞ; the latter denoting the nth minimal radius of d-noisy information for app: Clearly rn ðd; appÞXrn ð0; appÞ:

ð4Þ

Moreover, rn ð0; appÞknr=d ; see, e.g., [9, p. 34]. Hence rn ðd; appÞknr=d :

ð5Þ

Thus, to establish (3), we only need to prove that rn ðd; appÞkd:

ð6Þ

Let Nd be noisy information of cardinality at most n: By the results in [11, Chapter 2.7], there exists l 0 Af1; y; ng and nonadaptive information Nnon of cardinality l 0 d such that rðNd ; appÞX12 rðNnon d ; appÞ: By Plaskota [11, Lemma 2.8.2], rðNnon d ; appÞkd: Hence rðNd ; appÞkd: Since Nd is arbitrary information of cardinality at most n; we find that (6) holds. Using (4)–(6), we find that (3) holds, as claimed. We now claim that rn ð0Þkns=2d holds. Our approach follows that outlined in [5, pp. 260–261]. Let  1 ; 1Þ and k ¼ min y c ; 1  y1 Aðc1 ; 0 1 1 2 y1 c 2

ð7Þ

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and define f
 1 and

k
 k0 :

Now jjk
jjW s;N ðI 2d Þ ¼ k0 oc1 :

ð8Þ

It is easy to see that jjTk jjLin½Lp ðI d Þ pjjkjjLN ðI 2d Þ

8kAK:

ð9Þ

In particular, we have jjTk
jjLin½Lp ðI d Þ pk0 o1; so that jjðI  Tk
Þ1 jjLin½Lp ðI d Þ p

1 py1 c2 oc2 : 1  k0

ð10Þ

From (8) and (10), we see that k
AK: Since it is clear that f
ABW r;p ðI d Þ; we find that ½ f
; k
AF : Let N be noiseless information of cardinality at most n: Then we may write Nð½ f
; k
Þ ¼ ½z1 ; y; zl  for some lpn; where each zi is an evaluation of either f
or k
: Suppose that there are l 0 evaluations of k
: Without loss of generality, we may assume that these evaluations have the form zi ¼ k
ðxi ; yi Þ

ð1pipl 0 Þ:

From [2] (see also [9, p. 34]), we can find a function wABW s;N ðI 2d Þ such that 0pwðx; yÞpk0 wðxi ; yi Þ ¼ 0

8x; yAI d ;

ð1pipl 0 Þ;

jjwjjW s;N ðI 2d Þ ¼ 1; Z I 2d

y2 wðx; yÞ dx dyX s=2d ; 0 ðl Þ

where y2 is a positive constant that is independent of the points ðxi ; yi Þ and of l 0 : Let y3 ¼ minfð1  y1 Þc1 ; 1  c1 2  k0 g: Note that since y1 o1 and k0 p1  ðy1 c2 Þ1 ; we have k0 o1  c1 2 ; and so y3 40: We define k

¼ k0 þ y3 w:

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We claim that k

AK: Indeed, we have jjk

jjW s;N ðI 2d Þ p jjk0 jjW s;N ðI 2d Þ þ y3 jjwjjW s;N ðI 2d Þ ¼ k0 þ y3 p y1 c1 þ y3 pc1 : Moreover, jjTk

jjLin½Lp ðI d Þ pjjTk0 jjLin½Lp ðI d Þ þ y3 jjTw jjLin½Lp ðI d Þ pk0 þ y3 o1;

ð11Þ

and thus jjðI  Tk

Þ1 jjLin½Lp ðI d Þ p

1 pc2 : 1  ðk0 þ y3 Þ

Hence, k

AK: Letting f
 1; we let u
¼ Sð½ f
; k
Þ

u

¼ Sð½ f
; k

Þ:

and

Since ½ f
; k
; ½ f
; k

AF

with Nð½ f
; k
Þ ¼ Nð½ f
; k

Þ;

we have rðNÞX12jju
 u

jjLp ðI d Þ ;

ð12Þ

see, e.g., [13, pp. 45, 49]. We claim that u

41 on I d : Indeed, since (11) holds, the Neumann series ðI  Tk

Þ1 ¼

N X

Tkj



j¼0

converges in Lin½Lp ðI d Þ: Now Tkj

¼ Tkj



for jX1;

where fkj

gN j¼1 is defined inductively as ( k

ðx; yÞ if j ¼ 1;

kj ðx; yÞ ¼ R



I d k ðx; tÞ kj1 ðt; yÞ dt if jX2 Hence u

¼

N X

Tkj

f
:

j¼0

By induction, we find that kj

ðx; yÞXk0j

8x; yAI d ; 8jX1;

8x; yAI d :

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and thus for xAI d ; we have N Z N X X

u ðxÞ ¼ 1 þ kj

ðx; yÞ dyX1 þ k0j ¼ j¼1

Id

as claimed. Hence






u ðxÞ  u ðxÞ ¼ k0

j¼1

Z








½u ðyÞ  u ðyÞ dy þ y3

Id

1 41; 1  k0

Z

wðx; yÞu

ðyÞ dy:

Id

Since u

41 on I d and w40 on I 2d ; we find that Z Z ð1  k0 Þ ½u

ðxÞ  u
ðxÞ dx ¼ y3 wðx; yÞu

ðyÞ dy dx Id I 2d Z 4 y3 wðx; yÞ dy dx I 2d

X

y2 y3 ðl 0 Þs=2d

y2 y3 X s=2d ; n

the latter since l 0 pn: By Minkowski’s inequality, we have Z ð1  k0 Þ ½u

ðxÞ  u
ðxÞ dxpð1  k0 Þjju

 u
jjLp ðI d Þ : Id

Using the last two inequalities and (12), we get rðNÞX

y2 y3 : 2ð1  k0 Þns=2d

Since N is arbitrary information of cardinality at most n; inequality (7) holds, as claimed. From (3), we see that rn ðdÞkd; which, together with (7), implies that rn ðdÞkns=2d þ d: The theorem now follows immediately from this inequality and (3).

&

4. Some finite element spaces Now that we have a lower bound on the nth minimal radius for our problem, the next task will be to find a matching upper bond and an nth minimal error algorithm. This algorithm will be a modified finite element method using noisy information. Before describing the algorithm, we need to define some finite element spaces. In what follows, our notation is based on the standard one found in, e.g., [3,15, Chapter 5].

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Let mAZþ : For KDRd ; let ( ) X a aa x : xAK Qm ðKÞ ¼ 0pa1 ;y;ad pm

denote the polynomials of degree at most m in each variable, with the domain restricted to K: Here, we recall that xa ¼ xa11 yxadd for any multi-index a ¼ ða1 ; y; ad Þ: Clearly Qm ðKÞ is a function space over K; with dim Qm ðKÞ ¼ ðm þ 1Þd : In particular, we note that the space Qm ðI d Þ has a basis fˆs1 ; y; sˆa g consisting of tensor products. More precisely, let us fix a choice of interpolation nodes 0ox# 1 o?ox# m o1: We then define Y x#  x# j # ¼ ð0pipmÞ pˆ i ðxÞ # # 0pjpm xi  xj jai

as the usual one-dimensional Lagrange basis polynomials for Qm ðIÞ with respect to ðmþ1Þd the interpolation points 0ox# 1 o?ox# m o1: Now let faðiÞ gi¼1 be an enumeration of the multi-indices aAðZþ Þd ðiÞ ðiÞ ða1 ; y; ad Þ:

satisfying max1pjpd aj pm; we write aðiÞ ¼

We can set

sˆi ðx# 1 ; y; x# d Þ ¼

d Y j¼1

pˆ aðiÞ ðx# j Þ j

and ðiÞ ðiÞ xˆ i ¼ ðx# a1 ; y; x# ad Þ:

Then fˆs1 ; y; sˆðmþ1Þd g is a basis for Qm ðI d Þ such that sˆj ðxˆ i Þ ¼ di; j

for 1pi; jpðm þ 1Þd :

Associated with the space Qm ðI d Þ; we have the polynomial interpolation operator # : CðI d Þ-Qm ðI d Þ defined as P #v¼ P#

d ðmþ1Þ X

v#ðxˆ i Þˆsi

8#vACðI d Þ:

i¼1

Now let K be a cube in Rd whose sides are parallel to the coordinate axes. Then K can be written as the image of I d under an affine bijection FK : I d -K having the form ˆ ¼ hK xˆ þ bK FK ðxÞ

ˆ d; 8xAI

where hK is the length of any side of K and bK is the element in K closest to the origin, i.e., the smallest corner of K: We get a basis fs1;K ; y; sðmþ1Þd ;K g for Qm ðKÞ

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by taking sj;K ¼ sˆj 3FK1 ; that is, ˆ sj;K ðxÞ ¼ sˆj ðxÞ;

where xˆ ¼ FK1 ðxÞ ¼

x  bK hK

for 1pjpðm þ 1Þd : Defining xj;K ¼ FK ðxˆ j Þ for 1pjpðm þ 1Þd ; we find that sj;K ðxi;K Þ ¼ di; j

for 1pi; jpðm þ 1Þd :

Associated with the polynomial space Qm ðKÞ; we have an interpolation operator PK : CðKÞ-Qm ðKÞ defined as d ðmþ1Þ X PK v ¼ vðxj;K Þsj;K 8vACðKÞ; j¼1

so that # vÞðxÞ ˆ ðPK vÞðxÞ ¼ ðP#

for v# ¼ v3FK and xˆ ¼ FK1 ðxÞ:

We are finally ready to define finite element spaces. Choose h40 such that 1=h is an integer. Let Qh be a decomposition of I d into congruent cubes whose sides parallel the coordinate axes and have length h: Then o   Sh ¼ vALp ðI d Þ: v AQm ðKÞ for KAQh K

is our finite element space. Note that since jQh j ¼ hd ; we have   mþ1 d : nh :¼ dim Sh ¼ h

ð13Þ

We now construct a basis fs1 ; y; snh g for Qh : Let bK1 ; y; bKhd be an enumeration of the points fbK gKAQh by lexicographic ordering. This induces an enumeration K1 ; y; Khd of the cubes KAQh : We then let shd ði1Þþj ¼ si;Kj for 1pjphd ; 1pipðm þ 1Þd ; with each si;K being extended from K to I d as being zero outside K: Analogously, we let xhd ði1Þþj ¼ xi;Kj for 1pjphd ; 1pipðm þ 1Þd : We then find that sj ðxi Þ ¼ di; j for 1pi; jpnh : Associated with the finite element space Sh ; we have an interpolation operator Ph : CðI d Þ-S h ; defined as X PK v 8vACðI d Þ; Ph v ¼ KAQh

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where each PK v is extended from K to I d as being zero outside K: Alternatively, we may write nh X Ph v ¼ vðxj Þsj 8vACðI d Þ: j¼1

We have a second interpolation operator Ph#h : CðI 2d Þ-Sh #Sh ; defined as ðPh#h vÞðx; yÞ ¼ Ph ½x/Ph ðy/vðx; yÞÞ ¼

nh X

vðxi ; xj Þsj ðyÞsi ðxÞ

i; j¼1

for x; yAI d and vACðI 2d Þ: Remark 3. In the sequel, we shall often write si;h and xj;h rather than si and xj ; to indicate their dependence on h: We now present some standard error estimates, which will be useful in the sequel. Lemma 4. Let tX0 and qA½1; N: There exists M1 40 such that the following hold: (1) Let vAW t;q ðI d Þ: Then jjv  Ph vjjLq ðI d Þ pM1 hminfmþ1;tg jjvjjW t;q ðI d Þ : (2) Let wAW t;q ðI 2d Þ: Then jjw  Ph#h wjjLq ðI 2d Þ pM1 hminfmþ1;tg jjwjjW t;q ðI 2d Þ :

Proof. For KDRd ; let 8 9 < X = aa xa : xAK Pm ðKÞ ¼ : jajpm ; denote the polynomials of total degree at most m: Since Pm ðI d ÞDQm ðI d Þ; we see that # v ¼ v# for all vAPm ðI d Þ: Hence the local estimates of [3, pp. 118–122] hold. Since P# there are no inter-element continuity relations to deal with, the global estimates of [3] hold as well. This suffices to establish the lemma. & Remark 5. We have chosen our finite element space to be piecewise Qm : Alternatively, we could have chosen the space to be piecewise Pm : Making this change would not have changed the results of this paper. Moreover, this replacement would make for a smaller finite element space. The main reason for choosing piecewise Qm spaces is that this choice leads more easily to an explicit construction of the basis functions.

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Remark 6. The interpolation nodes 0ox# 1 o?ox# m o1 were not specified. Indeed, the main results of this paper are independent of how these nodes are chosen. However, the upper-bound constants of this paper may well be influenced by their choice. Hence, this issue may be important in practical computations. For qA½1; N and h40; let Ph : Lq ðI d Þ-Sh be the mapping defined by /Ph v; wS ¼ /v; wS

8vALq ðI d Þ; wASh :

ð14Þ

Here, /; S is the standard duality pairing Z vðxÞwðxÞ dx 8vALp ðI d Þ; wALp0 ðI d Þ /v; wS ¼ Id

with p0 ¼

p p1

denoting the exponent conjugate to p: We have: Lemma 7. Let qA½1; N: There exists pq 40 such that for any h40; jjPh vjjLq ðI d Þ ppq jjvjjLq ðI d Þ

8vALq ðI d Þ:

Proof. See, e.g., [15, pp. 177–178], and the references cited therein.

&

5. The noisy modified FEM We now define the noisy modified finite element method (noisy mfem). This is an algorithm using information consisting of noisy function evaluations. As mentioned in the Introduction, it would be somewhat more accurate to describe this method as the ‘‘mfem using noisy information,’’ but the conciseness of ‘‘noisy mfem’’ outweighs its mild inaccuracy. The easiest way to describe the noisy mfem is by following three steps. First, we describe the pure finite element method, which uses inner product information. Next, we describe the noise-free mfem, which uses noise-free standard information. Finally, we describe the noisy mfem, which uses noisy standard information. We first recall how the pure finite element method is defined. Let ½ f ; kAF and h40: Then the pure finite element method (pure fem) consists of finding uh ASh such that Bðuh ; w; kÞ ¼ / f ; wS

8wASh ;

where Bðv; w; kÞ ¼ /ðI  Tk Þv; wS

8vALp ðI d Þ; wALp0 ðI d Þ:

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Alternatively, we have ðI  Ph Tk Þuh ¼ Ph f : If we write uh ðxÞ ¼

nh X

uj sj;h ðxÞ

8xAI d ;

j¼1

then we see that the vector u ¼ ½u1 ; y; unh T is the solution of the linear system ðA  BÞu ¼ f; where ai; j ¼ /sj;h ; si;h S and

bi; j ¼ /Tk sj;h ; si;h S for 1pi; jpnh

and f ¼ ½/ f ; s1;h Sy/f ; snh ;h ST : Of course, the pure fem requires the calculation of / f ; si S and /Tk sj ; si S: These are weighted integrals of f and k: Since we are only using (noisy) standard information, such information about f and k is not available to us. Instead, we replace f and k by their interpolants. This gives us an approximation, the modified mfem, that uses only standard information. % More precisely, let h; h40: For ½ f ; kAF ; we define Bh% ðv; w; kÞ ¼ Bðv; w; Ph# % h% kÞ

8vALp ðI d Þ; wALp0 ðI d Þ

and let fh ðwÞ ¼ /Ph f ; wS

8wALp0 ðI d Þ:

Note that for vALp ðI d Þ and wALp0 ðI d Þ; we have /TPh# % h% k v; wS ¼

nh% X

kðxi;h% ; xj;h% Þ/sj;h% ; vS/si;h% ; wS;

i; j¼1

so that Bh% ðv; w; kÞ ¼ /v; wS 

nh% X

kðxi;h% ; xj;h% Þ/sj;h% ; vS/si;h% ; wS:

i; j¼1

Moreover fh ðwÞ ¼

nh X

f ðxj;h Þ/sj;h ; wS 8wALp0 ðI d Þ:

j¼1

The modified finite element method (mfem) consists of finding uh;h% ASh such that Bh% ðuh;h% ; w; kÞ ¼ fh ðwÞ

8wASh :

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If we write uh;h% ðxÞ ¼

nh X

uj sj;h ðxÞ

8xAI d ;

j¼1

then we see that the vector u ¼ ½u1 ; y; unh T is the solution of the linear system ðA  BÞu ¼ f: Here ai; j ¼ /sj;h ; si;h S and

bi; j ¼ /TPh# for 1pi; jpnh % h% k sj;h ; si;h S

and f ¼ ½ fh ðs1;h Þyfh ðsnh ;h ÞT : Of course, the mfem uses noise-free information. If we allow noisy evaluations in the % d40: For ½ f ; kAF ; we mfem, we get the noisy mfem. More precisely, let h; h; calculate f˜i;d AR such that j f ðxi;h Þ  f˜i;d jpd for 1pipnh and k˜i; j;d AR such that jkðxi;h ; xj;h Þ  k˜i; j;d jpd

for 1pi; jpnh% :

Let Tk;h;d % v ¼

nh% X

k˜i; j;d /sj;h% ; vSsi;h%

8vALp ðI d Þ

i; j¼1

and Ph;d f ¼

nh X

f˜j;d sj;h :

j¼1

For kAK; define a bilinear form Bh;d % ð; ; kÞ approximating Bh% ð; ; kÞ as Bh;d % v; wS % ðv; w; kÞ ¼ /v  Tk;h;d

8vALp ðI d Þ; wALp0 ðI d Þ

and a linear form fh;d approximating fh as fh;d ðwÞ ¼ /Ph;d f ; wS 8wALp0 ðI d Þ:

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The noisy modified finite element method (noisy mfem) consists of finding uh;h;d % ASh such that Bh;d % ; w; kÞ ¼ fh;d ðwÞ 8wASh : % ðuh;h;d Writing uh;h;d % ðxÞ ¼

nh X

8xAI d ;

uj sj;h ðxÞ

j¼1

we see that the vector u ¼ ½u1 ; y; unh T is the solution of the linear system ðA  BÞu ¼ f:

ð15Þ

Here ai; j ¼ /sj;h ; si;h S and

bi; j ¼ /Tk;h;d % sj;h ; si;h S for 1pi; jpnh

and f ¼ ½ fh;d ðs1;h Þyfh;d ðsnh ;h ÞT : Let

% h;d Nh;h;d % ðkÞ; % ð½ f ; kÞ ¼ ½Nh;d ð f Þ; N

where Nh;d ð f Þ ¼ ½f˜1;d ; y; f˜nh ;d  and % h;d % ð1Þ % ðn% h% Þ ðkÞ N % ðkÞ ¼ ½N % ðkÞ; y; Nh;d h;d with ˜ ˜ % ðiÞ N % ðkÞ ¼ ½ki;1;d ; y; ki;nh% ;d  for 1pipnh% : h;d If uh;h;d % is well defined, then we can write uh;h;d % ¼ fh;h;d % ðNh;h;d % ð½ f ; kÞÞ; where card Nh;h;d % ¼

n2h%

 þ nh ¼

mþ1 h%

 2d  m þ 1 d %2d þ ^h þ hd : h

6. Error analysis of the noisy MFEM In this section, we establish an error bound for the noisy modified fem. We do this as follows. First, we establish the uniform weak coercivity of the bilinear forms Bð; ; kÞ for kAK: Once we know that the bilinear forms are uniformly weakly coercive, we can obtain an abstract error estimate, as a variant of the First Strang Lemma (see, e.g., [3, p. 186]). The remaining task is then to estimate the various terms appearing in this abstract error estimate. So, the first task is to establish uniform weak coercivity. Before doing so, we establish two auxiliary lemmas.

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The first lemma shows that the inverses of certain operators are uniformly bounded. Let  1=minfmþ1;sg 1 h0 ¼ : 2c1 c2 M1 Recall that the adjoint of a linear transformation A : Lp ðI d Þ-Lp ðI d Þ of normed linear spaces is the linear operator A
: Lp0 ðI d Þ-Lp0 ðI d Þ satisfying /A
v; wS ¼ /v; AwS

8vALp ðI d Þ; wALp0 ðI d Þ:

In particular, for any kAK; we have Z kðx; yÞwðxÞ dx 8wALp0 ðI d Þ: ðTk
wÞðyÞ ¼ Id

Lemma 8. Let hAð0; h0  and kAK: Then I  TP
h#h k is invertible on Lp0 ðI d Þ; with jjðI  TP
h#h k Þ1 jjLin½Lp0 ðI d Þ p2c2 : Proof. Let hAð0; h0  and kAK: Note that since ðA
Þ1 ¼ ðA1 Þ
for any invertible linear transformation A; we find that I  Tk
is invertible and jjðI  Tk
Þ1 jjLin½Lp0 ðI d Þ pc2 : Let us write
I  TP
h#h k ¼ ðI  Tk
Þ þ TkP : h#h k

From (9) and Lemma 4, along with the definition of the class K; we find
minfmþ1;sg jj jjkjjW s;N ðI 2d Þ jjTkP d p jjk  Ph#h kjjL ðI 2d Þ pM1 h N h#h k Lin½Lp0 ðI Þ minfmþ1;sg

p M 1 h0

 c1 ¼

1 ; 2c2

and so

1 jjTkP jj jjLin½Lp0 ðI d Þ p d jjðI  Tk Þ h#h k Lin½Lp0 ðI Þ

1 1  c2 ¼ : 2c2 2

From this inequality and [7, Lemma 1.3.14] we see that I  TP
h#h k is invertible, with jjðI 

TP
h#h k Þ1 jjLin½Lp0 ðI d Þ p

jjðI  Tk
Þ1 jjLin½Lp0 ðI d Þ

1 jj 1  jjTkP jj d jjðI  Tk Þ Lin½Lp0 ðI d Þ h#h k Lin½Lp0 ðI Þ

p 2c2 ; as required.

&

Remark 9. Note that TP
h#h k : Sh -Sh : Hence if hAð0; h0 ; the mapping I  TP
h#h k is an invertible linear operator on Sh :

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Our second auxiliary lemma shows that certain inner products can be bounded from below by products of norms. Lemma 10. Let vALp ðI d Þ be nonzero. For any tAð0; jjvjjLp ðI d Þ Þ; there is a nonzero function gALp0 ðI d Þ such that /v; gSXðjjvjjLp ðI d Þ  tÞjjgjjLp0 ðI d Þ : Proof. Suppose first that poN: Let g ¼ ðsgn vÞjvjp1 : Then g is nonzero, with /v; gS ¼ jjvjjLp ðI d Þ jjgjjLp0 ðI d Þ ; which is a stronger result than that which we want to prove. Hence it only remains to show that the lemma holds when p ¼ N: We use an idea found in [1, p. 26]. For tAð0; jjvjjLN ðI d Þ Þ; let E ¼ fxAI d : jvðxÞj4jjvjjLN ðI d Þ  t g: From the definition of the essential supremum, meas E40: Let g ¼ ðsgn vÞwE be the characteristic function of E: Then g is a nonzero function, with Z jjgjjL1 ðI d Þ ¼ wE ðxÞ dx ¼ meas E: Id

Hence we have /v; gS ¼

Z E

jvðxÞj dxXðjjvjjLN ðI d Þ  tÞmeas E ¼ ðjjvjjLN ðI d Þ  tÞjjgjjL1 ðI d Þ :

Hence the lemma holds when p ¼ N: & We are now ready to prove uniform weak coercivity of the bilinear forms Bð; ; kÞ over all kAK: Lemma 11. There exist h1 40 and g40 such that the following holds: for any kAK; any hAð0; h1 ; and any vASh ; there exists nonzero wASh such that Bðv; w; kÞXgjjvjjLp ðI d Þ jjwjjLp0 ðI d Þ :

ð16Þ

Proof. Let kAK and hAð0; h0 : Let vASh : If v ¼ 0; then this inequality holds for any nonzero wASh : So we may restrict our attention to the case va0: By Lemma 10, there exists nonzero gALp0 ðI d Þ such that /v; gSX12jjvjjLp ðI d Þ jjgjjLp0 ðI d Þ :

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Recalling the definition of the orthogonal projector Ph from (14) and using the remark following Lemma 8, we see that w ¼ ðI  TP
h#h k Þ1 Ph g is a well defined element of Sh : Since vASh ; we clearly have /v; ðI  TP
h#h k ÞwS ¼ /v; gSX12jjvjjLp ðI d Þ jjgjjLp0 ðI d Þ : Moreover, from Lemmas 7 and 8, we have jjwjjLp0 ðI d Þ p jjðI  TP
h#h k Þ1 jjLin½Lp0 ðI d Þ jjPh gjjLp0 ðI d Þ p 2c2 jjPh gjjLp0 ðI d Þ p2pp0 c2 jjgjjLp0 ðI d Þ : Hence 1 /ðI  TPh#h k Þv; wSX12jjvjjLp ðI d Þ jjgjjLp0 ðI d Þ X jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ : 4pp0 c2 Since g and v are nonzero, this inequality implies that /ðI  TPh#h k Þv; wS is nonzero. Since the latter is linear in w; we see that wa0: Using (9) and Lemma 4, we find j/TkPh#h k v; wSjp jjTkPh#h k vjjLp ðI d Þ jjwjjLp0 ðI d Þ p jjk  Ph#h kjjLN ðI 2d Þ jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ p M1 hminfmþ1;sg jjkjjW s;N ðI 2d Þ jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ p c2 M1 hminfmþ1;sg jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ : Hence Bðv; w; kÞ ¼ /ðI  TPh#h k Þv; wS  /TkPh#h k v; wS   1 X  c2 M1 hminfmþ1;sg jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ : 4pp0 c2 Letting

(

h1 ¼ min

1 8pp0 c22 M1

)

1=minfmþ1;sg ; h0

and g¼

1 ; 8pp0 c2

we see that the desired estimate (16) holds for hAð0; h1 : & Since the bilinear forms Bð; ; kÞ are uniformly weakly coercive for kAK; we have the following variant of the First Strang Lemma found in [3, p. 186]:

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Lemma 12. Suppose there exist d0 Að0; 1 and h2 Að0; h1  such that the following holds: % for any dA½0; d0 ; any h; hAð0; h2 ; and any kAK; we have 1 jBðv; w; kÞ  Bh;d % ðv; w; kÞjp2 gjjvjjLp ðI d Þ jjwjjLp0 ðI d Þ

8v; wASh ;

where g is as in Lemma 11. Then there exists M2 40 such that the following hold for % any dA½0; d0  and any h; hAð0; h2 : (1) The noisy modified fem is well defined. That is, there exists a unique uh;h;d % ASh such that Bh;d % ðuh;h;d % ; wÞ ¼ fh;d ðwÞ

8wASh :

(2) Let u ¼ Sð½ f ; kÞ: Then jju  uh;h;d % jjLp0 ðI d Þ h pM2 inf jju  vjjLp ðI d Þ vASh

þ sup wASh

jBðv; w; kÞ  Bh;d % ðv; w; kÞj j/ f ; wS  fh;d ðwÞj þ jjwjjLp0 ðI d Þ jjwjjLp0 ðI d Þ

!# :

ð17Þ

Proof. See, e.g., [15, pp. 310–312] for the proof of a version having slightly more restrictive conditions. & We now estimate the quantities appearing on the right-hand side of (17). Lemma 13. There exists M3 40 such that minfmþ1;sg þ dÞjjvjjLp ðI d Þ jjwjjLp0 ðI d Þ jBðv; w; kÞ  Bh;d % ðv; w; kÞjpM3 ðh%

% and d; for any kAK; and for any v; wASh : Hence, it follows that for any positive h; h; minfmþ1;sg þ dÞ: jjTk  Tk;h;d % jjLin½Lp ðI d Þ pM3 ðh%

Proof. The second part of the theorem follows from the first part, along with the identities /ðTk  Tk;h;d % Þv; wS ¼ Bðv; w; kÞ  Bh;d % ðv; w; kÞ and jjðTk  Tk;h;d % vjjLp ðI d Þ ¼

sup wALp

0 ðI d Þ

/ðTk  Tk;h;d % Þv; wS : jjvjjLp0 ðI d Þ

So it only remains to prove the first inequality.

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% and d; along with kAK and v; wASh : Then Choose positive h; h; jBðv; w; kÞ  Bh;d % ðv; w; kÞjpjA1 j þ jA2 j;

ð18Þ

where A1 ¼ Bðv; w; kÞ  Bðv; w; Ph# % h% kÞ ¼ /TkPh# % h% k v; wS and A2 ¼ Bðv; w; Ph# % h% kÞ  Bh;d % ðv; w; kÞj ¼ /ðTPh# % Þv; wS: % h% k  Tk;h;d We first estimate jA1 j: Using (9) and Lemma 4, we find jA1 jp jjTkPh# % h% k jjLin½Lp ðI d Þ jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ p jjk  Ph# % h% kjjLN ðI 2d Þ jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ p c1 M1 h%minfmþ1;sg jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ :

ð19Þ

To estimate jA2 j; let zðx; yÞ ¼

nh% X

ðkðxi;h% ; xj;h% Þ  k˜i; j;d Þsj;h% ðyÞsi;h% ðxÞ:

i; j¼1

Then

Z  jA2 jp 

  zðx; yÞvðyÞwðxÞ dy dx Id Id Z Z p sup jzðx; yÞj jvðyÞj dy jwðxÞj dx Z

Id

x;yAI 2d

Id

p jjzjjLN ðI 2d Þ jjvjjLp ðI d Þ jjwjjLp0 ðI d Þ :

ð20Þ

Now for xAI d ; define supph% x as iAsupph% x

iff

iAf1; y; nh% g and x is in the support of si;h% :

ð21Þ

By construction of the basis functions for Sh% ; there exist positive constants s1 and % such that s2 ; independent of x; j; and h; jsupph% xjps1

ð22Þ

jjsj;h% jjLN ðI d Þ ps2 :

ð23Þ

and

Hence for any x; yAI d ; we have X jzðx; yÞjp jkðxi;h% ; xj;h% Þ  k˜i; j;d j jsj;h% ðyÞj jsi;h% ðxÞj iAsupph% x jAsupph% y

p sd sup jjsj;h% jj2LN ðI d Þ ps1 s22 d: 1pjpnh%

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Since x; yAI d are arbitrary, we thus have jjzjjLN ðI 2d Þ ps1 s22 d:

ð24Þ

Using this inequality in (20), we obtain jA2 jps1 s22 djjvjjLp ðI d Þ jjwjjLp0 ðI d Þ : Combining this result with (20), recalling decomposition (18), and letting M3 ¼ maxfc1 M1 ; s1 s22 g; we obtain the first inequality in the statement of the theorem, as required. & Lemma 14. There exists M4 40 such that j/ f ; wS  fh;d ðwÞjpM4 ðhminfmþ1;rg þ dÞjjwjjLp0 ðI d Þ for any positive h and d; for any f ABW r;p ðI d Þ; and for any wASh : Proof. Choose positive h and d; along with f ABW r;p ðI d Þ and wASh : Then j/ f ; wS  fh;d ðwÞjpjA3 j þ jA4 j;

ð25Þ

where A3 ¼ / f  Ph f ; wS and * A4 ¼

Ph f 

nh X

+ ˜ fj;d sj;h ; w :

j¼1

We first estimate jA3 j: Using Lemma 4, we have jA3 jpjj f  Ph f jjLp ðI d Þ jjwjjLp0 ðI d Þ pM1 hminfmþ1;rg jjwjjLp0 ðI d Þ : We now estimate jA4 j: We find   nh X    jA4 jp  ½ f ðxj;h Þ  f˜j;d sj;h  jjwjjLp0 ðI d Þ  j¼1  d Lp ðI Þ   nh X    p d  js j jjwjjLp0 ðI d Þ :  j¼1 j;h  Lp ðI d Þ

Now

  nh X    jsj;h j   j¼1 

Lp ðI d Þ

  nh X    p jsj;h j  j¼1 

: LN ðI d Þ

ð26Þ

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But for any xAI d ; we may use (21)–(23) to see that nh X

X

jsj;h ðxÞj ¼

jsj;h ðxÞjps1 s2 ;

jAsupph% x

j¼1

and thus jA4 jps1 s2 d: Using this inequality, along with (26), in (25), and setting M4 ¼ maxfM1 ; s1 s2 g; the desired result follows immediately.

&

The final preparatory step is to prove a ‘‘shift theorem,’’ which relates the smoothness of ðI  Tk Þ1 f to the smoothnesses of f and of k: Lemma 15. Let 0ptpminfr; sg: For kAK and f AW t;p ðI d Þ; we have jjðI  Tk Þ1 f jjW t;p ðI d Þ pð1 þ c2 c3 Þjj f jjW t;p ðI d Þ ; where

c3 ¼

8 > > < > > :

d þs

!1=p c1

d

if poN;

ð27Þ

if p ¼ N:

c1

Proof. Let kAK: First, we show that jjTk jjLin½Lp ðI d Þ;W s;p ðI d Þ pc3 ;

ð28Þ

with jj  jjLin½Lp ðI d Þ;W s;p ðI d Þ denoting the usual operator norm. We shall prove only the case poN; the case p ¼ N being analogous. Let a be a multi-index of order at most s: Then for any vALp ðI d Þ; we have Z    jð@ a Tk vÞðxÞj ¼  @xa kðx; yÞvðyÞ dy Id

p sup j@xa kðx; yÞjjjvjjLp ðI d Þ yAI d

p jjkjjW s;N ðI 2d Þ jjvjjLp ðI d Þ ; so that jj@ a Tk vjjLp ðI d Þ pjjkjjW s;N ðI 2d Þ jjvjjLp ðI d Þ :

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783

Since a is an arbitrary multi-index of order at most s in d variables, we obtain 2 31=p !1=p X d þs a 4 5 jj@ Tk vjjLp ðI d Þ p jjkjjW s;N ðI 2d Þ jjvjjLp ðI d Þ ; jjTk vjjW s;p ðI d Þ ¼ s jajps from which the desired result (28) follows. Now let f AW t;p ðI d Þ; and set u ¼ ðI  Tk Þ1 f : Since u ¼ f þ Tk u; we get jjujjW t;p ðI d Þ pjj f jjW t;p ðI d Þ þ jjTk ujjW t;p ðI d Þ : Now jjTk ujjW t;p ðI d Þ p jjTk jjLin½Lp ðI d Þ;W t;p ðI d Þ jjujjLp ðI d Þ p jjTk jjLin½Lp ðI d Þ;W s;p ðI d Þ jjujjLp ðI d Þ p c3 jjujjLp ðI d Þ p c3 jjðI  Tk Þ1 jjLin½Lp ðI d Þ jj f jjLp ðI d Þ p c2 c3 jj f jjLp ðI d Þ : Hence jjðI  Tk Þ1 f jjW t;p ðI d Þ pjj f jjLp ðI d Þ þ c2 c3 jj f jjLp ðI d Þ ¼ ð1 þ c2 c3 Þjj f jjLp ðI d Þ ; as required.

&

We are now ready to show that the noisy modified fem is well defined, as well as to establish an upper bound on its error. Theorem 16. Let the degree m of the finite element spaces Sh and Sh% be chosen as m ¼ maxfr; sg  1: Let h1 be as in Lemma 11. Choose positive h2 and d0 such that M3 ðhs2 þ d0 Þp12 g:

ð29Þ

% Then there exists M5 40 such that the following hold for hAð0; h1 ; hAð0; h2 ; and dA½0; d0 : (1) The noisy modified fem is well defined. (2) We have the error bound minfr;sg eðfh;h;d þ h%s þ dÞ: % ; Nh;h;d % ÞpM5 ðh

% Proof. Let hAð0; h1 ; hAð0; h2 ; and dA½0; d0 : Using Lemmas 12 and 13, we see that the noisy modified fem is well defined. It only remains to establish the error bound.

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For ½ f ; kAF ; let u ¼ Sð½ f ; kÞ and uh;h;d % ¼ fh;h;d % ðNh;h;d % ð½ f ; kÞÞ: Using Lemmas 4 and 15, and setting C4 ¼ M2 ð1 þ c2 c3 Þ; we find jju  Ph ujjLp ðI d Þ p M2 hminfr;sg jjujjW minfr;sg;p ðI d Þ p M2 ð1 þ c2 c3 Þhminfr;sg jj f jjW minfr;sg;p ðI d Þ p C4 hminfr;sg :

ð30Þ

Now let wASh : By the definition of c2 ; we find jjPh ujjLp ðI d Þ p jju  Ph ujjLp ðI d Þ þ jjujjLp ðI d Þ p C4 hminfr;sg þ c2 jj f jjLp ðI d Þ pC4 þ c2 ; and thus using Lemma 13, we find that s jBðv; w; kÞ  Bh;d % ðv; w; kÞjp M3 ðh% þ dÞjjPh ujjL

p ðI

dÞ

jjwjjLp0 ðI d Þ

p ðC4 þ c2 ÞM3 ðh%s þ dÞjjwjjLp0 ðI d Þ :

ð31Þ

(Here we use the fact that f ABW r;p ðI d Þ) Moreover using Lemma 14, we have j/ f ; wS  fh;d jpM4 ðhr þ dÞjjwjjLp0 ðI d Þ :

ð32Þ

Hence using (30)–(32) in Lemma 12, we get minfr;sg þ ðC4 þ c2 ÞM3 ðh%s þ dÞ þ M4 ðhr þ dÞÞ: jju  uh;h;d % jjLp ðI d Þ pM2 ðC4 h

Taking M5 ¼ M2 ðC4 þ ðC4 þ c2 ÞM3 þ M4 Þ; we get the desired error bound.

&

Remark 17. We have a wide amount of latitude in choosing parameters h2 and d0 such that (29) holds. One simple choice is to pick   g 1=s g h2 ¼ and d0 ¼ : 4M3 4M3

7. The noisy MFEM is a minimal error algorithm Let nAZþ : In this section, we show how to choose the meshsizes h and h% such that the noisy modified fem is an nth minimal error algorithm. We define integer parameters l and l%; as follows: (1) Suppose that so2r: In this case, we have so2 minfr; sg: Take pffiffiffiffiffiffiffiffiffiffi l ¼ Jns=ð2 minfr;sgÞ n and l% ¼ I n  l m:

ARTICLE IN PRESS A.G. Werschulz / Journal of Complexity 19 (2003) 758–798

(2) Suppose that s ¼ 2r: Take l ¼ J12 nn

and

l% ¼ I

785

qffiffiffiffiffiffi 1 2 nm:

(3) Suppose that s42r: Take l% ¼ Jnr=s n and l ¼ n  l%2 : With these definitions for l and l%; define minfr; sg h% ¼ : l%1=d Recalling that the degree m of our finite element spaces is given by h¼

minfr; sg l 1=d

and

m ¼ minfr; sg  1; we see that nh ¼ l

and

nh% ¼ l%

% let by (13). With these choices of h and h; Nn;d ¼ Nh;h;d %

and

fn;d ¼ fh;h;d % :

That is, for any ½ f ; kAF ; we have % l%2 ;d ðkÞ; Nn;d ð½ f ; kÞ ¼ ½Nl;d ð f Þ; N where Nl;d ð f Þ ¼ Nh;d ð f Þ and

% l%2 ;d ðkÞ ¼ N % h;d N % ðkÞ:

Since Nn;d ð½ f ; kÞ uses l%2 noisy evaluations of k and l of f ; we have card Nn;d ¼ l%2 þ lpn: We now have Theorem 18. Recall from (2) that nr s o m ¼ min ; : d 2d Let fn;d and Nn;d be as defined above. (1) There exists n
0 AZþ and d0 40 such that the fn;d is well defined for all nXn
0 and all dA½0; d0 : (2) There exists M6 40 such that eðfn;d ; Nn;d ÞpM6 ðnm þ dÞ for nXn
0 and dA½0; d0 : (3) The nth minimal radius satisfies

ð33Þ

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rn ðdÞ^nm þ d: (4) The information Nn;d is nth optimal information, and fh;d is an nth minimal error algorithm.

Proof. The first item follows from Theorem 16. Once we establish (33), the remaining items will then follow immediately from (33) and Theorem 2. Hence, it remains to prove (33). We prove (33) on a case-by-case basis. Suppose first that so2r: We then have h^l 1=d ^ns=ð2 minfr;sgdÞ

and

% l%1=d ^n1=ð2dÞ : h^

Since so2r; we have m ¼ s=ð2dÞ: Hence eðfn;d ; Nn;d Þ%hminfr;sg þ h%s þ d%ns=ð2dÞ þ d^nm þ d: Next, suppose that s ¼ 2r: We have % 1=d ^n1=ð2dÞ : h^l 1=d ^n1=d and h^l Since ro2r ¼ s; we have minfr; sg ¼ r: Thus eðfn;d ; Nn;d Þ%hminfr;sg þ h%s þ d%nr=d þ ns=ð2dÞ þ d^nm þ d: Finally, suppose that s42r: We have % l%1=d ^nr=ðdsÞ : h^l 1=d ^n1=d and h^ Since ro2ros; we have minfr; sg ¼ r: Thus eðfn;d ; Nn;d Þ%hminfr;sg þ h%s þ d%nr=d þ d: But since s42r; we have m ¼ r=d: Thus eðfn;d ; Nn;d Þ%nm þ d: Hence (33) holds in all three cases.

&

8. Two-grid implementation of the noisy MFEM We have just seen that fn;d is an nth minimal error algorithm. Its information cost is cðdÞn: Hence if we were only interested in informational complexity, then we would have a source of optimal algorithms, see, e.g., [13, Section 4.4]. Unfortunately, the combinatory cost of this algorithm is generally much worse than YðnÞ: Indeed, for any ½ f ; kAF and any nXn
0 ; this algorithm presents us with a linear system ðA  BÞu ¼ f: The matrix B is a full l l matrix, where ( ns=ð2 minfr;sgÞ if so2r; l^ n if sX2r: Hence, if we were to use Gaussian elimination to solve this linear system, the combinatory cost would be proportional to nk ; where

ARTICLE IN PRESS A.G. Werschulz / Journal of Complexity 19 (2003) 758–798

k¼

787

8 <

3s if so2r; 2 minfr; sg : 3 if sX2r:

Since kA½32; 3; the combinatory cost is not OðnÞ: Rather than using Gaussian elimination to directly solve the linear system ðA  BÞu ¼ f; we shall use a two-grid algorithm to obtain a sufficiently accurate approximation of the solution u: This will give us a nearly optimal approximation at nearly optimal cost. Our approach will closely follow that of [7]. For given n; we shall define l; l%; h and h% as at the beginning of Section 7. This will give us a linear system ðA  BÞu ¼ f whose solution we wish to approximate. Following an idea that can be traced back to [5], we let n
be a second integer, satisfying n
¼ Yðn1=3 Þ: If we were to set up the linear system corresponding to the noisy mfem using information of cardinality n
; *  BÞ* * Here, l
; l
; h
; and h
are the * u ¼ f: we would get an l
l
linear system ðA parameters for the noisy mfem using information of cardinality n
; as defined at the beginning of Section 7. Before describing the two-grid method, we need to introduce some prolongation and restriction operators, as described in Sections 5.2 and 5.3 of [7]. Let X ¼ Lp ðI d Þ;
Xl ¼ ðRl ; jj  jjcp Þ; and Xl
¼ ðRl ; jj  jjcp Þ: We define the canonical prolongation Ph : Xl -X as Ph v ¼

l X

vj sj;h

8v ¼ ½v1 yvl ARl :

j¼1

The canonical restriction Rh : X -Xl is defined as Rh w ¼ A1 ½/w; s1;h Sy/w; sl;h ST

8wAX :

Note that Ph and Rh are uniformly bounded mappings, i.e., there exist positive constants CP and CR such that jjPh jjLin½Xl ;X  pCP

and

jjRh jjLin½X ;Xl  pCR

8h40:

ð34Þ

Moreover Rh Ph ¼ I

and Ph Rh ¼ Ph :

ð35Þ

(See [7, p. 161].) We then define the intergrid prolongation operator p : Xl
-Xl and the intergrid restriction operator r : Xl -Xl
as p ¼ Rh Ph


and

r ¼ Rh
Ph :

We will also need to use the adjoint operator p
: Xl -Xl
; defined as p
v  w ¼ v  pw

8vAXl ; wAXl
:

We are now ready to define the two-grid iteration scheme. This is the variant ZGM0 found in [7, p. 179].

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function TGðn : Zþ ; A; B : Rl l ; f : Rl Þ : Rl ; begin if n
pn
0 then compute uARl such that ðA  BÞu ¼ f else begin u :¼ 0; for i :¼ 1 to 3 do begin Solve the linear system A*u ¼ f þ Bu; fPicard iterationg d :¼ p
ðA*u  f  B*uÞ; fcompute defectg *  BÞd * ¼ d; fcoarse-grid solutiong solve the system ðA u :¼ u  pd fcoarse-grid correctiong end end; TG :¼ u end; Finally, let $ n;d ¼ ½N % l%2 ;d ; N % l 2 ;d ; Nl;d  N be two-grid information of cardinality at most n: Let uˇ n;d ¼ Ph ½TGðn; A; B; fÞ ¼

l X

uj sj;h :

ð36Þ

j¼1

$ n;d ð½ f ; kÞ; and so we Then uˇ n;d depends on ½ f ; kAF only through the information N $ $ $ may write uˇ n;d ¼ fn;d ðNn;d ð½ f ; kÞÞ; where fn;d is an algorithm using the information $ n;d : We call f$ n;d the two-grid algorithm. N Remark 19. In much of what follows, we shall need to assume that nXn
0 ; where n
0 is as defined in Theorem 18, so that we will have proper error estimates. Since we will need these estimates to also hold for the coarse-grid computation, we will also need n
Xn
0 to hold. This explains the if-guard in the definition of the function TG: Remark 20. If we could somehow choose l
so that the coarse-grid problem reuses the information from the fine-grid problem, then the algorithm would use information of cardinality l þ l%2 ; rather than of cardinality l þ l%2 þ ðl
Þ2 : Even though both these expressions are proportional to n; it would be preferable to have information of smaller cardinality. Unfortunately, we do not see a way to reuse this information. Remark 21. In a practical implementation, one would probably use at least one more two-grid iteration, so that the for-loop would have i go from 1 to 4. This would

ARTICLE IN PRESS A.G. Werschulz / Journal of Complexity 19 (2003) 758–798

789

reduce some of the constants that appear below, see [7, Remark 5.4.7]. (However, this is not the only factor affecting influencing the best choice for the number of iterations.) Our first task is to analyze the cost of the two-grid algorithm. Before doing this, we prove the following Lemma 22. Let nAZþ : For vARl ; we can calculate Bv using at most OðnÞ operations. %

Proof. Let SARl l have ð ;% ; jÞ entry s ;% ;j ¼ /s ;% ;h% ; sj;h S

for 1p ;% pl%; 1pjpl;

and let C ¼ ½k˜%i; ;% ;d 1p%i; ;% pl%: We then have B ¼ ST CS: For vARl ; we calculate Bv as follows: (1) Let a ¼ SvARl%: Since each row of S has only Oð1Þ nonzero elements, this matrix/vector multiplication can be done in at most OðlÞ operations. (2) Let b ¼ CaARl%: This is the usual multiplication of an l% l% matrix by an l%-vector, which can be done in at most Oðl%2 Þ operations. (3) Let c ¼ ST bARl : Since each row of ST has only Oð1Þ nonzero elements, this matrix/vector multiplication can be done in at most OðlÞ operations. Then Bv ¼ c: Moreover, the cost of calculating c is clearly Oðl%2 þ lÞ ¼ OðnÞ operations, as required. & We then have Lemma 23. The cost of the two-grid algorithm satisfies $ n;d ; N $ n;d Þ%cðdÞn: cost ðf $ n;d has cardinality proportional to n: Proof. By construction, the information N Hence the information cost of the two-grid algorithm is at most cðdÞn: Hence, it remains to determine the combinatory cost. Let ½ f ; kAF : We need to find the cost of computing TGðn; A; B; fÞ: (1) We first do the Picard iteration. From Lemma 22, evaluating Bu costs OðnÞ; and hence the cost of evaluating z ¼ f þ Bu is also OðnÞ: Furthermore, the bandwidth of A is bounded, independent of n; since there are no interelement continuity requirements. Thus the cost of the Picard iteration step is OðnÞ operations.

ARTICLE IN PRESS A.G. Werschulz / Journal of Complexity 19 (2003) 758–798

790

(2) Next, we compute the defect. Since the number of elements in any row of A is bounded, the cost of evaluating A*u is OðlÞ operations. Using Lemma 22, we can calculate B*u in OðnÞ operations. Thus we can calculate w ¼ A*u  f  B*u in OðnÞ operations. It only remains to calculate p
w; which can clearly be done in OðlÞ operations. (3) To calculate the coarse-grid solution, we need to solve an n
n
linear system. Since n
¼ Yðn1=3 Þ; we can do this in OðnÞ operations. (4) The coarse-grid correction can clearly be done in OðlÞ operations. Thus we can compute TGðn; A; B; fÞ with a cost of at most Oðl%2 þ lÞ ¼ OðnÞ operations, as required. & Our next task is to analyze the error of the two-grid approximation. Before doing this, we need to do a little groundwork. Write Y ¼ W minfr;sg;N ðI d Þ: Let Yl ¼ ðRl ; jj  jjYl Þ; where jjvjjYl ¼

inf

vAR1 ðvÞ h

jjvjjY :

The norm jj  jjYl
and space Yl
are defined analogously. For future reference, we note that the linear system ðA  BÞu ¼ f may be rewritten in the form ðI  KÞu ¼ g; where K ¼ A1 B

and g ¼ A1 f:

* 1 B: * We have the following: * ¼A We will also have cause to refer to the matrix K Lemma 24. There exist positive constants CS ; CK ; CB ; CI ; and CC ; which are independent of n and d; such that the following hold: (1) Stability: jjðI  KÞ1 jjLin½X  pCS for all lXl
l (2) Discrete regularity: jjKjjLin½Yl  pCK : (3) Uniform boundedness of prolongations: jjpjjLin½Xl
;Xl  pCB : (4) Interpolation error: jjI  prjj pCI ðl
Þminfr;sg=d : Lin½Yl ;Xl 


s=d þ d: (5) Relative consistency: jjrK  Krjj * Lin½Yl ;Xl
 pCC ðl Þ

Proof. We first prove stability. Let f ¼ ½a1 yal T AXl and then define u ¼ ðI  KÞ1 f: Let u˘ ¼ Ph u and f˘ ¼ Ph f: Then ˘ wÞ ¼ f˘h;d ðwÞ 8wASh : B % ðu; h;d

Using Lemmas 11 and 13, there exists nonzero wASh such that ˘ wÞX12 gjjujj ˘ Lp ðI d Þ jjwjjLp0 ðI d Þ : Bh;d % ðu;

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Using Lemma 14, we easily find that ˘ j f˘hc ;d ðvÞjp3M4 jj fjj Lp ðI d Þ jjvjjLp0 ðI d Þ ; and thus 6M4 ˘ ˘ Lp ðI d Þ p jjujj jj fjjLp ðI d Þ : g

ð37Þ

˘ Using (34), we see that From (35), we have u ¼ Rh u: ˘ X: jjujjXl pCR jjujj P Since f˘ ¼ lj¼1 aj sj;h ; we use the discrete Ho¨lder inequality to find p !1=p Z X  l   ˘ ¼ jj fjj aj sj;h ðxÞ dx  X   d I j¼1 8 2 !1=p !1=p0 3p 91=p
ð38Þ

ð39Þ

where 2 Z y¼4

l X Id

0

!p=p0

31=p dx5

jsj;h ðxÞjp

:

j¼1

Now l X

0

0

jsj;h ðxÞjp p max jjsj;h jjpLN ðI d Þ jsupph xj; 1pjpl

j¼1

where supph is defined similarly to (21). By construction of the basis functions for Sh% ; there exist positive constants s1 and s2 ; independent of x; j; and c; such that max jjsj;h jjLN ðI d Þ ps1

1pjpl

and

jsupph xjps2 :

Hence 1=p0

yps1 s2 : Using (39), we find that ˘ ps1 s1=p0 jjfjj : jj fjj X Xl 2 1=p

0

Let CS ¼ 6M4 CR s1 s2 =g: Using (37), (38), and (40), we obtain jjðI  KÞ1 fjjXl ¼ jjujjXl pCS jjf l jjXc : Since fAXl is arbitrary, we find that part 1 holds, as required.

ð40Þ

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792

We next check that discrete regularity holds. From [7, Remark 5.2.3], we find that ð41Þ

K ¼ Rh Tk;h;d % Ph : Using the definition of the norm jj  jjYl ; we find that jjRh jjLin½Y ;Yl  ¼ 1; and so jjKjjLin½Xl ;Yl  p jjRh jjLin½Y ;Yl  jjTk;h; h;d % jjLin½Y ;X  jjPh jjLin½Xl ;X  p CP jjTk;h;d % jjLin½Y ;X  ; where CP is defined by (34). Now jjTk;h;d % jjLin½Y ;X  pjjTk jjLin½Y ;X  þ jjTk  Tk;h;d % jjLin½Y ;X  : From (28), we have jjTk jjLin½Y ;X  pc3 ; whereas Lemma 13 and Theorem 18 yield that s m þ d: jjTk  Tk;h;d % jjLin½Y ;X  %jjTk  Tk;h;d % jjLin½X  %h% þ d%n

ð42Þ

Combining the previous inequalities, we see that part 2 holds. To prove uniform boundedness of prolongations, we use Exercise 5.3.6(a) in [7, p. 171], finding that part 3 holds with CB ¼ CR CP : Next, we establish the interpolation error. Note that since (35) holds, we have I  pr ¼ Rh Ph  Rh Ph
Rh
Ph ¼ Rh ðI  Ph
ÞPh : Hence using Lemma 4, we find jjI  prjjLin½Yl ;Xl  p jjRh jjLin½X ;Xl  jjI  Phl
jjLin½Y ;X  jjPh jjLin½Yl ;Y  p CR CP M1 ðh
Þminfr;sg %ðl
Þminfr;sg=d ; so that part 4 holds with CI ¼ CR CP M1 : We now establish relative consistency, proceeding as in [7, p. 172]. Using (41), we have * þ Rh
EPh
; rKp ¼ ðrRh ÞTk;h;d % ðPh pÞ ¼ Rh
Tk;h;d % Ph
¼ K where E ¼ Tk;h
;h
;d  Tk;h;d % : Hence * ¼ rKðI  prÞ þ Rh
EPh
r: rK  Kr Now jjrKðI  prÞjjLin½Yl ;Xl  pCB CK CI ðl
Þminfr;sg=d : Moreover, jjRh
EPh
rjjLin½Yl ;Xl  pCR CP CB jjEjjLin½Y ;X  :

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793

Using the triangle inequality and the coarse-grid version of (42), we find jjEjjLin½Y ;X  p jjTk  Tk;h
;h
;d jjLin½Y ;X  þ jjTk  Tk;h;d % jjLin½Y ;X  % jjTk  Tk;h
;h
;d jjLin½Y ;X  % ðh
Þs þ d%ðl
Þs=d þ d: Combining these results, we see that part 5 holds, as claimed.

&

Using some of the ideas found in the proofs of [7, Theorems 5.5.7 and 5.6.4], we are now ready to estimate the distance between the exact solution u of the linear system ðI  KÞu ¼ f and the solution u* ¼ TGðn; A; B; fÞ produced by the two-grid method. Lemma 25. We have jj*u  ujjXl %ðnm þ dÞjjfjjXl : Proof. It is no loss of generality to assume that n is sufficiently large that we do not solve the linear system ðA  BÞu ¼ f directly. Let * 1 rðI  KÞK * MTG ¼ I  ðI  KÞ and * * 1 rðI  KÞKðI  KÞ1 g: c ¼ ðI  KÞ Using Lemma 24 and [7, Theorem 5.4.3], we have jjMTG jjLin½Xl  pCTG ððl
Þminfr;sg=d þ ðl
Þs=d þ dÞ; where CTG ¼ ðCI þ CB CS CC ÞCK : Arguing as in the proof of Theorem 18, we find that jjMTG jjLin½Xl  p%ðn
Þm þ d: Since n
¼ Yðn1=3 Þ; it follows that jjMTG jjLin½Xl  %nm=3 þ d: It is fairly easy to check (see also [7, Theorem 5.4.3]) that u* ¼ u* ð3Þ ; where u* ð0Þ ¼ 0; u* ðiÞ ¼ MTG u* ði1Þ þ c

ð1pip3Þ:

Moreover, it is also easy to see that u ¼ MTG u þ c; so that jju  u* ðiÞ jjXl pjjMTG jjLin½Xl  jju  u* ði1Þ jjXl :

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Combining these results, we find jj*u  ujjXl p jjMTG jj3Lin½Xl  jjujjXl %ðnm þ dÞjjujjXl % ðnm þ dÞjjfjjXl ; the latter following from part 1 of Lemma 24.

&

We are now ready to state and prove the main result of this section. Theorem 26. There exist positive constants M7 and M8 such that for any nXn
0 and dA½0; d0 ; we have $ n;d ÞpM8 ðnm þ dÞ eðf$ n;d ; N with $ n;d ; N $ n;d ÞpM7 cðdÞ n: costðf Proof. For ½ f ; kAF ; let un;d ¼ fn;d ðNn;d ð½ f ; kÞÞ

and

$ n;d ð½ f ; kÞÞ: uˇ n;d ¼ f$ n;d ðN

By Theorem 18 and Lemma 23, it suffices to show that jjun;d  uˇ n;d jjLp ðI d Þ %ðnm þ dÞjj f jjLp ðI d Þ :

ð43Þ

Now un;d ¼ Ph u; where u is the solution of the linear system ðA  BÞu ¼ f given by (15), and uˇ n;d ¼ Ph u* ; where u* ¼ TGðn; A; B; fÞ: Using (34) along with Lemma 25, we obtain jjun;d  uˇ n;d jjLp ðI d Þ pCP jju  u* jjXl %ðnmþd ÞjjfjjXl : Hence (43) holds if jjfjjXl %jj f jjLp ðI d Þ :

ð44Þ

For iAf1; y; lg; define ei ¼ / f ; si;h S  fh;d ðsi;h Þ: Let e ¼ ½e1 ; y; el T and f
¼ ½/ f ; s1;h S; y; / f ; sl;h ST : Then jjfjjXl pjjejjXl þ jjf
jjXl : Since 0

jjejjXl ¼ jjejjcp ðRl Þ pl 1=p jjejjcN ðRl Þ

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and 0

jei j%ðnm þ dÞjj f jjLp ðI d Þ jjsi;h jjLp0 ðI d Þ %ðnm þ dÞl 1=p jj f jjLp ðI d Þ ; we see that jjejjXl %ðnm þ dÞjj f jjLp ðI d Þ :

ð45Þ

On the other hand, we have Ph f
¼ Ph f ; so that f
¼ Rh Ph f
¼ Rh Ph f by (35). From (35) and Lemma 4, we obtain jjf
jjXl pCR jjPh f jjLp ðI d Þ pCR ð1 þ M1 Þjj f jjLp ðI d Þ : Using this inequality and (45), we obtain our desired result (44), which completes the proof of the theorem. &

9. Complexity In this section, we determine the e-complexity of the noisy Fredholm problem. We recall from (2) that nr s o m ¼ min ; : d 2d Our main result is Theorem 27. Let e40: There exist positive numbers C1 ; C2 ; and C3 ; depending only on the global parameters of the problem but independent of e; such that the following hold: (1) The problem complexity is bounded from below by & 1=m ’ 1 compðeÞX inf cðdÞ : 0odoC1 e C1 e  d (2) The problem complexity is bounded from above by & 1=m ’ 1 compðeÞpC2 inf cðdÞ : 0odoC3 e C3 e  d

ð46Þ

$ n;d using information N $ n;d ; The upper bound is attained by using the noisy mfem f where & 1=m ’ 1 n¼ ð47Þ C3 e  d with C3 ¼ M81 from Theorem 26 and where d is chosen to minimize the appropriate right-hand side appearing in (46).

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796

Proof. To prove the lower bound, suppose that f is an algorithm using noisy information Nd such that eðf; Nd Þpe: Then card Nd Xn; where n must be large enough to make rn ðdÞpe: Theorem 2 immediately tells us that we must choose doM01 e and that we must have & 1=m ’ 1 : nX M01 e  d The cost of any algorithm using n information evaluations must be at least ncðdÞ; and so & 1=m ’ 1 costðf; Nd ÞXcðdÞ : M01 e  d Since f and Nd are any algorithm and information for which eðf; Nd Þpe; we find that & 1=m ’ 1 compðeÞXcðdÞ : M01 e  d Finally, since d40 is arbitrary, we get the desired lower bound with C1 ¼ M01 : To prove the upper bound, let d40: If (47) holds, then we may use Theorem 26 to $ n;d ; N $ n;d Þpe: Moreover, we have see that eðf & 1=m ’ 1 $ n;d ; N $ n;d ÞðeÞpM7 cðdÞ costðf : M81 e  d Set C2 ¼ M7 and C3 ¼ M81 : Choosing d minimizing the right-hand side in these inequalities, the desired result follows. & The lower and upper bounds in Theorem 27 are very tight. For an error level e and a constant C; define the function ge;C : Rþþ -Rþþ as  1=m 1 ge;C ðdÞ ¼ cðdÞ 8d40; Ce  d and set g
e;C ¼

inf

0odoCe

ge;C ðdÞ:

By Theorem 27, we see that g
e;C1 pcompðeÞpC2 g
e;C2 : This inequality allows us to determine the complexity for various cost functions cðÞ: In particular, if the cost function cðÞ is differentiable, then the optimal d must satisfy g0e;C ðdÞ ¼ 0; i.e., we must have 

cðdÞ ¼ mðCe  dÞ: c0 ðdÞ

ð48Þ

ARTICLE IN PRESS A.G. Werschulz / Journal of Complexity 19 (2003) 758–798

797

As a specific example, consider the cost function cðdÞ ¼ dt ; where t40: We find that for e40; the optimal d is d
¼

Cmte ; mt þ 1

ð49Þ

so that g
e;C ^

     1 þ mt tþ1=m 1 t 1 tþ1=m : C mt e

Thus we see that the optimal d
is proportional to e; and that  tþ1=m 1 : compðeÞ^ e

Acknowledgments I thank W. Hackbusch, S. Heinrich, E. Novak, R. Israel, and S. Pereverzev for their various suggestions. I also thank H. Woz´niakowski for his comments on an early draft of this paper. Finally, I thank the two referees for their exceptionally careful reading of the version of this paper that was initially submitted to the Journal of Complexity.

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