What is the complexity of solution-restricted operator ... - CiteSeerX
What is the complexity of solution-restricted operator equations? Technical Report CUCS-009-95 Arthur G. Werschulz Department of Computer and Information Sciences, Fordham University Fordham College at Lincoln Center New York, NY 10023 and Department of Computer Science Columbia University New York, NY 10027 May 25, 1995 Abstract. We study the worst case complexity of operator equations = , where : ! is a bounded linear injection, is a Hilbert space, and is a normed linear space. Past work on the complexity of such problems has generally assumed that the class of problem elements to be the unit ball of . However, there are many problems for which this choice of yields unsatisfactory results. Mixed elliptic-hyperbolic problems are one example, the diculty being that our technical tools are not strong enoguh to give good complexity bounds. Ill-posed problems are another example, because we know that the complexity of computing nite-error approximations is in nite if is a ball in . In this paper, we pursue another idea. Rather than directly restrict the class of problem elements , we will consider problems that are solution-restricted, i.e., we restrict the class of solution elements . In particular, we assume that is the unit ball of a Hilbert space continuously embedded in . The main idea is that our problem can be reduced to the standard approximation problem of approximating the embedding of into . This allows us to characterize optimal information and algorithms for our problem. Then, we consider speci c applications. The rst application we consider is any problem for which and are standard Sobolev Hilbert spaces; we call this the \standard problem" since it includes many problems of practical interest. We show that nite element information and generalized Galerkin methods are nearly optimal for standard problems. We then look at elliptic boundary-value problems, Fredholm integral equations of the second kind, the Tricomi problem (a mixed hyperbolic-elliptic problem arising in the study of transonic ow), the inverse nite Laplace transform, and the backwards heat equation. (Note that with the exception of the backwards heat equation, all of these are standard problems. Moreover, the inverse nite Laplace transform and the backwards heat equation are ill-posed problems.) We determine the problem complexity and derive nearly optimal algorithms for all these problems. Lu
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This research was supported in part by the National Science Foundation under Grant CCR-91-01149.
1. Introduction
Operator equationsd Lu = f are among the most important problems of applied mathematics. We are interested in the "-complexity of such problems; that is, we want to nd the minimal cost of computing "-accurate approximations and to nd algorithms that yield such approximations with (nearly) minimal cost. Past work on the worst case complexity of such problems has generally assumed that the class F of problem elements f is a unit ball. As we shall see, this assumption is not strong enough for many classes of problems. In this paper, we assume instead that the class U of solution elements u is a unit ball. Such problems are called \solution-restricted" problems. We study the complexity of solution-restricted operator equations Lu = f in this paper. More precisely, suppose that L : G ! X is a bounded linear injection, where G is a Hilbert space and X is a normed linear spaces. Let F X be a xed class of problem elements f . We wish to solve Lu = f for f 2 F , our only knowledge of any f being the values of a nite set of linear functionals of f , each evaluation having cost c. Since we only have partial information about f , we can only calculate approximations of u = L?1f . Our goal is to calculate "-approximations (i.e., approximations with error at most ") with minimal cost. This is, of course, a problem of information-based complexity (IBC); see [22] for further discussion. We will solve our problem in a worst case setting, so that the error and cost of an algorithm are given by their maximum values over all f . Researchers in IBC have been most successful in getting good complexity bounds for operator equation problems (for F a ball in X ) whenever the solution operator L?1 has been bounded. One such class of problems is the solution of elliptic operator equations Lu = f , for which we have found a wealth of complexity-related results. For instance, one popular method for elliptic problems is the nite element method (FEM); we have been able to nd conditions that are necessary and sucient for the FEM to be a nearly optimal error or nearly optimal complexity algorithm. Many of these results use a \shift theorem," which says that if L is elliptic of order 2m, then f has smoothness r (i.e., r derivatives in some sense) i u has smoothness r + 2m. Similarly, since there is a shift theorem (with m = 0) for Fredholm integral equations of the second kind, we have been successful in proving results about the complexity of second-kind Fredholm problems, as well as characterizing nearly optimal FEMs for such problems. An exhaustive treatment of this subject may be found in [28, Chapters 5 and 6]. Unfortunately, there has been far less success to date in dealing with operator equations for which there is no shift theorem. For example, consider the Tricomi problem. This is a mixed elliptic-hyperbolic problem arising in the study of transonic ow across an airfoil, see [7, Chapter X] and [13]. Since there is no shift theorem for the Tricomi problem,1 we have not been able to use these techniques to obtain sharp complexity bounds for the Tricomi problem. Things are even worse when we consider the solution of ill-posed problems Lu = f , in which the solution element u does not depend continuously on the problem element f , e.g., when L is compact. We have a strong negative result on the worst case complexity of ill-posed problems when F is a ball in X , namely, that there exists no algorithm having The best global smoothness result we know for solutions of the Tricomi problem is that of [2], which only proves rst-order smoothness for a solution of a second-order problem, so there is a smoothness gap.
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nite error (see [26]). Hence the complexity of computing an "-approximation is in nite, no matter how large we choose " to be. This means we cannot solve ill-posed problems in a worst case setting. If we need to solve such problems, we must go to a di erent setting, such as the average case setting. For further discussion, see [23], [27], [28]. Summarizing, we see that if the class F is a ball in X , good worst case complexity bounds for operator equations Lu = f have so far eluded us, except for L enjoying special properties. If we wish to solve such problems in the worst case setting for L not satisfying these properties, we need to look at other classes F . This paper uses an idea of Tikhonov [20] that is often used in the solution of ill-posed problems (see also [15] for a fuller development, as well as the discussion in [19]). Instead of solving the problem Lu = f under the a priori assumption that f belongs to a known set F (typically a ball in X ), we assume that u belongs to a known set U . Thus for such a solution-restricted operator equation, we restrict the solution elements instead of the problem elements.2 The main point of this paper is that we can often get good complexity results for solutionrestricted problems, simply because much of the work can be rephrased in terms of the well-studied problem of approximating an identity embedding. We now outline our main results. In Section 2, we formally describe the problem to be solved. The class U will be the unit ball of a Hilbert space W that is continuously embedded in G. Part of our problem description includes a discussion of the class of permissible information operations. We will be mainly interested in two classes . The rst is the usual class of continuous linear functionals over X , which has been well-studied in previous work on IBC. The second class of permissible information operations is the class F of linear functionals on F that are bounded on F . This new class of information operations is quite natural for our problem, since it is the largest class of linear functionals de ned on F . In Section 3, we show that a solution-restricted operator equation can be reduced to the approximation problem of approximating the embedding of W into G. Since the approximation problem is a well-studied problem of information-based complexity, we can easily adapt known results for the approximation problem to our problem. As a result, we quickly determine optimal algorithms and information for our problem. Moreover, we nd that the minimal radii are the same for the classes and F . This means that in principle, there is no need to consider F . However, our introduction of F is no mere arti ce. Indeed, it is often easier to rst consider the problem for = F , and then to approximate F -information by -information, rather than to directly consider information at the start. Moreover, there are situations for which F -information is more natural than -information. In Section 4, we use these results to study several important problems. The rst problem we study is called the \standard problem," which is any problem for which G and W are q ( ) and H r ( ) (the \bd" meaning that certain standard Sobolev Hilbert spaces Hbd bd homogeneous boundary conditions may be imposed), with q < r and Rd. Note that since L and X are not speci ed, this problem is really a meta-problem, which includes many Of course, since our class of problem elements is now given by de nition of . However, need not be a ball in . 2
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= ?1 ( ), we do have an implicit L
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important examples. Also note that there is no requirement that L have a bounded inverse, so that the standard problem covers both well-posed and ill-posed problems. Hence we are q ( )solving any linear operator equation Lu = f , with error being measured in the Hbd r ( ) 1. Although the standard problem is norm, subject to the constraint that kukHbd quite general, we are able to develop complexity bounds and to nd optimal algorithms and information for the standard? problem. We nd that the nth minimal radius is (n?(r?q)=d ) and the "-complexity is c (1=")d=(r?q) . The optimal information is given by eigenvectors r ( ) into H q ( ). We show that the eigenproblem of E E , where E is the embedding of Hbd bd for E E can be expressed as a generalized eigenproblem for a partial di erential equation. Unfortunately, this eigenproblem usually does not have closed-form solutions. Hence, we need to nd more accessible nearly optimal information. We are able to show that nite element information is nearly optimal, and that generalized Galerkin methods turn out to be nearly optimal algorithms. In the remainder of Section 4, we analyze several important speci c applications. The rst class of applications is elliptic boundary-value problems. We consider a 2mth-order problem, with error measured in the energy norm, which is equivalent to the Sobolev kkH m( )-norm. This is then set as a standard problem with G = H0m ( ) and W = Hr ( ). ? The nth minimal error is (n?(r?m)=d ), and the "-complexity is c (1=")d=(r?m) . For = F , we nd that the usual nite element method is nearly optimal, the proof not requiring a shift theorem. This means that we can handle elliptic problems that do not admit a shift theorem.3 For the sake of completeness, we develop the results for = , although the method that results would probably not be used in practice. We next look at Fredholm integral equations of the second kind. This is set up as a standard problem with G = L2( ) and W? = H r ( ). We nd that the nth minimal error ? r=d d=r is (n ) and the "-complexity is c (1=") . Finite element information is nearly optimal, and we exhibit nearly optimal generalized Galerkin methods. Next, we look at the Tricomi problem. Once again, we set this up as a standard problem with G = L2 ( ) and W = H r ( ). Our results are essentially the same as for the Fredholm problem of the second kind. The nth minimal error is (n?r=d ) and the "-complexity is ? c (1=")d=r . We nd that nite element information and generalized Galerkin methods are nearly optimal. Our next application is the inverse nite Laplace transform, which is a Fredholm integral equation of the rst kind and hence is ill-posed. This particular problem arises in the study of \measurement of the distribution of an absorbing gas (such as ozone in the earth's atmosphere) from the spectrum of scattered light", see [21, pp. 12{13]. This may be set up as a standard problem with G = L2 ( ) and W =? H r ( ). We nd that the nth minimal error is (n?r=d ) and the "-complexity is c (1=")d=r . Once again, nite element information is nearly optimal, as are generalized Galerkin methods. To show that our techniques are not limited to standard problems, we close this paper by studying a problem that is not a standard problem. Our nal application is the heat equation running backwards in time, with nal data f . That is, we want to know what the This means that our analysis can be easily extended to include problems for which the coecients or the boundary of are not very smooth, or problems having boundary conditions of mixed Dirichlet-Neumann type.
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temperature distribution was at some time t = ?t0 in the past (where t0 > 0), where the temperature distribution at time t = 0 is given by the function f . Suppose that both the nal data and solution are measured in the L2-sense. Then the backwards heat equation is ill-posed, said ill-posedness being related to the second law of thermodynamics, see [12]. To express this as a solution-restricted problem, we let t1 > t0, and solve the problem under the condition that the solution at time ?t1 must have nite L2-norm, see also [10] for ? 2 2 (t1 ? t0 ), further discussion. We then nd that the n th minimal error is exp ? ( n + 1) p so that the "-complexity is c ( ln(1=")). Moreover, if we truncate the usual series representation for the solution of the heat equation, we get an optimal algorithm. For further discussion of the complexity of this problem, see [27]. We close this introduction by noting that one of our main assumptions has been that the calss of permissible information operations has been either F or . There is a third class that one could investigate. Suppose that X is a function space on some domain
Rd, as is generally the case in most applications. Then we could study the class std of standard information, i.e., evaluations of f (or some of its derivatives) at points in . At this time, we only have results for standard information under (what we feel are) unnecessarily-restrictive conditions on L, conditions that do not apply (for example) when L is compact, i.e., when the original problem is ill-posed. We hope that we will be able to coherently deal with standard information for solution-restricted operator equations in future work. 2. Problem description
Let G be a Hilbert space, and let X be a normed linear space. We assume that both G and X are in nite-dimensional. Let L : G ! X be a bounded linear injection, whose range D is dense in X . We de ne a solution operator S : D X ! G as
u = Sf () Lu = f
for u 2 G and f 2 D:
(2.1)
Note that if L does not have a closed range, then S is only densely de ned. Remark : Note that we restrict our attention to the Hilbert case. We do this for two reasons. The rst is for ease of exposition, while the second is that all the examples we consider are instances of the Hilbert case. We note in passing that many of the results contained here also hold for the case of general normed linear spaces. Let F be a balanced, convex subset of D, and let be a class of continuous linear functionals on F (more precisely, a class of functionals whose extensions to the linear hull of F are continuous linear functionals). Throughout this paper, we will be especially interested in the following classes : (1) = , the class of all continuous linear functionals de ned over X . This is a \standard" choice of for many problems arising in information-based complexity; see [22]. (2) = F , the class of all linear functionals on F that are bounded on F . Note that F , with equality when F is a ball in X . The strict inclusion F holds, e.g., when F is a compact subset of X . Our interest in using F instead of 4
is that we can use our a priori knowledge that f 2 F to allow us to expand the possible choice of information functionals from those that must be de ned over all of X to those that need be de ned only over F , which is a (possibly small) subset of X . Our abstract setting is the usual one of information-based complexity, see, e.g., [22] and [28]. We wish to compute approximate values of Sf for f 2 F , given a nite number of information values (f ) for some elements 2 . However, our point of departure will be to assume that there is a subset U G such that F = L(U ). Thus
f 2 F () u = Sf 2 U: (Thus, F is a subset of D.) Note that we are now restricting our solution elements u rather than our problem elements f . For this reason, we will call our problem a solution-restricted operator equation. In particular, we will assume in this paper that U is the unit ball of a Hilbert space W that is continuously embedded in G. That is,
U = f Ew : kwkW 1 g;
(2.2)
where the embedding mapping E : W ! G, de ned as Ew = w for w 2 W , is continuous and dense. To simplify the exposition in what follows, we assume without essential loss of generality that kE k 1. This assumption holds for many, but not all, cases of practical interest.4 Since Ew = w for w 2 W , we will often write w instead of Ew whenever this will simplify the presentation, provided that no confusion will result. We close this section by recalling some standard terminology from [22]. Let N denote the nonnegative integers. For any n 2 N, we say that N is information from of cardinality n if there exist linearly independent linear functionals 1; : : : ; n 2 such that
8f 2 F:
Nf = [1(f ); : : : ; n(f )]
(2.3)
We let n denote the class of information from whose cardinality is at most n. An algorithm using N is a mapping : N (F ) ! G, its error being given by
e( ; N ) = sup kSf ? (Nf )kG : f 2F
The radius of information N is given as
r(N ) = inf e( ; N );
the in mum being over all algorithms using N . An algorithm N for which e( N ; N ) = r(N ) is an optimal error algorithm using N .
For example, this assumption holds for the embedding r ( ) ! q ( ) of Sobolev spaces ( ), but it does not hold for the embeddings given by either the general statement of Sobolev's embedding theorem or by the Rellich-Kondrasov theorems (see, e.g., [6, pg. 114] for statements). 4
H
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The nth minimal radius in is de ned to be
r(n; ) = inf r(N ); N 2n
and information Nn 2 n is nth optimal information if r(Nn ) = r(n; ). (Note that since F is balanced and convex, we are only considering nonadaptive information of xed cardinality; see [22, pp. 57{60].) An optimal error algorithm using nth optimal information is said to be an nth minimal error algorithm. We turn these error-related concepts into complexity-related concepts by introducing a model of computation. Our model is the standard one of [22], see also [28, pg. 27]. For any 2 and any f 2 F , the cost of evaluating (f ) is c, and the cost of basic combinatory operations is 1. Typically, c 1. Then the cost of an algorithm using information N is given as cost( ; N ) = sup cost( ; N; f ); f 2F
where for f 2 F , we let cost( ; N; f ) denote the cost of computing (Nf ) under this model of computation. As an example, suppose that is a linear algorithm using N given P by (2.3), i.e., there exist g1; : : : ; gn 2 G such that (Nf ) = nj=1 j (f )gj for f 2 F ; then cost( ; N ) (c + 2)n ? 1. Finally, we say that for " > 0, the "-complexity in the class is comp("; ) = inf f cost( ; N ) : e( ; N ) " g; the in mum being over all algorithms using information N of nite cardinality from , and an algorithm " using information N" of nite cardinality from for which cost( "; N" ) = comp("; ) is an ("; )-optimal complexity algorithm. Since we cannot often nd optimal complexity algorithms, we will usually be happy to settle for nearly optimal complexity algorithms, which compute "-approximations with cost at most a constant multiple of the "-complexity, said constant being independent of ". One way of doing this involves the ("; )-cardinality number m("; ) = inf f n : r(n; ) " g: Suppose that N 2 n is information of cardinality n = m("; ) for which r(N ) = ", and that is a linear optimal error algorithm using N . Then
c m("; ) comp("; ) cost( ; N ) (c + 2)m("; ) ? 1; and is an ("; )-nearly optimal complexity algorithm for our problem. 6
3. Reduction to the approximation problem
Our main strategy for nding optimal algorithms for solution-restricted operator is to reduce such problems to the approximation problem of approximating the embedding E : W ! G. Let be a class of linear functionals on F , and let N be information from of nite cardinality. Suppose that is an algorithm using N . For any f 2 F , we have kSf ? (Nf )kG = kEw ? (NL w)kG ; (3.1) where Ew = Sf and NL = N LE: (3.2) Note that we are being somewhat cautious in explicitly using the embedding operator E , so we can keep closer track of the spaces in which various elements may be found. Also note that N and NL are information operators on the spaces X and W , respectively. Equation (3.1) tells us that the error of any algorithm for our problem equals the error of the same algorithm (using di erent information) for the approximation problem. In what follows, we will let eapp , rapp , mapp , and compapp denote the error, radius, cardinality number, and complexity for the approximation problem. Using [22, Chapter 3], we then nd the following: Theorem 3.1. Let be a class of linear functionals on F . (1) Let N be information from of nite cardinality. (a) For any algorithm using N , we have e( ; N ) = eapp( ; NL ); where NL is de ned by (3.2). (b) An algorithm is an optimal error algorithm using N for our problem i is an optimal error algorithm using NL for the approximation problem, and r(N ) = rapp (NL ): (2) For any nonnegative integer n, r(n; ) = rapp (n; L ); where L = E L : Hence, m("; ) = mapp("; L ): (3) An ("; L )-(nearly) optimal complexity algorithm for the approximation problem is an ("; )-(nearly) optimal complexity algorithm for our problem, and comp("; ) = compapp ("; L ): In particular, if NL 2 (L )n is information of cardinality n = mapp ("; L ) for which r(NL ) = " and is a linear optimal error algorithm using N , then c m("; ) comp("; ) cost( ; N ) (c + 2)m("; ) ? 1; and so is a nearly optimal complexity algorithm for our problem. 7
The results in Theorem 3.1 essentially hold even in the case where the spaces G and W are not Hilbert spaces. However, we are dealing only with the Hilbert case in this paper. Let us use this fact to derive specifc formulas for a linear optimal error algorithm. Let N be information of cardinality n. Then there exist linearly independent 1; : : : ; n 2 X such that Nf = [1(f ); : : : ; n(f )] 8f 2 F: (3.3) Let wj = E L j (1 j n): (3.4) Since f1 ; : : : ; fn are linearly independent and E L is an injection, we see that w1; : : : ; wn are linearly independent. De ne For f 2 F , let uN 2 SN satisfy
SN = span fw1; : : : ; wng:
i (LuN ) = i (f ) (1 i n): (3.5) Note that this is a generalized Galerkin method, i.e., a Galerkin method with di erent spaces of test and trial functions. Clearly the dependence of uN on f is only through the information Nf , so we may write uN = N (Nf ) 8f 2 F: (3.6) Lemma 3.1. Let N be given by (3.3). (1) The linear algorithm N given by (3.6) is an optimal error algorithm for N . (2) If fw1; : : : ; wng is a W -orthonormal set, then N (Nf ) =
n X j =1
j (f )wj =
n X j =1
hSf; wj iW wj
8f 2 F:
Proof: Using (3.4), we see that
i (LuN ) = huN ; wiiW and i (f ) = hSf; wiiW : Hence uN 2 SN satis es (3.5) i it satis es huN ; wi iW = hSf; wiiW (1 i n): (3.7): Now use [22, Theorem 5.5.3]. F As mentioned in Section 2, we will deal with the classes = and = of permissible information operations. In the remainder of this section, we show that the nth minimal radii are the same for these classes, being given by the (same) Gelfand n-width. We also construct nth optimal (or nearly optimal) information for these classes. First, we prove the following 8
2 Fn . For any > 0, there exists information N 2 n such that r(N ) (1 + ) r(N ): Since N 2 Fn , there exist linearly independent 1 ; : : : ; k 2 F (with k n)
Lemma 3.2. Let N Proof:
such that
N = [1 ; : : : ; k ]: Since each functional Sf 7! i (f ) is a bounded linear functional on W , there exist linearly independent w1; : : : ; wk 2 W such that i (f ) = hSf; wiiW for f 2 W . Without loss of generality, we may assume that w1; : : : ; wk are orthonormal. Now let > 0, and de ne = 1+r(rN(N) ) : Since LE : W ! X is an injection, we see that E L : X ! W is dense, and so there exist 1; ; : : : ; k; 2 X such that kwi ? E L i; kW k : We then de ne information N 2 n as N = [1; ; : : : ; k; ]: Recall that the radius of information for the Hilbert case is given by the simpli ed expression r(N ) = sup kwkG = sup kwkG: (3.8); Lw2ker N kwkW 1
w2ker(N )L kwkW 1
see [22, pg. 80]. Let Lw 2 ker N with kwkW 1. For 1 i k, we have (E L i; )(w) = i; (Lw) = 0 and so Let
jhwi; wiW j kwi ? E L i; kW kwkW k : w0 = w ?
Note that
k X j =1
hwj ; wiW wj ;
i (Lw0) = E Li (w0) = hwi ; w0iW = 0 and so Lw0 2 ker N . Moreover
kw0kW kwkW +
k X j =1
(3.9) (1 i k);
jhwj ; wiW j kwj kW 1 + : 9
It now follows that
kw0kG sup kvkG = (1 + ) r(N ): Lv2ker N kvkW 1+
From (3.9) and the de nition of , it now follows that
kwkG kw0kG +
k X j =1
jhwj ; wiW j kwj kG (1 + ) r(N ) + = (1 + ) r(N ):
Since w is an arbitrary element in the W -unit ball such that Lw 2 ker N , the lemma follows immediately from (3.8). Recall that the Gelfand n-width of the embedding E is de ned as ? dn E (W ); G = ninf n supn kwkG; W 2W w2W kwkW 1
where W n is the class of W -subspaces whose codimension is at most n. From [18, Corollary II.7.3], there always exists an nth optimal Gelfand subspace of E , i.e., a subspace W n 2 W n at which this in mum is attained. Without loss of generality, we may assume that W n = ker[h; w1 iW ; : : : ; h; wniW ]; (3.10) where fw1; : : : ; wng is a W -orthonormal set. Theorem 3.2.
(1) For any n 2 N,
?
r(n; F ) = r(n; ) = dn E (W ); G : (2) Suppose that = F . Let w1; : : : ; wn be as in (3.10). De ne information Nn 2 F by Nnf = [hSf; w1iW ; : : : ; hSf; wniW ] 8f 2 F: Then ? r(Nn ) = r(n; ) = dn E (W ); G ; i.e., Nn 2 Fn is nth optimal information. (3) Suppose that = . For > 0, let ? 2 dn E?(W ); G ; = 1 + 2 dn E (W ); G and choose 1; ; : : : ; n; 2 X such that kwi ? E L i; kW n : De ne information Nn; 2 n by Nn; = [1; ; : : : ; n; ]: Then r(Nn; ) (1 + ) r(n; ); so that Nn; 2 is nth nearly optimal information. 10
Proof: Parts (1) and (2) follows immediately from Lemma 2 and the observation that if = F , then L = . We need only prove part (3). For > 0, we have
n) = 1 + r(N r(Nn ) by part (1). Using the notation of Lemma 3.2, we see that for N = Nn , we have N = Nn; , and so ? r(Nn; ) (1 + ) r(Nn ) = (1 + ) r(n; ) = (1 + ) dn E (W ); G ; as required. ? n Suppose for a moment that E is not a compact embedding. Then limn!1 d E (W ); G is strictly positive (see [18, Proposition II.7.4]). This means that the nth minimal radius is bounded away from zero for the classes F and . In short, if E is not compact, then the problem is not convergent, i.e., we cannot get arbitrarily good approximations at nite cost. Hence, in the remainder of this paper, we shall assume that W is compactly embedded in G. This implies that the space W has an orthonormal basis consisting of eigenvectors of E E . Thus there exist 1 2 > 0 with limn!1 j = 0, and a complete orthonormal basis fzj g1 j =1 for W such that E Ezj = j2zj for j = 1; 2; : : : : We then nd that for any n 2 N, ? dn E (W ); G = n+1; (3.11) see [18, Theorem IV.2.2]. We rst look at optimal information and algorithms for the class F . Let Nn f = [hSf; z1iW ; : : : ; hSf; zniW ] 8f 2 F; (3.12) where z1; : : : ; zn are the eigenvectors of E E corresponding to the n largest eigenvalues of E E . Clearly we have Nn 2 Fn . De ne an algorithm n using Nn as
n(Nn f ) = Theorem 3.3.
n X j =1
hSf; zj iW zj
8f 2 F:
(3.13)
(1) For any n 2 N, we have ? e( n ; Nn) = r(Nn ) = r(n; F ) = dn E (W ); G = n+1: Hence Nn is nth optimal information and n is an nth minimal error algorithm. (2) Let " > 0. Then the "-cardinality number for F is given by m("; F ) = inf f integers n 0 : n+1 " g : Moreover, let Nn 2 Fn be the information given by (3.12) and let n be the algorithm given by (3.13), with n = m("; ). Then c m("; F ) comp("; F ) cost( n; Nn) (c + 2)m("; F ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. 11
Proof: Immediate from (3.11), along with Theorems 3.1 and 3.2, as well as Lemma 3.1. We now consider optimal information and algorithms for the class . Let > 0, and choose 1; ; : : : ; n; 2 X such that
kzj ? L j; kG nn+1 ;
(3.14)
where we recall that z1; : : : ; zn are the eigenvectors of E E corresponding to the n largest eigenvalues of E E . Our information Nn; 2 n is de ned as
Nn; f = [1; (f ); : : : ; n; (f )]
8f 2 F:
(3.15)
De ne an algorithm n; using Nn; as
n; (Nn; f ) =
n X j =1
j; (f )zj
8f 2 F:
(3.16)
We then have Theorem 3.4.
(1) For any n 2 N and any > 0, we have
e( n; ; Nn; ) (1 + ) n+1: Hence Nn; is nth nearly optimal information and n; is an nth nearly minimal error algorithm. (2) Let " > 0. Then the "-cardinality number for is given by
m("; ) = inf f integers n 0 : n+1 " g : Suppose that n is strictly monotonically decreasing with n. Let Nn; 2 Fn be the information given by (3.15) and let n; be the algorithm given by (3.16), with n = m("; ) + 1 and satisfying 0 < <
n ? 1: n+1 Then ?
c m("; F ) comp("; ) cost( n; ; Nn; ) (c + 2) m("; ) + 1 ? 1: Hence for c 1, the algorithm n; using information Nn; is a nearly optimal complexity algorithm. 12
Proof: Note that part (2) follows from Theorem 3.1 and part (1). So we need only prove
part (1). To do this, let n 2 N, and let > 0. Choose f 2 F . For any j , we have
jhSf; zj iW ? j; (f )j = jhSf; zj ? L j; iG j kSf kGkzj ? L j; kG nn+1 : Recalling the de nition of Nn and n from Theorem 3.3, we then have
k n(Nnf ) ? n; (Nn; f )kG
n X j =1
jhSf; zj iW ? j; (f )jkzj kG n+1:
So
kSf ? n; (Nn; f )kG kSf ? n(Nnf )kG + k n(Nn f ) ? n; (Nn; f )kG (1 + ) n+1: Taking the supremum over all such f , we nd the desired bound on e( n; ; Nn; ). Now that we have proved this bound, the rest of the result follows from (3.11) and Theorem 3.2, as well as Lemma 3.1. Remark : This result tells us that n; is an nth nearly minimal error algorithm in . Of course, this implies that n; is an nearly optimal error algorithm using the nth nearly optimal information Nn; . Note that we decided to use the algorithm n; , instead of the algorithm Nn; , i.e., the optimal error algorithm using Nn; given by Lemma 3.1. We did this because the algorithm n; is simpler than the optimal error algorithm Nn; , since the latter would have required either an orthogonalization of the vectors E L 1; ; : : : ; E L n; or the solution of an n n linear system. Note that the increased error when using n; instead of Nn; is small compared to the error of n; and to the nth minimal radius. All things considered, it appears better to use a simpler nearly optimal error algorithm than a more complicated optimal error algorithm in this situation. Remark : The strict monotonicity assumption in Theorem 3.4 is not necessary, but only used to simplify the statement of the theorem. A more general (and more complicated) statement is possible for the case where E E has multiple eigenvalues, provided that the multiplicity of the eigenvalues does not increase super-exponentially. Since we will not need such a result, we will not pursue this further. 4. Applications
In this section, we apply the previous results to several problems. We rst look at a common situation, namely, a problem in which the spaces G and W are the Sobolev spaces q ( ) and H r ( ), respectively. Here, q < r, and the \bd" indicates that the spaces Hbd bd may satisfy certain homogeneous boundary conditions. Any such problem will be called a \standard problem." We develop detailed results for standard problems. Once we have these results, we can use them to study speci c instances of the standard problem. In particular, we will consider elliptic boundary-value problems, Fredholm integral equations of the second kind, mixed elliptic-hyperbolic problems, and the inverse nite Laplace transform. For all these problems, we nd the problem complexity and derive nearly optimal 13
algorithms. In particular, we discuss the optimality of Galerkin algorithms using nite element information. We then conclude by looking at the heat equation running backwards in time, an application that is not an instance of the standard problem. Suppose we know an a priori L2-bound on the solution of the backwards heat equation at time t = ?t1 and that we want to solve the equation at time t = ?t0, where 0 < t0 < t1. The problem element is \ nal data", i.e., the solution at time t = 0. We then nd that by truncating the standard series representation of the solution, we get an optimal algorithm. Note that two of our applications (the inverse nite Laplace transform and the backwards heat equation) are ill-posed problems. Hence we see that the techniques of this paper are powerful enough for us to determine that the "-complexity of ill-posed solution-restricted problems is nite. In what follows we use the standard terminology and notation for multi-indices, as well as Sobolev spaces, norms, and inner products. For details, consult any standard reference on elliptic boundary-value problems and nite element methods, such as [1], [3], [7, Chapter IV], [9], or [16]. The letter C will denote a generic constant whose value may change from one place to the next. All O-, -, -, and -estimates will be independent of n or ", depending on the context. Finally, we note that in the speci c applications we consider here, the space X is also a Sobolev Hilbert space, along with G and W . Hence, an information operation from can be represented as an inner product over X . We shall do this consistently, without further comment.
4.1. The standard problem.
Our rst problem is really a meta-problem, since it includes many important practical problems as particular instances. Let Rd be a suciently smooth simply-connected q ( ) and H r ( ) be region, which is bounded. Given q and r with q < r, we let Hbd r bd( ) and closed subspaces of H q ( ) and H r ( ), respectively, for which C01( ) Hbd r ( ) ,! H q ( ). That is, functions in these spaces may satisfy homogeneous boundary Hbd bd q ( )-functions are also satis ed by functions conditions, and the conditions satis ed by Hbd r ( ). in Hbd q ( ) and W = H r ( ), so that the We now consider any problem for which G = Hbd q r ( ) ! H ( ) is the usual inclusion embedding and bd F is the unit ball mapping E : Hbd r ( ). Since this kindbdof problem will be useful in later applications, we will call it of Hbd the standard problem. Although we have speci ed neither the space X nor the mapping L, we can still discover much about such a problem. In particular, we show that optimal information in F is given by the solution of a generalized elliptic eigenproblem whenever q; r 2 N. We then show that for any r and q, the nth minimal radius is proportional to n?(r?q)=d , so that the "-complexity is proportional to (1=")d=(r?q) . Since it is not generally possible to nd a closed-form solution of this eigenproblem, we need to consider nearly optimal that is easier to obtain. We show that nite element information of degree k is nearly optimal if k r ? 1. Having dealt with the case = F , we then prove analogous results for the case = . We rst show that the eigenvectors and eigenvectors of E E are solutions of a generalized elliptic eigenproblem when q; r 2 N. For any s 2 N, we de ne the partial di erential 14
operator Ps as
Ps v =
X
j js
(?1)j jD2 v
From [1, Theorem 10.2], we see that for s j 2s ? 1, there exist partial di erential operators s;j of exact order j , such that the Green's formula
hz; wiH s ( ) = hPsz; wiL2 ( ) +
s X j =0
h s;2s?1?j (z); @j wiL2 (@ )
8 z; w 2 C 1( ) (4.1.1)
holds, with @ denoting the outward-pointing normal derivative on @ . We then have r ( ) into H q ( ). For 0 j r ? 1, let Lemma 4.1.1. Let E be the embedding of Hbd bd ? ;j =
r;2r?1?j ? ?2 q;2q?1?j for 0 j q ? 1; r;2r?1?j for q j r ? 1:
Then E Ez = 2z i z and are a solution of the generalized elliptic eigenproblem
Pr z = ?2Pq z
in ;
(4.1.2)
(0 j r ? 1):
(4.1.3)
subject to the homogeneous boundary conditions
? ;j (z) @ = 0 Proof: : We have E Ez = 2 z i
2hz; wiHbdr ( ) = hz; wiHbdq ( ) ;
8 w 2 C 1( ):
(4.1.4)
Using (4.1.1) and (4.1.4), we see that E Ez = 2z i r?1 X ? 2 h[Pr ? Pq ]z; wiL2 ( ) = h? ;j (z); @j wiL2 (@ ) j =0
8w 2 C 1( ):
Choosing w with compact support, we nd that
h[Pr ? ?2Pq ]z; wiL2 ( ) = 0
8w 2 C01( ):
Hence z and are weak solutions of the eigenproblem (4.1.2). Since is smooth, we see that weak and smooth solutions of this eigenproblem coincide. Moreover, by appropriate choices of w, it follows that the boundary conditions (4.1.3) are satis ed. Note that (4.1.2){(4.1.3) is a generalized eigenproblem, akin to the generalized eigenproblems Az = Bz studied in computational linear algebra. It reduces to a standard elliptic eigenproblem whenever q = 0, i.e., whenever G = L2( ). We illustrate this construction by two one-dimensional cases. 15
Example (q = 0): Suppose that d = 1 and = I = [0; 1], with q = 0. Then the eigenfunctions zj and eigenvalues j of E E are solutions of the eigenproblem q X i=0
(?1)i z(2i)(s) = ?2y(s)
8 s 2 [0; 1];
j X i ( q ? j +2 i ) (?1) z (0) = (?1)i z(q?j+2i) (1) = 0 i=0 i=0
j X
(0 j q ? 1):
When q = 0 and r = 1, we easily nd that
1 if j = 1 zj (s) = p ? 2 cos (j ? 1)s if j 2 and that
1
j = p : 2 1 + (j ? 1)2 Unfortunately, the exact solution of the eigenproblem is unknown for arbitrary r. Example (q = 1, r = 2): Again suppose that d = 1 and = I = [0; 1], but now with q = 1 and r = 2. Then the eigenfunctions zj and eigenvalues j of E E are solutions of the generalized eigenproblem
z ? z00 + z0000 = 2(z ? z00 );
(4.1.5)
subject to the boundary conditions
z00 (0) = z00 (1) = 0; z000 (0) + ( ?2 ? 1)z0 (0) = z000 (1) + ( ?2 ? 1)z0 (1) = 0:
(4.1.6)
Using Mathematica,pwe quickly found that apbasis for the solution spacepof (4.1.5) is given by f ; = exp(x i )g, where = 21 2 4 + 4 2 ? 6 and = 21 2(1 ? 2), and so
zj = C1
+;+ + C2 ?;+ + C3 +;? + C4 ?;? :
However, we were unsuccessful in nding values for the weights C1; : : : ; C4 and for such that the boundary conditions (4.1.6) hold. 4.1.1. Results for the case = F . We now suppose that = F . Using the solution to the eigenproblem in Lemma 4.1.1, we can now nd nth optimal information in F , along with an nth minimal error algorithm. Moreover, we can get a tight bound on the nth minimal radius of information in F . 16
Theorem 4.1.1.1. Let zj and j be the j th eigenfunction and eigenvalue satisfying (4.1.2), subject to the boundary conditions (4.1.3). Let
Nn f = [hSf; z1iHbdr ( ); : : : ; hSf; zniHbdr ( )] and
n(Nn f ) =
n X j =1
hSf; zj iHbdr ( )zj
8f 2 F
8f 2 F:
Then Nn is nth optimal information in F and n is an nth minimal error algorithm, with
e( n; Nn) = r(n; F ) = n+1 = (n?(r?q)=d ): Proof: Using Lemma 4.1.1, along with Theorem 3.3, we immediately have ? ?
r ( ) ; H q ( ) : e( n; Nn) = r(n; F ) = n+1 = dn E Hbd bd
Following the proof of [28, Theorem 5.4.1], we see that ? ?
r ( ) ; H q ( ) = (n?(r?q)=d ): dn E Hbd bd
Thus the nth minimal radius is proportional to n?(r?q)=d . We now consider the "complexity of our standard problem in the class F . Using Theorems 3.3 and 4.1.1.1, we have Theorem 4.1.1.2. Let " > 0. Then ?
comp("; F ) = c (1=")d=(r?q) : Moreover, let Nn and n be the information and algorithm given by Theorem 4.1.1.1. Then c m("; F ) comp("; F ) cost( n; Nn) (c + 2)m("; F ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. Since we cannot usually nd a closed form solution to the eigenproblem (4.1.2){(4.1.3), we need to nd other kinds of information that will be nearly as good. To do this, we will look at nite element information. q ( ), whose degree is k . Let Sn;k be an n-dimensional nite element subspace of Hbd That is, there is a triangulation Tn of such that
Sn;k = s 2 Hbdq ( ) : s K 2 Pk (K ) 8 K 2 Tn ; dim Sn;k = n: 17
(Here, Pk (K ) is the space of polynomials of degree at most k, considered as functions de ned over the region K .) We will further require that fTng1 n=1 be a quasi-uniform family of triangulations of . This means that lim sup sup hK < 1; n!1 K 2Tn K where
hK = diam K
and
K = supf diam B : spheres B containing K g for any n 2 N and any K 2 Tn. (See [3], [7, Chapter XII], [16], or [28, Chapter 5 and Appendix] for further properties of nite element spaces.) We can now de ne our information. For any n 2 N, let Nn;k f = [hSf; g1iHbdq ( ); : : : ; hSf; gniHbdq ( )]
8f 2 F;
(4.1.1.1)
where fg1; : : : ; gng is a basis for Sn;k . Of course, Nn;k 2 Fn . We say that Nn;k is nite element information (FEI). Now consider the following algorithm: For f 2 F , let un;k 2 Sn;k satisfy (1 i n)
hun;k ; giiHbdq ( ) = hSf; giiHbdq ( ):
(4.1.1.2)
Since un;k depends on f only through the information Nn;k f , we may write
un;k = n;k (Nn;k f ): Clearly, n;k is a Galerkin method using test and trial space Sn;k . Note that we can determine the coecients a = [ 1; : : : ; n] of un;k (with respect to the basis g1; : : : ; gn) by q ( ) and i = hSf; gj iH q ( ) . solving a linear system Ka = b, where j;j = hgj ; giiHbd bd Remark : Note that we refer to n;k as a Galerkin method using the FEI Nn;k . However, n;k is not a \ nite element method" in the usual sense of the term, since the information q ( ) instead of the form i (f ) used by the nite element used is of the form hSf; giiHbd method. Our main error estimate for the Galerkin method using FEI is Theorem 4.1.1.3. Let k r ? 1. Then for any n, we have ?
e( n;k ; Nn;k ) = (n?(r?q)=d ) = r(n; F ) : Thus if k r ? 1, then the information Nn;k is nth nearly optimal information in F , and the algorithm n;k is an nth nearly minimal error algorithm. 18
Proof: Let f
q ( )-projection of Sf onto S . By [28, 2 F . Then n;k (Nn;k f ) is the Hbd n;k
Lemma 5.4.3], there is a positive constant C , independent of n and f , such that
kSf ? n;k (Nn;k f )kHbdq ( ) = inf kSf ? skHbdq ( ) Cn?(minfk+1;rg?q)=d kSf kHbdr ( ): s2Sn;k
Since k + 1 r, we thus nd that
e( n;k ; Nn;k ) = O(n?(r?q)=d ): Using Theorem 4.1.1.1, we have
e( n;k ; Nn;k ) r(n; F ) = (n?(r?q)=d ); completing the proof of the theorem. Remark : Note that we do not claim that n;k is an optimal error algorithm using Nn;k . If we use the prescription of Lemma 3.1, we would construct an optimal error algorithm Nn;k using Nn;k as follows. For f 2 F , let uNn;k 2 E Sn;k satisfy
huNn;k ; E giiHbdr ( ) = hSf; E giiHbdr ( )
(1 i n):
Since uNn;k depends on f only through the information Nn;k f , we may write uNn;k = Nn;k (Nn;k f ). Then Nn;k is an optimal error algorithm using Nn;k . Remark : It is possible to describe somewhat di erent nite element information such that the resulting Galerkin method (using the same space of test and trial functions) is an optimal error algorithm using this new FEI. To do this, let S^n;k be an n-dimensional r ( ), whose degree is k , with the spaces S^n;k being based on a nite element subspace Hbd quasi-uniform family of triangulations of . Then our FEI has the form
N^n;k f = [hSf; w1iHbdr ( ); : : : ; hSf; wniHbdr ( )]
8f 2 F;
where fw1; : : : ; wng is a basis for S^n;k . Now consider the following algorithm: For f 2 F , let u^n;k 2 S^n;k satisfy
hu^n;k; wi iHbdr ( ) = hSf; wiiHbdr ( ): Since u^n;k depends on f only through the information Nn;k f , we may write
u^n;k = ^n;k (N^n;k f ): Clearly, ^n;k is a Galerkin method using test and trial space S^n;k . Moreover, it is possible to show that e( ^n;k ; N^n;k ) = r(N^n;k ) = (n?(r?q)=d ): 19
(The proof is slightly more involved than that of Theorem 4.1.1.3, requiring the use of the r ( ), we duality estimate in [3, Theorem 2.3.1].) Note that since S^n;k is a subspace of Hbd automatically have k r ? 1 (see [28, Lemma 5.4.2]). It might seem that since ^n;k is an optimal error algorithm, one would prefer using the algorithm ^n;k instead of the algorithm n;k. However, note that if we use ^n;k, r ( ). In then we need to construct a nite element space S^n;k that is a subspace of Hbd practical situations, this would require more initial precomputation than constructing a q ( ). Moreover, the errors of and nite element space Sn;k that is a subspace of Hbd n;k ^n;k are roughly the same. Hence any gain that might be realized in using ^n;k will be o set by the loss involved in the additional precomputation. For this reason, we prefer to use n;k . We now determine the cost of using our Galerkin method to compute "-approximations for our standard problem. Let us denote this cost by costGal ("; F ) = minf cost( n;k ; Nn;k ) : e( n;k ; Nn;k ) " g: We then have
Theorem 4.1.1.4. Let " > 0. Then
?
costGal ("; F ) = c (1=")d=(r?q) : Hence the Galerkin method using FEI is a nearly optimal complexity algorithm when = F . Proof: 4.1.1.3, we see that e( n;k ; Nn;k ) " holds if and only if we choose ? From Theorem d= ( r ? q ) n = (1=") . Now use Theorem 4.1.1.2. 4.1.2. Results for the case = . Now, we suppose that = , i.e., continuous linear functionals are permissible information. We can nd optimal information, minimal error algorithms, and optimal complexity algorithms for the case = by using the results for the case = F , along with Theorem 3.4. We rst look at information based on the eigenvectors of E E . Theorem 4.1.2.1. Let zj and j be the j th eigenfunction and eigenvalue satisfying (4.1.2), subject to the boundary conditions (4.1.3). For any n 2 N and any > 0, choose 1; ; : : : ; n; 2 X such that kzj ? L j; kHbdq ( ) nn+1 : Let Nn; f = [1; (f ); : : : ; n; (f )] 8f 2 F; and n X n; (Nn; f ) = j; (f )zj 8f 2 F: j =1
20
Then
e( n; ; Nn; ) (1 + ) n+1 = (n?(r?q)=d )
and
r(n; ) = r(n; F ) = n+1 = (n?(r?q)=d ): Hence Nn; is nth nearly optimal information in and n; is an nth nearly minimal error algorithm. Proof: Immediate from Theorems 3.4 and 4.1.1.1. Having determined nearly optimal information and nearly minimal algorithms for the case = , we can now determine the problem complexity and nd nearly optimal complexity algorithms. Theorem 4.1.2.2.
(1) For any " > 0, we have ?
comp("; ) = c (1=")d=(r?q) : (2) Suppose that the eigenvalues n are strictly monotonically decreasing in n. Then for any " > 0, we have ?
comp("; ) = c (1=")d=(r?q) : Let Nn; 2 n be the information given by (3.15) and let n; be the algorithm given by (3.16), with n = m("; ) + 1 and satisfying
1 0 < <
n ? 1 n + n n+1
(r?q)=d
? 1:
Then ?
c m("; ) comp("; ) cost( n; ; Nn; ) (c + 2) m("; ) + 1 ? 1: Hence for c 1, the algorithm n; using information Nn; is a nearly optimal complexity algorithm. Proof: This follows immediately from Theorems 3.4 and 4.1.2.1. Remark : Note that the assumption that the n are strictly monotonically decreasing in n
generally will not hold unless d = 1. For example, suppose that q = 0 and r = 0; then the n are eigenvalues of the operator (1 ? )?1 , where is the Laplacian. This operator has multiple eigenvalues, unless d = 1, and so Theorem 4.1.2.2 cannot be immediately applied. However, it is possible to modify this theorem to account for the case of multiple eigenvalues. Since we will not need this modi ed version of this theorem in what follows, we do not feel that it is important to state the modi ed theorem. 21
As in the previous section, the optimal information and algorithm of Theorem 4.1.2.2 are usually not available in closed form. Hence, we need to look at alternative nearly optimal information and algorithms. In particular, we now consider modi ed Galerkin methods for computing "-approximations to our problem. As in the previous section, we let Sn;k q ( ), whose degree is k , with the denote an n-dimensional nite element subspace of Hbd corresponding family fTng1 n=1 of triangulations being quasi-uniform. We let fg1 ; : : : ; gn g again be a basis for Sn;k . However, we require that this basis satisfy the condition that there exist a positive constant C , independent of n, such that X n
j =1
2 j
1=2
n
X Cn1=2
j gj
L2 ( ) j =1
8 1; : : : ; n 2 R:
(4.1.2.1)
This condition is satis ed by the usual nite element basis functions having small support; see [5] and [28, pp. 216 .] for details. Since the range of L is dense in X , there exist 1; : : : ; n 2 X such that (4.1.2.2) kgi ? L i kHbdq ( ) Cn?((r?q)=d+1):
We can now de ne our information and algorithm. For any n 2 N, let N~n;k = [1(f ); : : : ; n(f )] 8f 2 F: Note that N~n;k 2 n is an approximation of the nite element information Nn;k 2 Fn de ned in the previous section, and so we refer to Nn;k as modi ed nite element information, or modi ed FEI, for short. Then for f 2 F , we seek u~n;k 2 Sn;k satisfying hu~n;k ; giiHbdq ( ) = i(f ): (1 i n) (4.1.2.3) Since u~n;k depends on f only through the information N~n;k f , we may write
u~n;k = ~n;k (N~n;k f ): We call ~n;k an modi ed Galerkin method using modi ed FEI. Remark : It is easy to see that the Galerkin and modi ed Galerkin methods may be reduced to the solution of n n linear systems whose solution gives the (respective) coecients of un;k or u~n;k with respect to the basis fg1; : : : ; gng. The only di erence between the Galerkin and modi ed Galerkin methods is that the latter method uses i(f ) to approxq ( ) appearing in the de nition of the former method. However, the imate the hSf; giiHbd same coecient matrix is used for both algorithms. Our main error estimate is Theorem 4.1.2.3. Let k r ? 1. Then for any n 2 N, we have ? e( ~n;k ; N~n;k ) = (n?(r?q)=d ) = r(n; ) : Thus if k r ? 1, then the information N~n;k is nth nearly optimal information in , and the algorithm ~n;k is an nth nearly minimal error algorithm. 22
2 N. For fP2n F , let un;k = n;k (Nn;k f ) and let u~n;k = ~n;k (N~n;k f ). Writing en;k = un;k ?u~n;k = j=1 "j gj , we may use the discrete Cauchy-Schwarz inequality Proof: Let n
to see that
ken;k
k2 q
Hbd ( ) =
n X j =1
"j hen;k ; gj iHbdq ( )
X n
j =1
"2 j
1=2 X n
j =1
hen;k ; gj
i2 q
Hbd ( )
1=2
: (4.1.2.4)
By (4.1.2.1), there is a positive constant C such that
n X j =1
"2 j
1=2
Cn1=2ken;k kL2( ) Cn1=2ken;k kHbdq ( ):
(4.1.2.5)
Moreover, we may use (4.1.1.2), (4.1.2.3), and (4.1.2.2) to see that
jhen;k ; gj iHbdq ( )j = jhSf; gj iHbdq ( ) ? j (f )j = jhSf; gj ? L j iHbdq ( )j kSf kHbdq ( )kgj ? L j kHbdq ( ) Cn?((r?q)=d+1):
(4.1.2.6)
Using (4.1.2.5) and (4.1.2.6) in (4.1.2.4), we nd that
ken;k kHbdq ( ) Cn?(r?q)=d for some positive constant C . Hence
kSf ? ~n;k(N~n;k f )kH q ( ) kSf ? n;k(Nn;k f )kHbdq ( ) + ken;k kHbdq ( ) Cn?(r?q)=d for a positive constant C . Since f 2 F is arbitrary, we nd that
e( ~n;k ; N~n;k ) = O(n?(r?q)=d ): Using Theorem 4.1.2.1, we have
e( n;k ; Nn;k ) r(n; ) = (n?(r?q)=d ); completing the proof of the theorem. We now determine the "-complexity of the standard problem in the class , as well as the cost of using our modi ed Galerkin method to compute "-approximations. Let us denote this cost by costmod-Gal("; F ) = minf cost( ~n;k ; N~n;k ) : e( ~n;k ; N~n;k ) " g: We then have 23
Theorem 4.1.2.4. Let " > 0. Then ?
comp("; ) = c (1=")d=(r?q) and
?
costmod-Gal ("; ) = c (1=")d=(r?q) : Hence the modi ed Galerkin method using modi ed FEI is a nearly optimal complexity algorithm when = . ? Proof: From Theorem 4.1.2.3, we see that e( ~n;k ; N~n;k ) " holds i n = (1=")d=(r?q) . Thus ? comp("; ) = c O (1=")d=(r?q) Now use Theorem 4.1.2.2.
4.2. Elliptic boundary-value problems.
We now consider the complexity of solution-restricted elliptic boundary-value problems. For the sake of brevity, we provide neither de nitions of standard vocabulary nor lists of standard results in the study of elliptic problems. The interested reader should consult any standard reference (such as [1], [3], [7, Chapter V], [9], or [16]) for more details. Previous results on the complexity of elliptic problems may be be found in [28, Chapter 5], and the references found therein. To simplify the exposition, we will follow the approach taken in [25]. However, more complicated problems could have been handled as well. Let Rd be a smooth, bounded, simply-connected region. Recall that H0m ( ) is the space of all H m ( )-functions whose normal derivatives of order less than m vanish at the boundary of . Let X Z B(v; w) = a ; D vD w j j;j jm
be a symmetric H0m ( )-coercive bilinear form. Then B is an inner product on H0m ( ), yielding an energy norm k kB de ned by p
kvkB = B(v; v)
8 v 2 H0m ( )
that is equivalent to the usual norm k k on H0m ( ). Note that if we let
Lv =
X
(?1)j jD (a ; D v);
j j;j jm
then L is a uniformly strongly elliptic operator of order 2m. Moreover, we have
B(v; w) = hLv; wiL2 ( )
8 v; w 2 H0m ( );
where (as usual) the L2( )-inner product may be interpreted as the duality pairing of H0m ( ) and its dual space H ?m ( ). 24
We now show how our problem may be expressed in terms of the general framework that r ( ) = f v 2 H r ( ) : v 2 H m ( ) g. We we have developed. Let r m, and let W = Hbd 0 then take G = H0m ( ) and X = H ?m ( ). The class F of problem elements is now the unit ball of H r ( ). Note that the Lax-Milgram lemma implies that the range of L is H ?m ( ). So L : H0m ( ) ! H ?m ( ) is a bijection. Hence S = L?1 : H ?m ( ) ! H0m ( ). Hence our solution operator S : H ?m ( ) ! H0m ( ) is given as B(Sf; v) = hf; viL2 ( ) 8f 2 H ?m ( ); v 2 H0m ( ): (4.2.1) Again, the Lax-Milgram lemma tells us that S is well-de ned. For any f 2 H ?m ( ), we have that u = Sf is the variational solution of
Lu = f in
@j u = 0 on @ (0 j m ? 1); a 2mth-order elliptic boundary problem satisfying homogeneous Dirichlet boundary conditions. Note that our problem (4.2.1) is a standard problem (with q = m), and so we can apply the results in Section 4.1. We rst look at the case = F . For any n 2 N, let Sn;k be an n-dimensional nite element subspace of H0m ( ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by Nn;k f = [hf; g1 iL2 ( ); : : : ; hf; gn iL2 ( )]: For f 2 F , let un;k = n;k (Nn;k f ) be the Galerkin method given by B(un;k ; gi ) = hf; gi iL2 ( ) (1 i n): (4.2.2) Note that the Galerkin algorithm n;k is the standard nite element method (FEM). Theorem 4.2.1. Let = F . (1) For any n 2 N, we have r(n; F ) = (n?(r?m)=d ): (2) Let k r ? 1. For any n 2 N, we have e( n;k ; Nn;k ) = (n?(r?m)=d ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ? comp("; F ) = c (1=")d=(r?m) : ? (4) Let k r ? 1. For any " > 0, let n = (1=")d=(r?m) . Then ? costGal ("; F ) = c (1=")d=(r?m) ; and so the FEM n;k using FEI Nn;k is a nearly optimal complexity algorithm. 25
Proof: Recall that B (; ) is the inner product on H0m ( ). But B (Sf; gi ) = hf; gi iL2 ( )
by (4.2.1). Hence Nn;k and n;k as de ned in the statement of the theorem are the same as in (4.1.1.1) and (4.1.1.2). The result now follows from Theorems 4.1.1.3 and 4.1.1.4. So we see that the standard nite element method of degree k r ? 1 is nearly optimal for a 2mth-order elliptic problem, if the solution elements are constrained to lie in the unit ball of H r ( ). This result should not be too surprising, given the known results about optimality of FEMs for elliptic problems, see [28, Chapter 5] and the results cited therein. The novelty in this result lies in the fact that we did not need to use a shift theorem (i.e., a result saying that if f has r ? 2m derivatives, then Sf has r derivatives) to prove the optimality of the FEM. We now look at the case = , mainly for the sake of completeness. It will turn out that FEI for this problem (which is a priori only F -information) is really continuous linear information. Rather than use the standard H ?m ( ) inner product on H ?m ( ), it will be more convenient to consider H ?m ( ) as the dual of H0m ( ) under the energy norm, i.e., we use the norm (4.2.3) kvkB0 = supm hv;kwwikL2 ( ) 8 v 2 H ?m ( ): B w2H0 ( ) We let h; iB0 denote the corresponding inner product. Since the energy norm k kB is equivalent to the usual H0m ( )-norm, it follows that this norm k kB0 is equivalent to the usual H ?m ( )-norm. Lemma 4.2.1.
(1) For any v; w 2 H ?m ( ), we have
hv; wiB0 = B(Sv; Sw): (2) L = S . Proof: From (4.2.1) and (4.2.3), we nd
hv; wiL2 ( ) = sup B(Sv; w) = kSvk ; kvkB0 = sup B v2H0m( ) kwkB v2H0m( ) kwkB the last holding because B(; ) is the inner product corresponding to the norm k kB . Part (1) now follows immediately. To prove part (2), note that we must have
hLv; wiB0 = B(v; L w) But
8 v 2 H0m ( ); w 2 H ?m ( ):
hLv; wiB0 = B(SLv; Sw) = B(v; Sw)
Comparing these last two, we see that L = S . 26
8 v 2 H0m ( ); w 2 H ?m ( ):
Now suppose that Nn;k is FEI as in the statement of Theorem 4.2.1. If for 1 i n, we let vi = Lgi , then we have L vi = Svi = gi by Lemma 4.2.1. Thus we nd that
hf; giiL2 ( ) = hf; vi iB0
(1 i n):
We now see that
Nn;k f = [hf; v1 iB0 ; : : : ; hf; vn iB0 ]: For f 2 F , it now follows that un;k = n;k (Nn;k f ) satis es B(un;k ; gi) = hf; vi iB0
(1 i n):
(4.2.4)
So the information Nn;k de ned in Theorem 4.2.1 is continuous linear information. Theorem 4.2.1 now tells us that the FEM is a nearly optimal error algorithm and a nearly optimal complexity algorithm using continuous linear information. Of course, Theorems 4.1.1.3 and tell us that r(n; ) = r(n; F ) = (n?(r?m)=d ) and comp("; ) = c ? 4.1.1.4 (1=")d=(r?m) . Hence (4.2.2) and (4.2.4) are two formulations of the FEM algorithm using FEI. The rst formulation shows that this FEI is information from F , whereas the second shows that it actually is information from . However, the B0 -inner product is more complicated than the L2( )-inner product. So in practice, we would probably rather use (4.2.2) than (4.2.4).
4.3. Fredholm problems of the second kind.
We now look at the complexity of solution-restricted Fredholm integral equations of the second kind. For previous work on the complexity of Fredholm problems of the second kind, see [24] and [28, Section 6.3]. Let Rd be a smooth, bounded, simply-connected region. Suppose that k : ! R satis es Z Z jk(x; y)j2 dx dy < 1:
Then the operator K : L2( ) ! L2( ), de ned as (Kv)(x) =
Z
k(x; y)v(y) dy
8 v 2 L2( );
is an integral operator with a Hilbert-Schmidt kernel, and is thus compact. Assume that 1 is not an eigenvalue of (K K )1=2 . For f : ! R, we wish to nd (approximations to) u : ! R satisfying
u(x) ?
Z
k(x; y)u(y) dy = f (x)
8 x 2 ;
(4.3.1)
a Fredholm integral equation of the second kind. We formally describe our problem by taking X = G = L2( ) and W = H r ( ) for some r 0. Hence the class F of problem elements is the unit ball of H r ( ). Let L = I ? K . Then L is a bounded bijection of G onto X , and so S = L?1 is a bounded bijection of X 27
onto G. Note that our problem (4.3.1) is a standard problem (with q = 0), and so we can apply the results in Section 4.1. We rst look at the case = F . For any n 2 N, let Sn;k be an n-dimensional nite element subspace of L2( ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by
Nn;k f = [hSf; g1iL2 ( ); : : : ; hSf; gniL2 ( )]: For f 2 F , let un;k = n;k (Nn;k f ) be the Galerkin method given by
hun;k ; gi iL2 ( ) = hSf; giiL2 ( )
(1 i n):
(4.3.2)
We then have
Theorem 4.3.1. Let = F .
(1) For any n 2 N, we have
r(n; F ) = (n?r=d ): (2) Let k r ? 1. For any n 2 N, we have
e( n;k ; Nn;k ) = (n?r=d ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; F ) = c (1=")d=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")d=r . Then ?
costGal ("; F ) = c (1=")d=r ; and so the Galerkin method n;k using FEI Nn;k is a nearly optimal complexity algorithm for the Fredholm problem of the second kind. Proof: Immediate from Theorems 4.1.1.3 and 4.1.1.4. Remark : The algorithm n;k de ned by (4.3.2) is not a nite element method, a marked
contrast with what happened when we were looking at elliptic boundary-value problems. The only possibility we would have for using the approach in the previous section, in which the Galerkin method n;k turned out to be an FEM, would be to assume that K is symmetric and kK k < 1. If this were the case, then B(v; w) = hLv; wiL2 ( ) would be p an inner product on L2( ), with the norm k kB = B(; ) being equivalent to the usual L2( )-norm. Under these hypotheses, it then turns out that the Galerkin method (4.1.1.2) is the standard FEM with test and trial spaces Sn;k . However, we often need to solve our 28
problem (I ? K )u = f for non-symmetric K or kK k 1, so this approach is not generally applicable. Note that linear functionals hSf; giiL2 ( ) are hard to directly evaluate, because of their dependence on Sf . However, since S : X ! G is a bounded bijection, we can write
hSf; giiL2 ( ) = hf; S gi iL2 ( ); where S = (L )?1 = (I ? K )?1 . In principle, we can then consider the computation of S g1; : : : ; S gn as precomputation. We now see that
Nn;k f = [hf; S g1iL2 ( ); : : : ; hf; S gniL2 ( ) ]: For f 2 F , it now follows that un;k = n;k (Nn;k f ) satis es
hun;k ; giiL2 ( ) = hf; S giiL2 ( )
(1 i n):
(4.3.3)
So the information Nn;k de ned in Theorem 4.3.1 is continuous linear information. Moreover, the Galerkin algorithm (4.3.3) is a nearly optimal error algorithm and a nearly optimal complexity algorithm using continuous linear information. Of course, Theorems 4.1.1.3 ? F ? r=d and 4.1.1.4 tell us that r(n; ) = r(n; ) = (n ) and comp("; ) = c (1=")d=r . Hence (4.3.2) and (4.3.3) are two formulations of the Galerkin algorithm using FEI. The rst formulation shows that this FEI is information from F , whereas the second shows that it actually is information from . Moreover, the rst formulation uses functionals hSf; giiL2 ( ), whereas the second uses hf; S giiL2 ( ). Since the formulation (4.3.2) is simpler than the formulation (4.3.3), as well as making it clear that we are using continuous linear information. So in practice, we would probably rather use (4.3.3) than (4.3.2). Unfortunately, there is still one diculty with the Galerkin method, even if we decide to use the simpler formulation (4.3.2). Even though S g1; : : : ; S gn are well-de ned, they may not be easy to calculate. For this reason, we will look at modi ed Galerkin methods for the solution-restricted Fredholm problem of the second kind. Once again, we let Sn;k be a nite element subspace of L2( ), of dimension n and degree k. Let fg1; : : : ; gng be a basis of Sn;k satisfying (4.1.2.1). Next, we let r~ 2 f1; : : : ; rg, and let Sn~ ;k~ be a nite element subspace of H r~( ), of dimension n~ and degree k~. Of course, we must have k~ r~, since Sn~ ;k~ H r~( ). For any n 2 N, let n~ = (n(r?r~)=r~+d=(2~r)): (4.3.4) For 1 i n, let vi 2 Sn~ ;k~ satisfy
hL vi ; siL2 ( ) = hgi ; siL2 ( ): We then have 29
(4.3.5)
Lemma 4.3.1. For 1 i n, we have
kgi ? L vikL2 ( ) Cn?(r=d+1): Proof: Let 1 i n. We rst note that vi is an Sn~ ;k~ - nite element approximation (in the usual sense) of S gi . Using [28, Theorem 6.3.3.2] and (4.3.5), it follows that
kgi ? Lvi kL2 ( ) kL k kS gi ? vi kL2 ( ) C n~ ?r~=dkS gikH r~( ) C kS k n~?r~=d kgikH r~( ): (Here kL k and kS k respectively denote the L2( ) and H r~( ) operator norms. See [28, Theorem 6.3.1.1] for a proof that kS k is nite.) Recall that the spaces Sn;k are based on a quasi-uniform family of triangulations. For 0 j r, we may use [28, Lemma A.2.3.4] to see that the inverse inequalities
jgijH j ( ) Cn(~r?j)=d kgikL2 ( ) hold, where
jvjH j ( ) =
X
j j=j
kD vk2L2( )
1=2
8 v 2 H j ( )
is the H j ( )-seminorm. So
kgi kH r~( ) =
X r~
j =0
jg j2 j
i H ( )
1=2
Cnr~=d kgikL2 ( ) Cnr~=d?1=2
the latter since gi is a bounded function whose support has volume (n?1). Combining these results, we see that
kL vikL2 ( ) C n~ ?r~=d nr~=d?1=2 : Using this inequality along with (4.3.4), the lemma follows. We can now de ne our information and algorithm. For any n 2 N, let
N~n;k = [hf; v1 iL2 ( ); : : : ; hf; vn iL2 ( )]
8f 2 F:
Then for f 2 F , we seek u~n;k = ~n;k (N~n;k f ) satisfying
hu~n;k ; giiL2 ( ) = hf; vi iL2 ( ) We then have 30
(1 i n):
Theorem 4.3.2. Let = .
(1) For any n 2 N, we have
r(n; ) = (n?r=d ): (2) Let k r ? 1. For any n 2 N, we have
e( ~n;k ; N~n;k ) = (n?r=d ); so that ~n;k is an nth nearly minimal error algorithm, and N~n;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; ) = c (1=")d=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")d=r . Then ?
costmod-Gal("; ) = c (1=")d=r ; and so the modi ed Galerkin method ~n;k using modi ed FEI N~n;k is a nearly optimal complexity algorithm for the Fredholm problem of the second kind. Proof: Using Lemma 4.3.1, we see that inequality (4.1.2.2) holds. Now we can apply
Theorems 4.1.2.3 and 4.1.2.4. We close this section by discussing the optimal choice of r~ in (4.3.4), as well as the amount of preprocessing required in computing v1 ; : : : ; vn . Clearly, v1 ; : : : ; vn are independent of any problem element f , and so their calculation may be be considered precomputation. However, in practice, we would like to compute them as cheaply as possible. There are (at least) two con icting reasons why it may be dicult or expensive to calculate v1 ; : : : ; vn satisfying (4.3.5) with n~ given by (4.3.4). The rst is that we need Sn~ ;k~ to be a subspace of H r~( ). On the one hand, we want to simplify the task of designing the basis functions of the nite element space Sn~ ;k~ over a reference element; this tells us that we should choose r~ as small as possible, i.e., r~ = 1. On the other hand, we want to minimize amount of work required to calculate v1; : : : ; vn once we have designed these basis functions from (4.3.4), this criterion tells to choose r~ as large as possible, i.e., r~ = r. The question is now one of which criterion to use. To solve this conundrum, we note that we only design the reference element basis functions once, independent of n, whereas the calculation of v1; : : : ; vn depends on n. We are probably willing to expend the extra e ort involved in designing the basis functions (which only needs to be done once), thereby saving cost arising in the calculation of v1 ; : : : ; vn for various n. In other words, we feel that it would be preferable to choose r~ = r. Thus we choose a nite element subspace Sn~ ;k~ of H r ( ), where k~ is the degree of the subspace and the dimension n~ of the subspace satis es n~ = nd=(2r). 31
We now discuss the cost of computing v1; : : : ; vn. Assuming we choose r~ = r, the previous analysis implies that the cost of computing v1; : : : ; vn is (nd=(2r)+1). Since this cost grows faster than n, on-the- y calculation of v1 ; : : : ; vn will dwarf the remainder of the calculation of ~n;k(N~n;k f ) for a problem element f . However, it is possible to precompute v1; : : : ; vn, since they are independent of any f . If we decide to compute "-approximations for many f 2 F , with a xed value of ", then we may consider the cost of this precomputation as an overhead whose cost we can ignore.
4.4. Mixed elliptic-hyperbolic problems.
In this section, we look at the complexity of a solution-restricted Tricomi problem. This is a simple mixed hyperbolic-elliptic problem, which arises in the study of two-dimensional transonic ow across an airfoil; see [7, Chapter X] and [13] for further discussion. Let ?0 be a simple curve in the region y > 0 of the two-dimensional (x; y)-plane, intersecting the x-axis only at the points A(?1; 0) and B(1; 0). Let ?1 and ?2 be given by ( ?1 + 32 (?y)3=2 (x; y) 2 ?1 ; x= 1 ? 32 (?y)3=2 (x; y) 2 ?2 : Note that ?1 and ?2 intersect at the point C (0; yC ), where yC = ?( 23 )2=3 =: ?1:31037. Let
R2 be the region whose boundary is ?0 [ ?1 [ ?2 . De ne a partial di erential operator L by
Lu = yuxx + uyy : L is called the Tricomi operator. Note that (1) L is elliptic (but not strongly elliptic) in the elliptic region E = f (x; y) 2 : y > 0 g. This corresponds to subsonic ow. (2) L is parabolic on the parabolic line J = f (x; y) 2 : y = 0 g. This corresponds to sonic ow. (3) L is hyperbolic in the hyperbolic region E = f (x; y) 2 : y < 0 g. This corresponds to supersonic ow. We also note that ?1 and ?2 are the characteristic curves of L, respectively emanating from A and B. Let f : ! R. The Tricomi problem is to nd a function u : ! R satisfying Lu = f in ; (4.4.1) u = 0 on ?0 [ ?1 : Since the domain is divided by the sonic line into elliptic and hyperbolic regions, the Tricomi problem is an example of a mixed elliptic-hyperbolic problem. Remark : Note that we only prescribe boundary data for the Tricomi problem on part of @ . The simple explanation for this is that since the Tricomi problem is hyperbolic in H , prescribing boundary data on one of the characteristic lines that bounds H is sucient for solvability on H , whereas requiring the solution to satisfy given boundary 32
values on two characteristic lines would overly constrain the problem in H . Note that in principle, once we have solved the problem in H , we could solve the remaining degenerate elliptic problem in E . We now show how to express (4.4.1) as a standard problem. First, we let H 2 (bd) = f v 2 H 2( ) : v = 0 on ?0 [ ?1 g and H 2 (bd)+ = f v 2 H 2 ( ) : v = 0 on ?0 [ ?2 g: (The notations bd and bd+ respectively refer to boundary conditions and adjoint boundary conditions; see [4, pg. 71].) Then hLv; wiL2 ( ) = hv; LwiL2 ( ) 8 v 2 H 2 (bd); w 2 H 2(bd)+ ; (4.4.2) see [4, pg. 308]. Let H ?2 (bd) denote the dual space of H 2(bd), which is a Hilbert space under the norm hv; wiL2 ( ) : kvk?H2(bd) = sup w2H 2 (bd)+ kwkH 2( ) We can then de ne a linear transformation L : L2( ) ! H ?2 (bd) by letting Lv = hv; LiL2 ( ) 8 v 2 L2( ): Using (4.4.2), we see that there is a positive constant C for which kLvk?H2(bd) C kvkL2( ) 8 v 2 L2( ): Thus the bounded linear transformation L : L2( ) ! H ?2 (bd) is a weak extension of the original Tricomi operator L. Using [4, Theorem IV.3.1], we nd that L is a dense injection. Hence, we see that the problem (4.4.1) may be expressed as a standard problem, with X = H ?2 (bd), G = L2 ( ), and W = H r ( ) for some r 0. Note that q = 0. Once again, the class F of problem elements is the unit ball of H r ( ). Since L : L2 ( ) ! H ?2 (bd) is de ned weakly, we must also de ne S = L?1 weakly. That is, for f 2 F , we require that u = Sf satisfy hu; LviL2 ( ) = hf; viL2 ( ) 8 v 2 H 2(bd)+: Note that this weak de nition of S allows us to avoid the issue of boundary values for L2( )-functions. We now apply the results of Section 4.1. First, we look at the case = F . For any n 2 N, we once again let Sn;k be an n-dimensional nite element subspace of L2( ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by Nn;k f = [hSf; g1iL2 ( ); : : : ; hSf; gniL2 ( )]: For f 2 F , let un;k = n;k (Nn;k f ) 2 Sn;k be the Galerkin approximation given by hun;k ; gi iL2 ( ) = hSf; giiL2 ( ) (1 i n): (4.4.3) We then have 33
Theorem 4.4.1. Let = F .
(1) For any n 2 N, we have
r(n; F ) = (n?r=2 ): (2) Let k r ? 1. For any n 2 N, we have
e( n;k ; Nn;k ) = (n?r=2 ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; F ) = c (1=")2=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")2=r . Then ?
costGal("; F ) = c (1=")2=r ; and so the Galerkin method n;k using FEI Nn;k is a nearly optimal complexity algorithm for the Tricomi problem.
Proof: Immediate from Theorems 4.1.1.3 and 4.1.1.4.
Once again, we see that linear functionals hSf; giiL2 ( ) are hard to directly evaluate, because of their apparent dependence on Sf . To overcome this diculty, we describe an auxilliary adjoint problem. Even though (4.4.2) tells us that L is formally self-adjoint, we will now write L for the adjoint operator.5 For g : ! R, we wish to nd a function v : ! R satisfying Lv = g in ; (4.4.4) v = 0 on ?0 [ ?2 : This problem is called the adjoint Tricomi problem. We need to nd the proper weak formulation of this adjoint problem. To do this, let H ?2(bd)+ denote the dual space of H 2 (bd)+, which is a Hilbert space under the norm
kvk?H2(bd)+ = sup2 hkv;wwk iL22 ( ) : H ( ) w2H (bd) De ne a linear transformation L : L2( ) ! H ?2 (bd)+ by letting
L v = hL; viL2 ( ) 5
8 v 2 L2( ):
We do this because and , when considered as operators with domain 2 ( ), have di erent codomains. L
L
L
34
Using (4.4.2), we see that there is a positive constant C for which
kL vk?H2(bd)+ C kvkL2( )
8 v 2 L2( ): Thus the bounded linear transformation L : L2( ) ! H ?2 (bd)+ is a weak extension of
the adjoint Tricomi operator L . From [4, Theorem IV.3.1], it follows that L is a dense injection. Hence, we have de ned L weakly. This means that we can de ne S = (L )?1 weakly. For g 2 F , we require that v = S g satisfy
hLu; viL2 ( ) = hu; giL2 ( )
8 u 2 H 2 (bd):
This is the weak formulation of the adjoint Tricomi problem. Using (4.4.2), we nd that
hSf; giL2( ) = hf; S giL2( )
8f; g 2 L2( ):
In particular, we have
hSf; giiL2 ( ) = hf; S giiL2 ( )
(1 i n):
This means that we can once again consider the computation of S g1; : : : ; S gn as precomputation. Hence we have
Nn;k f = [hf; S g1iL2 ( ); : : : ; hf; S gniL2 ( ) ]: For f 2 F , it now follows that un;k = n;k (Nn;k f ) satis es hun;k ; giiL2 ( ) = hf; S giiL2 ( ) (1 i n): (4.4.5) So the information Nn;k de ned in (4.4.3) is continuous linear information. Moreover, the Galerkin algorithm (4.4.5) is a nearly optimal error algorithm and a nearly optimal complexity algorithm using continuous linear information. Of course, Theorems ? 4.1.1.3 F ? r= 2 and 4.1.1.4 tell us that r(n; ) = r(n; ) = (n ) and comp("; ) = c (1=")2=r . We now see that (4.4.3) and (4.4.5) are two formulations of the Galerkin algorithm using FEI, the rst showing that this FEI is information from F and the second showing that it actually is from . Moreover, the rst formulation uses functionals hSf; giiL2 ( ), whereas the second uses hf; S giiL2 ( ). As was the case in the previous section, we prefer to use (4.4.5) instead of (4.4.3). As in the previous section we see that S g1; : : : ; S gn may be hard to calculate, even though they are well-de ned. Hence we need to once again consider modi ed Galerkin methods for our problem. Of course, the only issue that we need to resolve is how to calculate v1 ; : : : ; vn such that L vi is suciently close to gi for 1 i n. One idea is to let Sn~ ;k+2 be an n~ -dimensional nite element subspace of L2( ) having degree k + 2, where n~ = (nr+2 ): (4.4.6) 35
Suppose that we choose vi solving the least squares problem
kgi ? Lvi kL2 ( ) = inf kgi ? L skL2 ( ); s2Sn~ ;k+2
i.e., vi 2 Sn~ ;k+2 satis es
8 s 2 Sn~ ;k+2:
hLvi ; LsiL2 ( ) = hgi ; LsiL2 ( )
We are now ready to de ne our information and algorithm. For any n 2 N, let
N~n;k = [hf; v1 iL2 ( ); : : : ; hf; vn iL2 ( )]
8f 2 F:
Then for f 2 F , we seek u~n;k = ~n;k (N~n;k f ) satisfying
hu~n;k ; giiL2 ( ) = hf; vi iL2 ( )
(1 i n):
We then have
Theorem 4.4.2. Let = .
(1) For any n 2 N, we have
r(n; ) = (n?r=2 ): (2) Let k r ? 1. For any n 2 N, we have
e( ~n;k ; N~n;k ) = (n?r=2 ); so that ~n;k is an nth nearly minimal error algorithm, and N~n;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; ) = c (1=")2=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")2=r . Then ?
costmod-Gal ("; ) = c (1=")2=r ; and so the modi ed Galerkin method ~n;k using modi ed FEI N~n;k is a nearly optimal complexity algorithm for the Tricomi problem. 36
Proof: Since Lv 2 Pk for v 2 Pk+2 , one can exactly solve the problem vi = S gi , except possibly on the boundary elements. But these boundary elements have area (~n?1=2 ) = (n?(r=2+1)). Hence kgi ? Lvi kL ( ) Cn?(r=2+1) for 1 i n. Now use Theo2
rems 4.1.2.3 and 4.1.2.4. We once again note that on-the- y computation of v1 ; : : : ; vn may be expensive. Indeed, from (4.4.6), we see that v1; : : : ; vn may be calculated with cost (nr+3 ), which of course greatly outweighs the cost of calculating u~n;k . However, we once again point out that v1; : : : ; vn are independent of any f 2 F . So, if we precompute v1 ; : : : ; vn and if we do not charge for this precomputation (since it is independent of any problem element), we can ignore the cost of the precomputation. Remark : Note that we used a very weak error bound to show that (4.4.6) implies that v1; : : : ; vn are suciently accurate. This was motivated by our lack of a shift theorem for the Tricomi problem, so that we cannot assume enough global smoothness in S gi (where gi is piecewise polynomial) to use the error estimates in 2. It is quite possible that our estimate is overly-pessimistic, and that we can nd suciently accurate piecewise polynomial approximations to S gi using fewer degrees of freedom.
4.5. Inverse nite Laplace transform.
In this section, we look at the complexity of a solution-restricted inverse Laplace transform. This is an example of a Fredholm integral equation of the rst kind, and is thus an ill-posed problem. This problem arises in remote sensing problems of geomathematics; see [21] for discussion and further examples. Without loss of generality, we assume that our functions are de ned over the unit interval I = [0; 1]. De ne an operator L : L2(I ) ! L2 (I ) as (Lu)(s) =
Z 1
0
e?stu(t) dt
(0 s 1)
for u 2 L2(I ). Thus Lu is the nite Laplace transform of u. We are interested in the inverse nite Laplace transform problem : for f 2 L2 (I ), nd u 2 L2(I ) such that Lu = f , i.e., such that Z 1 e?st u(t) dt = f (s) (0 s 1): (4.5.1) 0
From [14], we see that L : L2(I ) ! L2 (I ) is an injection. Since L is self-adjoint as an operator on L2(I ), we see that the range of L is dense in L2 (I ). Hence the solution operator S = L?1 is densely de ned in L2(I ). Thus we can express our problem (4.5.1) as a standard problem if we choose G = X = L2(I ) and W = H r (I ) for some r 0. Note that L is compact. Thus the problem of nding u 2 L2(I ) satisfying Lu = f is ill-posed for f 2 L2(I ). Remark : The problem of inverting L is in fact very ill-posed. From [28, pp. 198-199], we nd that L : L2(I ) ! H r (I ) is a compact injection with dense range for any r 0. Thus the (densely de ned) solution operator S = L?1 : H r (I ) ! L2(I ) is unbounded, no matter how big we choose r to be. Hence the results of [26] imply that if we choose our problem elements to be the unit ball of a Hilbert Sobolev space H r (I ), then the error of 37
any nite-cost algorithm is in nite, no matter how large we choose r to be. Simply stated, this means that restricting the problem elements for (4.5.1) will not work. This explains why we are interested in a solution-restricted version of this problem. Having expressed (4.5.1) as a solution-restricted problem, we can now use the results of Section 4.1. First, we look at the case = F . For any n 2 N, let Sn;k be an ndimensional nite element subspace of L2(I ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by
Nn;k f = [hSf; g1iL2 (I ); : : : ; hSf; gniL2 (I )]: For f 2 F , let un;k = n;k (Nn;k f ) be the Galerkin method given by
hun;k; gi iL2 (I ) = hSf; giiL2 (I )
(1 i n):
(4.5.2)
We then have
Theorem 4.5.1. Let = F .
(1) For any n 2 N, we have
r(n; F ) = (n?r ): (2) Let k r ? 1. For any n 2 N, we have e( n;k ; Nn;k ) = (n?r ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; F ) = c (1=")1=r :
?
(4) Let k r ? 1. For any " > 0, let n = (1=")1=r . Then ?
costGal("; F ) = c (1=")1=r ; and so the Galerkin method n;k using FEI Nn;k is a nearly optimal complexity algorithm for the inverse nite Laplace transform problem. Proof: Immediate from Theorems 4.1.1.3 and 4.1.1.4.
As with problems that we have considered previously, we prefer to avoid using the functionals hSf; giiL2 (I ) making up the nite element information Nn;k . We would rather use functionals of the form hf; vi iL2 (I ). In the discussion that follows, we will take advantage of the self-adjointness of L as an operator on L2 (I ), which implies that S is self-adjoint in L2(I ). Thus we write L and S instead of L and S in what follows. Note that we cannot rewrite the functionals hSf; giiL2 (I ) in the form hf; SgiiL2 (I ). This is because functions belonging to the domain of S (i.e., the range of L) are in nitely 38
di erentiable, while the functions gi are only piecewise smooth. So the nearly optimal Galerkin method using FEI for the inverse nite Laplace transform, which (of course) uses F -information, cannot be cosmetically rewritten as a method using -information. This is an important di erence between this problem and the other problems that we have studied. This being the case, we now let = , and look for nearly optimal -information, using our general results in Section 4.1.2. The main idea is to choose v1; : : : ; vn 2 X such that Lv1 ; : : : ; Lvn are suciently good approximations of g1 ; : : : ; gn. To do this, we use ideas based on those in [28, pp. 225{226]. We assume that the nite element space Sn;k is de ned over a uniform partition of I . Suppose that Sn;k H p(I ) for some positive integer p, so that k p. Recall that fg1; : : : ; gng is a standard basis for Sn;k , having small supports. Our task is to nd functions fv1; : : : ; vng such that (4.1.2.2) holds. To do this, we let R and m be parameters depending on n, p, and r. We will give criteria for choosing R and m later. For any j 2 N, we let Gj (x)P= Pj (2x ? 1), where Pj is the usual Legendre polynomial of degree j . We write Gj (x) = jl=0 j;lxl . Next, for any l 2 N, we let (
wl =
? (?1)j jl Rl+1 if t 2 [j=R; (j + 1)=R) for some j 2 f0; : : : lg, 0 if t 2 [(l + 1)=R; 1]
We then let
vi =
m X
j X
j =0
l=0
(2j + 1)hgi ; Gj iL2 (I )
j;l wl
for 1 i n. Then we have Lemma 4.5.1. For any n and k , there exist R and m such that
kLvi ? gikL2 (I ) Cn?(r+1) Proof: Let i
(1 i n):
(4.5.3)
2 f1; : : : ; ng. Since the support of gi has length (n?1 ), there exists a
positive constant A such that
kgikL2 (I ) An?1=2
Since Sn;k is de ned over a uniform partition of I , we know that the inverse inequality [28, Lemma A.2.3.4] holds, and so there exists B > 0 such that
kgi(p)kL2(I ) BA?1 npkgikL2 (I ) Bnp+1=2: We claim that for (4.5.3) to hold, it suces to choose
R pA nr+1=2 3C
and
m 4e 2CB 39
1=p
n(r+p+1=2)=p:
(4.5.4)
Indeed, let
gi;m =
m X j =0
(2j + 1)hgi ; Gj iL2 (I )Gj
be the mth Legendre series approximation of gi . From [8, Theorem 1.3.2], we have e p kgi ? gi;m kL2 (I ) 4m kgi(p)kL2 (I ): Following [28, pp. 225{226], we nd that e p 1 A kLvi ? gikL2 (I ) 2Rp3 n1=2 + B 4m np?1=2: Using (4.5.4), we now nd that (4.5.3) holds, as claimed. We are now ready to de ne our information and algorithm. For any n 2 N, let v1; : : : ; vn be as de ned in Lemma 4.5.1. De ne information N~n;k as N~n;k = [hf; v1 iL2 ( ); : : : ; hf; vn iL2 ( )] 8f 2 F: The modi ed Galerkin algorithm ~n;k is as follows: For f 2 F , we seek u~n;k = ~n;k (N~n;k f ) satisfying hu~n;k ; giiL2 ( ) = hf; vi iL2 ( ) (1 i n): We then have Theorem 4.5.2. Let = . (1) For any n 2 N, we have r(n; ) = (n?r ): (2) Let k r ? 1. For any n 2 N, we have e( ~n;k ; N~n;k ) = (n?r ); so that ~n;k is an nth nearly minimal error algorithm, and N~n;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; ) = c (1=")1=r :
?
(4) Let k r ? 1. For any " > 0, let n = (1=")1=r . Then ?
costmod-Gal ("; ) = c (1=")1=r ; and so the modi ed Galerkin method ~n;k using modi ed FEI N~n;k is a nearly optimal complexity algorithm for the inverse nite Laplace transform problem. 40
Proof: Immediate from Theorems 4.1.2.3 and 4.1.2.4, along with Lemma 4.5.1.
We brie y discuss the choice of p. Clearly the larger we make p, the smaller m needs to be. The optimal choice is then to make p = k. Since the best choice for k is to let k = r ? 1, we see that the best choice for p is p = r ? 1. It then follows that m = (n ), where = 2 + 3=(2r ? 2). Since m grows faster than n2, the cost of computing v1; : : : ; vn on-the- y grows faster than n3 , and is therefore impractical. However, since v1; : : : ; vn are independent of any f , they may be precomputed. If we wish to compute "-approximations for many f 2 F , with a xed value of ", then we can safely ignore the cost of this precomputation.
4.6. The backwards heat equation.
In our nal application, we look at the complexity of the heat equation running backwards in time. This is one of the most famous classical examples of an ill-posed problem. Further discussion and references may be found in [10], [11], [12], [17], and [19]. Let I = [0; 1]. For any positive integer j , let
p
8 x 2 I: (4.6.1) Then fsj g1 j =1 is an orthonormal basis for L2 (I ). For t 2 R, we de ne a positive-de nite sj (x) = 2 sin jx
self-adjoint operator Ht in L2(I ) by letting
Ht f =
1 X j =1
e?2 j2 t hf; sj iL2 (I )sj
8f 2 L2(I ):
(4.6.2)
Note that (1) If t > 0, then Ht is compact. (2) H0 is the identity map. (3) If t < 0, then Ht is a densely-de ned unbounded operator. (4) We have the semigroup properties
Ht H = Ht+ H?t = Ht?1
8 t; 2 R; 8 t 2 R:
(5) For any t 2 R, let u(; t) = Ht f . Then u(x; t) satis es
@u = @ 2 u @t @x2 u(; 0) = f u(0; ) = u(1; ) = 0
in I R; in I; in R:
Thus Ht f is the solution of the heat equation at time t, with u at time t = 0 being given by f . We will call these conditions at time t = 0 initial conditions if t > 0 and nal conditions if t < 0. 41
We wish to solve this problem at time t = ?t0 , where t0 > 0. Since the elapsed time is negative, we call this problem a backwards heat equation. The operator H?t0 is a unbounded in L2 (I ), and so this problem is ill-posed. We shall formulate our problem as a solution-restricted operator equation. Our restriction will be to assume that the solution is L2(I )-bounded at some time t = ?t1 in the past, where t0 < t1. It is well-known that the ill-posed backwards heat equation becomes well-posed under this hypothesis, see [10]. Note that the approach taken here is similar to that in [27]. To express this problem as a solution-restricted operator equation, we let X = G = L2(I ) and de ne L : G ! X be de ned as L = Ht0 . Then Lu = f i f is the solution of a heat equation at time t = t0 , with initial conditions given by u at time t = 0. From the previous comments, we see that L is a compact injection with dense range. Our solution operator is now S = L?1 : G ! X . From the semigroup properties, we see that S = H?t0 , i.e., u = Sf is the solution of a heat equation at time t = ?t0, with nal conditions given by f at time t = 0. Since S is an unbounded ? operator, we are trying to solve an ill-posed problem. Finally, we let W = Ht1 ?t0 L2(I ) , which is a Hilbert space under the norm
kwkW = kH?(t1?t0 )wkL2 (I ) 8 w 2 W: Since t1 > t0, we see that kkL2(I ) kkW , and so the identity embedding E : W ! L2(I ) is continuous, with kE k 1. Hence, the spaces X , G, and W , along with the operator L : G ! X , de ne a solution-restricted operator equation. It is straightforward to check that the class F of problem elements is the set of all f 2 L2(I ) for which H?t1 f belongs
to the unit ball of L2(I ). Hence this solution-restricted problem is the solution of the backwards heat equation at time t = ?t0 with prescribed nal data at time t = 0, under the constraint of a known bound on the solution at the earlier time t = ?t1. We will use the results contained in Theorem 3.3. This requires us to nd an orthonormal basis for W consisting of eigenvectors of E E , as well as the corresponding eigenvalues
12 22 > 0. For j = 1; 2; : : : , we will let
zj = Ht1 ?t0 sj = e?2 j2 (t1?t0 )sj :
(4.6.3)
Then fzj g1 j =1 is an orthonormal basis for W . Lemma 4.6.1. For any positive integer j , we have
E Ezj = j2zj ; where
j = e?2 j2 (t1?t0 )
(4.6.4)
and zj is given by (4.6.3). Proof: We rst claim that E E = H2(t1 ?t0 ) . Indeed, let v; w 2 W . Since H?(t1 ?t0 ) is self-adjoint on L2 (I ), we have
hv; wiL2 (I ) = hEv; EwiL2 (I ) = hv; E EwiW = hH?(t1 ?t0 )v; H?(t1 ?t0)E EwiL2 (I ) = hv; H?2(t1 ?t0 )E EwiL2 (I ): 42
Thus H?2(t1?t0 )E E is the identity operator on W , and so E E = H??2(1 t1?t0 ) = H2(t1?t0 ), as claimed. Next, we claim that for any w 2 W and any index j , we have hw; zj iW zj = hw; sj iL2 (I )sj : (4.6.5) Indeed, since H?(t1?t0 )sj is a scalar multiple of sj and H?(t1?t0 ) is self-adjoint, we have hw; zj iW zj = hH?(t1 ?t0 )w; H?(t1 ?t0)zj iL2 (I )zj = hH?(t1 ?t0 )w; sj iL2 (I )Ht1 ?t0 sj = hw; H?(t1 ?t0 )2sj iL2 (I )Ht1 ?t0 sj = hw; sj iL2 (I )sj ; as claimed. Now since E E = H2(t1?t0 ), we may use (4.6.2) and (4.6.5) to nd that 1 1 X X 2 j 2 (t1 ?t0 ) ? 2 E Ew = H2(t1?t0 )w = e hw; sj iL2 (I )sj = e?22 j2 (t1?t0)hw; zj iW zj : j =1
j =1
The result follows immediately. We are now ready to apply the results in Theorem 3.3. First, we consider the case = F . For any n 2 N, we de ne information Nn as Nnf = [hSf; z1iW ; : : : ; hSf; zniW ]; where z1 ; : : : ; zn are given by (4.6.3), and an algorithm n using Nn as
n(Nn f ) = We then have
n X j =1
hSf; zj iW zj
Theorem 4.6.1. Let = F .
8f 2 F:
(1) For any n 2 N, we have r(n; F ) = e( n; Nn ) = e?2 (n+1)(t1?t0 ): Hence Nn is nth optimal information in F , and n is an nth minimal error algorithm. (2) For any " > 0, the "-cardinality number m("; F ) = inf f integers n 0 : n+1 " g is given by (& ' ) r 1 1 F m("; ) = max pt ? t ln " ? 1; 0 : 1 0 (3) For " > 0, let n = m("; F ). Then c m("; F ) comp("; F ) cost( n; Nn) (c + 2)m("; F ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. 43
Proof: Immediate from Lemma 4.6.1 and Theorem 3.4.
?p Hence we see that comp("; F ) = c ln(1=") , giving the complexity of the solutionrestricted backwards heat equation when = F . We now look at the case = . For any n 2 N, we de ne information N~n 2 n as
N~nf = [hf; s1 iL2 (I ); : : : ; hf; sn iL2 (I )]; where s1 ; : : : ; sn are given by (4.6.1), and an algorithm ~n using N~n as n
X ~n(N~nf ) = e2 j2 t0 hf; sj iL2 (I )sj
j =1
8f 2 F:
Note that the algorithm ~n is the n-term truncation of the standard series representation H?t0 of the solution to the heat equation at time t = ?t0. We then have Theorem 4.6.2. Let = . (1) For any n 2 N, we have r(n; ) = e( n; Nn ) = e?2 (n+1)(t1?t0 ): Hence Nn is nth optimal information in , and n is an nth minimal error algorithm. (2) For any " > 0, the "-cardinality number m("; ) = inf f integers n 0 : n+1 " g is given by (& ' ) r 1 1 m("; ) = max pt ? t ln " ? 1; 0 : 1 0 (3) For " > 0, let n = m("; ). Then c m("; ) comp("; ) cost( n; Nn) (c + 2)m("; ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. Proof: Let j 2 f1; : : : ; ng. Using (4.6.5) and the self-adjointness of S = H?t0 , we nd
hSf; zj iW zj = hH?t0 f; sj iL2 (I )sj = hf; H?t0 sj iL2 (I )sj = e2 j2 t0 hf; sj iL2 (I )sj : ~ ) = n(Nnf ) for any f 2 F . The theorem now follows immediately from Thus ~n(Nf Theorem 4.6.1.
Hence, we have shown that the optimal F -information is actually -information, and that we get a minimal error algorithm by truncating the standard series representation for ?p the solution of the backwards heat equation. Moreover, comp("; ) = c ln(1=") , giving the complexity of the solution-restricted backwards heat equation when = . 44
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28.
, The Computational Complexity of Di erential and Integral Equations: An InformationBased Approach, Oxford University Press, Oxford, 1991.
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This research was supported in part by the National Science Foundation under Grant CCR-91-01149.
1. Introduction
Operator equationsd Lu = f are among the most important problems of applied mathematics. We are interested in the "-complexity of such problems; that is, we want to nd the minimal cost of computing "-accurate approximations and to nd algorithms that yield such approximations with (nearly) minimal cost. Past work on the worst case complexity of such problems has generally assumed that the class F of problem elements f is a unit ball. As we shall see, this assumption is not strong enough for many classes of problems. In this paper, we assume instead that the class U of solution elements u is a unit ball. Such problems are called \solution-restricted" problems. We study the complexity of solution-restricted operator equations Lu = f in this paper. More precisely, suppose that L : G ! X is a bounded linear injection, where G is a Hilbert space and X is a normed linear spaces. Let F X be a xed class of problem elements f . We wish to solve Lu = f for f 2 F , our only knowledge of any f being the values of a nite set of linear functionals of f , each evaluation having cost c. Since we only have partial information about f , we can only calculate approximations of u = L?1f . Our goal is to calculate "-approximations (i.e., approximations with error at most ") with minimal cost. This is, of course, a problem of information-based complexity (IBC); see [22] for further discussion. We will solve our problem in a worst case setting, so that the error and cost of an algorithm are given by their maximum values over all f . Researchers in IBC have been most successful in getting good complexity bounds for operator equation problems (for F a ball in X ) whenever the solution operator L?1 has been bounded. One such class of problems is the solution of elliptic operator equations Lu = f , for which we have found a wealth of complexity-related results. For instance, one popular method for elliptic problems is the nite element method (FEM); we have been able to nd conditions that are necessary and sucient for the FEM to be a nearly optimal error or nearly optimal complexity algorithm. Many of these results use a \shift theorem," which says that if L is elliptic of order 2m, then f has smoothness r (i.e., r derivatives in some sense) i u has smoothness r + 2m. Similarly, since there is a shift theorem (with m = 0) for Fredholm integral equations of the second kind, we have been successful in proving results about the complexity of second-kind Fredholm problems, as well as characterizing nearly optimal FEMs for such problems. An exhaustive treatment of this subject may be found in [28, Chapters 5 and 6]. Unfortunately, there has been far less success to date in dealing with operator equations for which there is no shift theorem. For example, consider the Tricomi problem. This is a mixed elliptic-hyperbolic problem arising in the study of transonic ow across an airfoil, see [7, Chapter X] and [13]. Since there is no shift theorem for the Tricomi problem,1 we have not been able to use these techniques to obtain sharp complexity bounds for the Tricomi problem. Things are even worse when we consider the solution of ill-posed problems Lu = f , in which the solution element u does not depend continuously on the problem element f , e.g., when L is compact. We have a strong negative result on the worst case complexity of ill-posed problems when F is a ball in X , namely, that there exists no algorithm having The best global smoothness result we know for solutions of the Tricomi problem is that of [2], which only proves rst-order smoothness for a solution of a second-order problem, so there is a smoothness gap.
1
nite error (see [26]). Hence the complexity of computing an "-approximation is in nite, no matter how large we choose " to be. This means we cannot solve ill-posed problems in a worst case setting. If we need to solve such problems, we must go to a di erent setting, such as the average case setting. For further discussion, see [23], [27], [28]. Summarizing, we see that if the class F is a ball in X , good worst case complexity bounds for operator equations Lu = f have so far eluded us, except for L enjoying special properties. If we wish to solve such problems in the worst case setting for L not satisfying these properties, we need to look at other classes F . This paper uses an idea of Tikhonov [20] that is often used in the solution of ill-posed problems (see also [15] for a fuller development, as well as the discussion in [19]). Instead of solving the problem Lu = f under the a priori assumption that f belongs to a known set F (typically a ball in X ), we assume that u belongs to a known set U . Thus for such a solution-restricted operator equation, we restrict the solution elements instead of the problem elements.2 The main point of this paper is that we can often get good complexity results for solutionrestricted problems, simply because much of the work can be rephrased in terms of the well-studied problem of approximating an identity embedding. We now outline our main results. In Section 2, we formally describe the problem to be solved. The class U will be the unit ball of a Hilbert space W that is continuously embedded in G. Part of our problem description includes a discussion of the class of permissible information operations. We will be mainly interested in two classes . The rst is the usual class of continuous linear functionals over X , which has been well-studied in previous work on IBC. The second class of permissible information operations is the class F of linear functionals on F that are bounded on F . This new class of information operations is quite natural for our problem, since it is the largest class of linear functionals de ned on F . In Section 3, we show that a solution-restricted operator equation can be reduced to the approximation problem of approximating the embedding of W into G. Since the approximation problem is a well-studied problem of information-based complexity, we can easily adapt known results for the approximation problem to our problem. As a result, we quickly determine optimal algorithms and information for our problem. Moreover, we nd that the minimal radii are the same for the classes and F . This means that in principle, there is no need to consider F . However, our introduction of F is no mere arti ce. Indeed, it is often easier to rst consider the problem for = F , and then to approximate F -information by -information, rather than to directly consider information at the start. Moreover, there are situations for which F -information is more natural than -information. In Section 4, we use these results to study several important problems. The rst problem we study is called the \standard problem," which is any problem for which G and W are q ( ) and H r ( ) (the \bd" meaning that certain standard Sobolev Hilbert spaces Hbd bd homogeneous boundary conditions may be imposed), with q < r and Rd. Note that since L and X are not speci ed, this problem is really a meta-problem, which includes many Of course, since our class of problem elements is now given by de nition of . However, need not be a ball in . 2
F
F
F
X
2
F
= ?1 ( ), we do have an implicit L
U
important examples. Also note that there is no requirement that L have a bounded inverse, so that the standard problem covers both well-posed and ill-posed problems. Hence we are q ( )solving any linear operator equation Lu = f , with error being measured in the Hbd r ( ) 1. Although the standard problem is norm, subject to the constraint that kukHbd quite general, we are able to develop complexity bounds and to nd optimal algorithms and information for the standard? problem. We nd that the nth minimal radius is (n?(r?q)=d ) and the "-complexity is c (1=")d=(r?q) . The optimal information is given by eigenvectors r ( ) into H q ( ). We show that the eigenproblem of E E , where E is the embedding of Hbd bd for E E can be expressed as a generalized eigenproblem for a partial di erential equation. Unfortunately, this eigenproblem usually does not have closed-form solutions. Hence, we need to nd more accessible nearly optimal information. We are able to show that nite element information is nearly optimal, and that generalized Galerkin methods turn out to be nearly optimal algorithms. In the remainder of Section 4, we analyze several important speci c applications. The rst class of applications is elliptic boundary-value problems. We consider a 2mth-order problem, with error measured in the energy norm, which is equivalent to the Sobolev kkH m( )-norm. This is then set as a standard problem with G = H0m ( ) and W = Hr ( ). ? The nth minimal error is (n?(r?m)=d ), and the "-complexity is c (1=")d=(r?m) . For = F , we nd that the usual nite element method is nearly optimal, the proof not requiring a shift theorem. This means that we can handle elliptic problems that do not admit a shift theorem.3 For the sake of completeness, we develop the results for = , although the method that results would probably not be used in practice. We next look at Fredholm integral equations of the second kind. This is set up as a standard problem with G = L2( ) and W? = H r ( ). We nd that the nth minimal error ? r=d d=r is (n ) and the "-complexity is c (1=") . Finite element information is nearly optimal, and we exhibit nearly optimal generalized Galerkin methods. Next, we look at the Tricomi problem. Once again, we set this up as a standard problem with G = L2 ( ) and W = H r ( ). Our results are essentially the same as for the Fredholm problem of the second kind. The nth minimal error is (n?r=d ) and the "-complexity is ? c (1=")d=r . We nd that nite element information and generalized Galerkin methods are nearly optimal. Our next application is the inverse nite Laplace transform, which is a Fredholm integral equation of the rst kind and hence is ill-posed. This particular problem arises in the study of \measurement of the distribution of an absorbing gas (such as ozone in the earth's atmosphere) from the spectrum of scattered light", see [21, pp. 12{13]. This may be set up as a standard problem with G = L2 ( ) and W =? H r ( ). We nd that the nth minimal error is (n?r=d ) and the "-complexity is c (1=")d=r . Once again, nite element information is nearly optimal, as are generalized Galerkin methods. To show that our techniques are not limited to standard problems, we close this paper by studying a problem that is not a standard problem. Our nal application is the heat equation running backwards in time, with nal data f . That is, we want to know what the This means that our analysis can be easily extended to include problems for which the coecients or the boundary of are not very smooth, or problems having boundary conditions of mixed Dirichlet-Neumann type.
3
3
temperature distribution was at some time t = ?t0 in the past (where t0 > 0), where the temperature distribution at time t = 0 is given by the function f . Suppose that both the nal data and solution are measured in the L2-sense. Then the backwards heat equation is ill-posed, said ill-posedness being related to the second law of thermodynamics, see [12]. To express this as a solution-restricted problem, we let t1 > t0, and solve the problem under the condition that the solution at time ?t1 must have nite L2-norm, see also [10] for ? 2 2 (t1 ? t0 ), further discussion. We then nd that the n th minimal error is exp ? ( n + 1) p so that the "-complexity is c ( ln(1=")). Moreover, if we truncate the usual series representation for the solution of the heat equation, we get an optimal algorithm. For further discussion of the complexity of this problem, see [27]. We close this introduction by noting that one of our main assumptions has been that the calss of permissible information operations has been either F or . There is a third class that one could investigate. Suppose that X is a function space on some domain
Rd, as is generally the case in most applications. Then we could study the class std of standard information, i.e., evaluations of f (or some of its derivatives) at points in . At this time, we only have results for standard information under (what we feel are) unnecessarily-restrictive conditions on L, conditions that do not apply (for example) when L is compact, i.e., when the original problem is ill-posed. We hope that we will be able to coherently deal with standard information for solution-restricted operator equations in future work. 2. Problem description
Let G be a Hilbert space, and let X be a normed linear space. We assume that both G and X are in nite-dimensional. Let L : G ! X be a bounded linear injection, whose range D is dense in X . We de ne a solution operator S : D X ! G as
u = Sf () Lu = f
for u 2 G and f 2 D:
(2.1)
Note that if L does not have a closed range, then S is only densely de ned. Remark : Note that we restrict our attention to the Hilbert case. We do this for two reasons. The rst is for ease of exposition, while the second is that all the examples we consider are instances of the Hilbert case. We note in passing that many of the results contained here also hold for the case of general normed linear spaces. Let F be a balanced, convex subset of D, and let be a class of continuous linear functionals on F (more precisely, a class of functionals whose extensions to the linear hull of F are continuous linear functionals). Throughout this paper, we will be especially interested in the following classes : (1) = , the class of all continuous linear functionals de ned over X . This is a \standard" choice of for many problems arising in information-based complexity; see [22]. (2) = F , the class of all linear functionals on F that are bounded on F . Note that F , with equality when F is a ball in X . The strict inclusion F holds, e.g., when F is a compact subset of X . Our interest in using F instead of 4
is that we can use our a priori knowledge that f 2 F to allow us to expand the possible choice of information functionals from those that must be de ned over all of X to those that need be de ned only over F , which is a (possibly small) subset of X . Our abstract setting is the usual one of information-based complexity, see, e.g., [22] and [28]. We wish to compute approximate values of Sf for f 2 F , given a nite number of information values (f ) for some elements 2 . However, our point of departure will be to assume that there is a subset U G such that F = L(U ). Thus
f 2 F () u = Sf 2 U: (Thus, F is a subset of D.) Note that we are now restricting our solution elements u rather than our problem elements f . For this reason, we will call our problem a solution-restricted operator equation. In particular, we will assume in this paper that U is the unit ball of a Hilbert space W that is continuously embedded in G. That is,
U = f Ew : kwkW 1 g;
(2.2)
where the embedding mapping E : W ! G, de ned as Ew = w for w 2 W , is continuous and dense. To simplify the exposition in what follows, we assume without essential loss of generality that kE k 1. This assumption holds for many, but not all, cases of practical interest.4 Since Ew = w for w 2 W , we will often write w instead of Ew whenever this will simplify the presentation, provided that no confusion will result. We close this section by recalling some standard terminology from [22]. Let N denote the nonnegative integers. For any n 2 N, we say that N is information from of cardinality n if there exist linearly independent linear functionals 1; : : : ; n 2 such that
8f 2 F:
Nf = [1(f ); : : : ; n(f )]
(2.3)
We let n denote the class of information from whose cardinality is at most n. An algorithm using N is a mapping : N (F ) ! G, its error being given by
e( ; N ) = sup kSf ? (Nf )kG : f 2F
The radius of information N is given as
r(N ) = inf e( ; N );
the in mum being over all algorithms using N . An algorithm N for which e( N ; N ) = r(N ) is an optimal error algorithm using N .
For example, this assumption holds for the embedding r ( ) ! q ( ) of Sobolev spaces ( ), but it does not hold for the embeddings given by either the general statement of Sobolev's embedding theorem or by the Rellich-Kondrasov theorems (see, e.g., [6, pg. 114] for statements). 4
H
5
,
H
q < r
The nth minimal radius in is de ned to be
r(n; ) = inf r(N ); N 2n
and information Nn 2 n is nth optimal information if r(Nn ) = r(n; ). (Note that since F is balanced and convex, we are only considering nonadaptive information of xed cardinality; see [22, pp. 57{60].) An optimal error algorithm using nth optimal information is said to be an nth minimal error algorithm. We turn these error-related concepts into complexity-related concepts by introducing a model of computation. Our model is the standard one of [22], see also [28, pg. 27]. For any 2 and any f 2 F , the cost of evaluating (f ) is c, and the cost of basic combinatory operations is 1. Typically, c 1. Then the cost of an algorithm using information N is given as cost( ; N ) = sup cost( ; N; f ); f 2F
where for f 2 F , we let cost( ; N; f ) denote the cost of computing (Nf ) under this model of computation. As an example, suppose that is a linear algorithm using N given P by (2.3), i.e., there exist g1; : : : ; gn 2 G such that (Nf ) = nj=1 j (f )gj for f 2 F ; then cost( ; N ) (c + 2)n ? 1. Finally, we say that for " > 0, the "-complexity in the class is comp("; ) = inf f cost( ; N ) : e( ; N ) " g; the in mum being over all algorithms using information N of nite cardinality from , and an algorithm " using information N" of nite cardinality from for which cost( "; N" ) = comp("; ) is an ("; )-optimal complexity algorithm. Since we cannot often nd optimal complexity algorithms, we will usually be happy to settle for nearly optimal complexity algorithms, which compute "-approximations with cost at most a constant multiple of the "-complexity, said constant being independent of ". One way of doing this involves the ("; )-cardinality number m("; ) = inf f n : r(n; ) " g: Suppose that N 2 n is information of cardinality n = m("; ) for which r(N ) = ", and that is a linear optimal error algorithm using N . Then
c m("; ) comp("; ) cost( ; N ) (c + 2)m("; ) ? 1; and is an ("; )-nearly optimal complexity algorithm for our problem. 6
3. Reduction to the approximation problem
Our main strategy for nding optimal algorithms for solution-restricted operator is to reduce such problems to the approximation problem of approximating the embedding E : W ! G. Let be a class of linear functionals on F , and let N be information from of nite cardinality. Suppose that is an algorithm using N . For any f 2 F , we have kSf ? (Nf )kG = kEw ? (NL w)kG ; (3.1) where Ew = Sf and NL = N LE: (3.2) Note that we are being somewhat cautious in explicitly using the embedding operator E , so we can keep closer track of the spaces in which various elements may be found. Also note that N and NL are information operators on the spaces X and W , respectively. Equation (3.1) tells us that the error of any algorithm for our problem equals the error of the same algorithm (using di erent information) for the approximation problem. In what follows, we will let eapp , rapp , mapp , and compapp denote the error, radius, cardinality number, and complexity for the approximation problem. Using [22, Chapter 3], we then nd the following: Theorem 3.1. Let be a class of linear functionals on F . (1) Let N be information from of nite cardinality. (a) For any algorithm using N , we have e( ; N ) = eapp( ; NL ); where NL is de ned by (3.2). (b) An algorithm is an optimal error algorithm using N for our problem i is an optimal error algorithm using NL for the approximation problem, and r(N ) = rapp (NL ): (2) For any nonnegative integer n, r(n; ) = rapp (n; L ); where L = E L : Hence, m("; ) = mapp("; L ): (3) An ("; L )-(nearly) optimal complexity algorithm for the approximation problem is an ("; )-(nearly) optimal complexity algorithm for our problem, and comp("; ) = compapp ("; L ): In particular, if NL 2 (L )n is information of cardinality n = mapp ("; L ) for which r(NL ) = " and is a linear optimal error algorithm using N , then c m("; ) comp("; ) cost( ; N ) (c + 2)m("; ) ? 1; and so is a nearly optimal complexity algorithm for our problem. 7
The results in Theorem 3.1 essentially hold even in the case where the spaces G and W are not Hilbert spaces. However, we are dealing only with the Hilbert case in this paper. Let us use this fact to derive specifc formulas for a linear optimal error algorithm. Let N be information of cardinality n. Then there exist linearly independent 1; : : : ; n 2 X such that Nf = [1(f ); : : : ; n(f )] 8f 2 F: (3.3) Let wj = E L j (1 j n): (3.4) Since f1 ; : : : ; fn are linearly independent and E L is an injection, we see that w1; : : : ; wn are linearly independent. De ne For f 2 F , let uN 2 SN satisfy
SN = span fw1; : : : ; wng:
i (LuN ) = i (f ) (1 i n): (3.5) Note that this is a generalized Galerkin method, i.e., a Galerkin method with di erent spaces of test and trial functions. Clearly the dependence of uN on f is only through the information Nf , so we may write uN = N (Nf ) 8f 2 F: (3.6) Lemma 3.1. Let N be given by (3.3). (1) The linear algorithm N given by (3.6) is an optimal error algorithm for N . (2) If fw1; : : : ; wng is a W -orthonormal set, then N (Nf ) =
n X j =1
j (f )wj =
n X j =1
hSf; wj iW wj
8f 2 F:
Proof: Using (3.4), we see that
i (LuN ) = huN ; wiiW and i (f ) = hSf; wiiW : Hence uN 2 SN satis es (3.5) i it satis es huN ; wi iW = hSf; wiiW (1 i n): (3.7): Now use [22, Theorem 5.5.3]. F As mentioned in Section 2, we will deal with the classes = and = of permissible information operations. In the remainder of this section, we show that the nth minimal radii are the same for these classes, being given by the (same) Gelfand n-width. We also construct nth optimal (or nearly optimal) information for these classes. First, we prove the following 8
2 Fn . For any > 0, there exists information N 2 n such that r(N ) (1 + ) r(N ): Since N 2 Fn , there exist linearly independent 1 ; : : : ; k 2 F (with k n)
Lemma 3.2. Let N Proof:
such that
N = [1 ; : : : ; k ]: Since each functional Sf 7! i (f ) is a bounded linear functional on W , there exist linearly independent w1; : : : ; wk 2 W such that i (f ) = hSf; wiiW for f 2 W . Without loss of generality, we may assume that w1; : : : ; wk are orthonormal. Now let > 0, and de ne = 1+r(rN(N) ) : Since LE : W ! X is an injection, we see that E L : X ! W is dense, and so there exist 1; ; : : : ; k; 2 X such that kwi ? E L i; kW k : We then de ne information N 2 n as N = [1; ; : : : ; k; ]: Recall that the radius of information for the Hilbert case is given by the simpli ed expression r(N ) = sup kwkG = sup kwkG: (3.8); Lw2ker N kwkW 1
w2ker(N )L kwkW 1
see [22, pg. 80]. Let Lw 2 ker N with kwkW 1. For 1 i k, we have (E L i; )(w) = i; (Lw) = 0 and so Let
jhwi; wiW j kwi ? E L i; kW kwkW k : w0 = w ?
Note that
k X j =1
hwj ; wiW wj ;
i (Lw0) = E Li (w0) = hwi ; w0iW = 0 and so Lw0 2 ker N . Moreover
kw0kW kwkW +
k X j =1
(3.9) (1 i k);
jhwj ; wiW j kwj kW 1 + : 9
It now follows that
kw0kG sup kvkG = (1 + ) r(N ): Lv2ker N kvkW 1+
From (3.9) and the de nition of , it now follows that
kwkG kw0kG +
k X j =1
jhwj ; wiW j kwj kG (1 + ) r(N ) + = (1 + ) r(N ):
Since w is an arbitrary element in the W -unit ball such that Lw 2 ker N , the lemma follows immediately from (3.8). Recall that the Gelfand n-width of the embedding E is de ned as ? dn E (W ); G = ninf n supn kwkG; W 2W w2W kwkW 1
where W n is the class of W -subspaces whose codimension is at most n. From [18, Corollary II.7.3], there always exists an nth optimal Gelfand subspace of E , i.e., a subspace W n 2 W n at which this in mum is attained. Without loss of generality, we may assume that W n = ker[h; w1 iW ; : : : ; h; wniW ]; (3.10) where fw1; : : : ; wng is a W -orthonormal set. Theorem 3.2.
(1) For any n 2 N,
?
r(n; F ) = r(n; ) = dn E (W ); G : (2) Suppose that = F . Let w1; : : : ; wn be as in (3.10). De ne information Nn 2 F by Nnf = [hSf; w1iW ; : : : ; hSf; wniW ] 8f 2 F: Then ? r(Nn ) = r(n; ) = dn E (W ); G ; i.e., Nn 2 Fn is nth optimal information. (3) Suppose that = . For > 0, let ? 2 dn E?(W ); G ; = 1 + 2 dn E (W ); G and choose 1; ; : : : ; n; 2 X such that kwi ? E L i; kW n : De ne information Nn; 2 n by Nn; = [1; ; : : : ; n; ]: Then r(Nn; ) (1 + ) r(n; ); so that Nn; 2 is nth nearly optimal information. 10
Proof: Parts (1) and (2) follows immediately from Lemma 2 and the observation that if = F , then L = . We need only prove part (3). For > 0, we have
n) = 1 + r(N r(Nn ) by part (1). Using the notation of Lemma 3.2, we see that for N = Nn , we have N = Nn; , and so ? r(Nn; ) (1 + ) r(Nn ) = (1 + ) r(n; ) = (1 + ) dn E (W ); G ; as required. ? n Suppose for a moment that E is not a compact embedding. Then limn!1 d E (W ); G is strictly positive (see [18, Proposition II.7.4]). This means that the nth minimal radius is bounded away from zero for the classes F and . In short, if E is not compact, then the problem is not convergent, i.e., we cannot get arbitrarily good approximations at nite cost. Hence, in the remainder of this paper, we shall assume that W is compactly embedded in G. This implies that the space W has an orthonormal basis consisting of eigenvectors of E E . Thus there exist 1 2 > 0 with limn!1 j = 0, and a complete orthonormal basis fzj g1 j =1 for W such that E Ezj = j2zj for j = 1; 2; : : : : We then nd that for any n 2 N, ? dn E (W ); G = n+1; (3.11) see [18, Theorem IV.2.2]. We rst look at optimal information and algorithms for the class F . Let Nn f = [hSf; z1iW ; : : : ; hSf; zniW ] 8f 2 F; (3.12) where z1; : : : ; zn are the eigenvectors of E E corresponding to the n largest eigenvalues of E E . Clearly we have Nn 2 Fn . De ne an algorithm n using Nn as
n(Nn f ) = Theorem 3.3.
n X j =1
hSf; zj iW zj
8f 2 F:
(3.13)
(1) For any n 2 N, we have ? e( n ; Nn) = r(Nn ) = r(n; F ) = dn E (W ); G = n+1: Hence Nn is nth optimal information and n is an nth minimal error algorithm. (2) Let " > 0. Then the "-cardinality number for F is given by m("; F ) = inf f integers n 0 : n+1 " g : Moreover, let Nn 2 Fn be the information given by (3.12) and let n be the algorithm given by (3.13), with n = m("; ). Then c m("; F ) comp("; F ) cost( n; Nn) (c + 2)m("; F ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. 11
Proof: Immediate from (3.11), along with Theorems 3.1 and 3.2, as well as Lemma 3.1. We now consider optimal information and algorithms for the class . Let > 0, and choose 1; ; : : : ; n; 2 X such that
kzj ? L j; kG nn+1 ;
(3.14)
where we recall that z1; : : : ; zn are the eigenvectors of E E corresponding to the n largest eigenvalues of E E . Our information Nn; 2 n is de ned as
Nn; f = [1; (f ); : : : ; n; (f )]
8f 2 F:
(3.15)
De ne an algorithm n; using Nn; as
n; (Nn; f ) =
n X j =1
j; (f )zj
8f 2 F:
(3.16)
We then have Theorem 3.4.
(1) For any n 2 N and any > 0, we have
e( n; ; Nn; ) (1 + ) n+1: Hence Nn; is nth nearly optimal information and n; is an nth nearly minimal error algorithm. (2) Let " > 0. Then the "-cardinality number for is given by
m("; ) = inf f integers n 0 : n+1 " g : Suppose that n is strictly monotonically decreasing with n. Let Nn; 2 Fn be the information given by (3.15) and let n; be the algorithm given by (3.16), with n = m("; ) + 1 and satisfying 0 < <
n ? 1: n+1 Then ?
c m("; F ) comp("; ) cost( n; ; Nn; ) (c + 2) m("; ) + 1 ? 1: Hence for c 1, the algorithm n; using information Nn; is a nearly optimal complexity algorithm. 12
Proof: Note that part (2) follows from Theorem 3.1 and part (1). So we need only prove
part (1). To do this, let n 2 N, and let > 0. Choose f 2 F . For any j , we have
jhSf; zj iW ? j; (f )j = jhSf; zj ? L j; iG j kSf kGkzj ? L j; kG nn+1 : Recalling the de nition of Nn and n from Theorem 3.3, we then have
k n(Nnf ) ? n; (Nn; f )kG
n X j =1
jhSf; zj iW ? j; (f )jkzj kG n+1:
So
kSf ? n; (Nn; f )kG kSf ? n(Nnf )kG + k n(Nn f ) ? n; (Nn; f )kG (1 + ) n+1: Taking the supremum over all such f , we nd the desired bound on e( n; ; Nn; ). Now that we have proved this bound, the rest of the result follows from (3.11) and Theorem 3.2, as well as Lemma 3.1. Remark : This result tells us that n; is an nth nearly minimal error algorithm in . Of course, this implies that n; is an nearly optimal error algorithm using the nth nearly optimal information Nn; . Note that we decided to use the algorithm n; , instead of the algorithm Nn; , i.e., the optimal error algorithm using Nn; given by Lemma 3.1. We did this because the algorithm n; is simpler than the optimal error algorithm Nn; , since the latter would have required either an orthogonalization of the vectors E L 1; ; : : : ; E L n; or the solution of an n n linear system. Note that the increased error when using n; instead of Nn; is small compared to the error of n; and to the nth minimal radius. All things considered, it appears better to use a simpler nearly optimal error algorithm than a more complicated optimal error algorithm in this situation. Remark : The strict monotonicity assumption in Theorem 3.4 is not necessary, but only used to simplify the statement of the theorem. A more general (and more complicated) statement is possible for the case where E E has multiple eigenvalues, provided that the multiplicity of the eigenvalues does not increase super-exponentially. Since we will not need such a result, we will not pursue this further. 4. Applications
In this section, we apply the previous results to several problems. We rst look at a common situation, namely, a problem in which the spaces G and W are the Sobolev spaces q ( ) and H r ( ), respectively. Here, q < r, and the \bd" indicates that the spaces Hbd bd may satisfy certain homogeneous boundary conditions. Any such problem will be called a \standard problem." We develop detailed results for standard problems. Once we have these results, we can use them to study speci c instances of the standard problem. In particular, we will consider elliptic boundary-value problems, Fredholm integral equations of the second kind, mixed elliptic-hyperbolic problems, and the inverse nite Laplace transform. For all these problems, we nd the problem complexity and derive nearly optimal 13
algorithms. In particular, we discuss the optimality of Galerkin algorithms using nite element information. We then conclude by looking at the heat equation running backwards in time, an application that is not an instance of the standard problem. Suppose we know an a priori L2-bound on the solution of the backwards heat equation at time t = ?t1 and that we want to solve the equation at time t = ?t0, where 0 < t0 < t1. The problem element is \ nal data", i.e., the solution at time t = 0. We then nd that by truncating the standard series representation of the solution, we get an optimal algorithm. Note that two of our applications (the inverse nite Laplace transform and the backwards heat equation) are ill-posed problems. Hence we see that the techniques of this paper are powerful enough for us to determine that the "-complexity of ill-posed solution-restricted problems is nite. In what follows we use the standard terminology and notation for multi-indices, as well as Sobolev spaces, norms, and inner products. For details, consult any standard reference on elliptic boundary-value problems and nite element methods, such as [1], [3], [7, Chapter IV], [9], or [16]. The letter C will denote a generic constant whose value may change from one place to the next. All O-, -, -, and -estimates will be independent of n or ", depending on the context. Finally, we note that in the speci c applications we consider here, the space X is also a Sobolev Hilbert space, along with G and W . Hence, an information operation from can be represented as an inner product over X . We shall do this consistently, without further comment.
4.1. The standard problem.
Our rst problem is really a meta-problem, since it includes many important practical problems as particular instances. Let Rd be a suciently smooth simply-connected q ( ) and H r ( ) be region, which is bounded. Given q and r with q < r, we let Hbd r bd( ) and closed subspaces of H q ( ) and H r ( ), respectively, for which C01( ) Hbd r ( ) ,! H q ( ). That is, functions in these spaces may satisfy homogeneous boundary Hbd bd q ( )-functions are also satis ed by functions conditions, and the conditions satis ed by Hbd r ( ). in Hbd q ( ) and W = H r ( ), so that the We now consider any problem for which G = Hbd q r ( ) ! H ( ) is the usual inclusion embedding and bd F is the unit ball mapping E : Hbd r ( ). Since this kindbdof problem will be useful in later applications, we will call it of Hbd the standard problem. Although we have speci ed neither the space X nor the mapping L, we can still discover much about such a problem. In particular, we show that optimal information in F is given by the solution of a generalized elliptic eigenproblem whenever q; r 2 N. We then show that for any r and q, the nth minimal radius is proportional to n?(r?q)=d , so that the "-complexity is proportional to (1=")d=(r?q) . Since it is not generally possible to nd a closed-form solution of this eigenproblem, we need to consider nearly optimal that is easier to obtain. We show that nite element information of degree k is nearly optimal if k r ? 1. Having dealt with the case = F , we then prove analogous results for the case = . We rst show that the eigenvectors and eigenvectors of E E are solutions of a generalized elliptic eigenproblem when q; r 2 N. For any s 2 N, we de ne the partial di erential 14
operator Ps as
Ps v =
X
j js
(?1)j jD2 v
From [1, Theorem 10.2], we see that for s j 2s ? 1, there exist partial di erential operators s;j of exact order j , such that the Green's formula
hz; wiH s ( ) = hPsz; wiL2 ( ) +
s X j =0
h s;2s?1?j (z); @j wiL2 (@ )
8 z; w 2 C 1( ) (4.1.1)
holds, with @ denoting the outward-pointing normal derivative on @ . We then have r ( ) into H q ( ). For 0 j r ? 1, let Lemma 4.1.1. Let E be the embedding of Hbd bd ? ;j =
r;2r?1?j ? ?2 q;2q?1?j for 0 j q ? 1; r;2r?1?j for q j r ? 1:
Then E Ez = 2z i z and are a solution of the generalized elliptic eigenproblem
Pr z = ?2Pq z
in ;
(4.1.2)
(0 j r ? 1):
(4.1.3)
subject to the homogeneous boundary conditions
? ;j (z) @ = 0 Proof: : We have E Ez = 2 z i
2hz; wiHbdr ( ) = hz; wiHbdq ( ) ;
8 w 2 C 1( ):
(4.1.4)
Using (4.1.1) and (4.1.4), we see that E Ez = 2z i r?1 X ? 2 h[Pr ? Pq ]z; wiL2 ( ) = h? ;j (z); @j wiL2 (@ ) j =0
8w 2 C 1( ):
Choosing w with compact support, we nd that
h[Pr ? ?2Pq ]z; wiL2 ( ) = 0
8w 2 C01( ):
Hence z and are weak solutions of the eigenproblem (4.1.2). Since is smooth, we see that weak and smooth solutions of this eigenproblem coincide. Moreover, by appropriate choices of w, it follows that the boundary conditions (4.1.3) are satis ed. Note that (4.1.2){(4.1.3) is a generalized eigenproblem, akin to the generalized eigenproblems Az = Bz studied in computational linear algebra. It reduces to a standard elliptic eigenproblem whenever q = 0, i.e., whenever G = L2( ). We illustrate this construction by two one-dimensional cases. 15
Example (q = 0): Suppose that d = 1 and = I = [0; 1], with q = 0. Then the eigenfunctions zj and eigenvalues j of E E are solutions of the eigenproblem q X i=0
(?1)i z(2i)(s) = ?2y(s)
8 s 2 [0; 1];
j X i ( q ? j +2 i ) (?1) z (0) = (?1)i z(q?j+2i) (1) = 0 i=0 i=0
j X
(0 j q ? 1):
When q = 0 and r = 1, we easily nd that
1 if j = 1 zj (s) = p ? 2 cos (j ? 1)s if j 2 and that
1
j = p : 2 1 + (j ? 1)2 Unfortunately, the exact solution of the eigenproblem is unknown for arbitrary r. Example (q = 1, r = 2): Again suppose that d = 1 and = I = [0; 1], but now with q = 1 and r = 2. Then the eigenfunctions zj and eigenvalues j of E E are solutions of the generalized eigenproblem
z ? z00 + z0000 = 2(z ? z00 );
(4.1.5)
subject to the boundary conditions
z00 (0) = z00 (1) = 0; z000 (0) + ( ?2 ? 1)z0 (0) = z000 (1) + ( ?2 ? 1)z0 (1) = 0:
(4.1.6)
Using Mathematica,pwe quickly found that apbasis for the solution spacepof (4.1.5) is given by f ; = exp(x i )g, where = 21 2 4 + 4 2 ? 6 and = 21 2(1 ? 2), and so
zj = C1
+;+ + C2 ?;+ + C3 +;? + C4 ?;? :
However, we were unsuccessful in nding values for the weights C1; : : : ; C4 and for such that the boundary conditions (4.1.6) hold. 4.1.1. Results for the case = F . We now suppose that = F . Using the solution to the eigenproblem in Lemma 4.1.1, we can now nd nth optimal information in F , along with an nth minimal error algorithm. Moreover, we can get a tight bound on the nth minimal radius of information in F . 16
Theorem 4.1.1.1. Let zj and j be the j th eigenfunction and eigenvalue satisfying (4.1.2), subject to the boundary conditions (4.1.3). Let
Nn f = [hSf; z1iHbdr ( ); : : : ; hSf; zniHbdr ( )] and
n(Nn f ) =
n X j =1
hSf; zj iHbdr ( )zj
8f 2 F
8f 2 F:
Then Nn is nth optimal information in F and n is an nth minimal error algorithm, with
e( n; Nn) = r(n; F ) = n+1 = (n?(r?q)=d ): Proof: Using Lemma 4.1.1, along with Theorem 3.3, we immediately have ? ?
r ( ) ; H q ( ) : e( n; Nn) = r(n; F ) = n+1 = dn E Hbd bd
Following the proof of [28, Theorem 5.4.1], we see that ? ?
r ( ) ; H q ( ) = (n?(r?q)=d ): dn E Hbd bd
Thus the nth minimal radius is proportional to n?(r?q)=d . We now consider the "complexity of our standard problem in the class F . Using Theorems 3.3 and 4.1.1.1, we have Theorem 4.1.1.2. Let " > 0. Then ?
comp("; F ) = c (1=")d=(r?q) : Moreover, let Nn and n be the information and algorithm given by Theorem 4.1.1.1. Then c m("; F ) comp("; F ) cost( n; Nn) (c + 2)m("; F ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. Since we cannot usually nd a closed form solution to the eigenproblem (4.1.2){(4.1.3), we need to nd other kinds of information that will be nearly as good. To do this, we will look at nite element information. q ( ), whose degree is k . Let Sn;k be an n-dimensional nite element subspace of Hbd That is, there is a triangulation Tn of such that
Sn;k = s 2 Hbdq ( ) : s K 2 Pk (K ) 8 K 2 Tn ; dim Sn;k = n: 17
(Here, Pk (K ) is the space of polynomials of degree at most k, considered as functions de ned over the region K .) We will further require that fTng1 n=1 be a quasi-uniform family of triangulations of . This means that lim sup sup hK < 1; n!1 K 2Tn K where
hK = diam K
and
K = supf diam B : spheres B containing K g for any n 2 N and any K 2 Tn. (See [3], [7, Chapter XII], [16], or [28, Chapter 5 and Appendix] for further properties of nite element spaces.) We can now de ne our information. For any n 2 N, let Nn;k f = [hSf; g1iHbdq ( ); : : : ; hSf; gniHbdq ( )]
8f 2 F;
(4.1.1.1)
where fg1; : : : ; gng is a basis for Sn;k . Of course, Nn;k 2 Fn . We say that Nn;k is nite element information (FEI). Now consider the following algorithm: For f 2 F , let un;k 2 Sn;k satisfy (1 i n)
hun;k ; giiHbdq ( ) = hSf; giiHbdq ( ):
(4.1.1.2)
Since un;k depends on f only through the information Nn;k f , we may write
un;k = n;k (Nn;k f ): Clearly, n;k is a Galerkin method using test and trial space Sn;k . Note that we can determine the coecients a = [ 1; : : : ; n] of un;k (with respect to the basis g1; : : : ; gn) by q ( ) and i = hSf; gj iH q ( ) . solving a linear system Ka = b, where j;j = hgj ; giiHbd bd Remark : Note that we refer to n;k as a Galerkin method using the FEI Nn;k . However, n;k is not a \ nite element method" in the usual sense of the term, since the information q ( ) instead of the form i (f ) used by the nite element used is of the form hSf; giiHbd method. Our main error estimate for the Galerkin method using FEI is Theorem 4.1.1.3. Let k r ? 1. Then for any n, we have ?
e( n;k ; Nn;k ) = (n?(r?q)=d ) = r(n; F ) : Thus if k r ? 1, then the information Nn;k is nth nearly optimal information in F , and the algorithm n;k is an nth nearly minimal error algorithm. 18
Proof: Let f
q ( )-projection of Sf onto S . By [28, 2 F . Then n;k (Nn;k f ) is the Hbd n;k
Lemma 5.4.3], there is a positive constant C , independent of n and f , such that
kSf ? n;k (Nn;k f )kHbdq ( ) = inf kSf ? skHbdq ( ) Cn?(minfk+1;rg?q)=d kSf kHbdr ( ): s2Sn;k
Since k + 1 r, we thus nd that
e( n;k ; Nn;k ) = O(n?(r?q)=d ): Using Theorem 4.1.1.1, we have
e( n;k ; Nn;k ) r(n; F ) = (n?(r?q)=d ); completing the proof of the theorem. Remark : Note that we do not claim that n;k is an optimal error algorithm using Nn;k . If we use the prescription of Lemma 3.1, we would construct an optimal error algorithm Nn;k using Nn;k as follows. For f 2 F , let uNn;k 2 E Sn;k satisfy
huNn;k ; E giiHbdr ( ) = hSf; E giiHbdr ( )
(1 i n):
Since uNn;k depends on f only through the information Nn;k f , we may write uNn;k = Nn;k (Nn;k f ). Then Nn;k is an optimal error algorithm using Nn;k . Remark : It is possible to describe somewhat di erent nite element information such that the resulting Galerkin method (using the same space of test and trial functions) is an optimal error algorithm using this new FEI. To do this, let S^n;k be an n-dimensional r ( ), whose degree is k , with the spaces S^n;k being based on a nite element subspace Hbd quasi-uniform family of triangulations of . Then our FEI has the form
N^n;k f = [hSf; w1iHbdr ( ); : : : ; hSf; wniHbdr ( )]
8f 2 F;
where fw1; : : : ; wng is a basis for S^n;k . Now consider the following algorithm: For f 2 F , let u^n;k 2 S^n;k satisfy
hu^n;k; wi iHbdr ( ) = hSf; wiiHbdr ( ): Since u^n;k depends on f only through the information Nn;k f , we may write
u^n;k = ^n;k (N^n;k f ): Clearly, ^n;k is a Galerkin method using test and trial space S^n;k . Moreover, it is possible to show that e( ^n;k ; N^n;k ) = r(N^n;k ) = (n?(r?q)=d ): 19
(The proof is slightly more involved than that of Theorem 4.1.1.3, requiring the use of the r ( ), we duality estimate in [3, Theorem 2.3.1].) Note that since S^n;k is a subspace of Hbd automatically have k r ? 1 (see [28, Lemma 5.4.2]). It might seem that since ^n;k is an optimal error algorithm, one would prefer using the algorithm ^n;k instead of the algorithm n;k. However, note that if we use ^n;k, r ( ). In then we need to construct a nite element space S^n;k that is a subspace of Hbd practical situations, this would require more initial precomputation than constructing a q ( ). Moreover, the errors of and nite element space Sn;k that is a subspace of Hbd n;k ^n;k are roughly the same. Hence any gain that might be realized in using ^n;k will be o set by the loss involved in the additional precomputation. For this reason, we prefer to use n;k . We now determine the cost of using our Galerkin method to compute "-approximations for our standard problem. Let us denote this cost by costGal ("; F ) = minf cost( n;k ; Nn;k ) : e( n;k ; Nn;k ) " g: We then have
Theorem 4.1.1.4. Let " > 0. Then
?
costGal ("; F ) = c (1=")d=(r?q) : Hence the Galerkin method using FEI is a nearly optimal complexity algorithm when = F . Proof: 4.1.1.3, we see that e( n;k ; Nn;k ) " holds if and only if we choose ? From Theorem d= ( r ? q ) n = (1=") . Now use Theorem 4.1.1.2. 4.1.2. Results for the case = . Now, we suppose that = , i.e., continuous linear functionals are permissible information. We can nd optimal information, minimal error algorithms, and optimal complexity algorithms for the case = by using the results for the case = F , along with Theorem 3.4. We rst look at information based on the eigenvectors of E E . Theorem 4.1.2.1. Let zj and j be the j th eigenfunction and eigenvalue satisfying (4.1.2), subject to the boundary conditions (4.1.3). For any n 2 N and any > 0, choose 1; ; : : : ; n; 2 X such that kzj ? L j; kHbdq ( ) nn+1 : Let Nn; f = [1; (f ); : : : ; n; (f )] 8f 2 F; and n X n; (Nn; f ) = j; (f )zj 8f 2 F: j =1
20
Then
e( n; ; Nn; ) (1 + ) n+1 = (n?(r?q)=d )
and
r(n; ) = r(n; F ) = n+1 = (n?(r?q)=d ): Hence Nn; is nth nearly optimal information in and n; is an nth nearly minimal error algorithm. Proof: Immediate from Theorems 3.4 and 4.1.1.1. Having determined nearly optimal information and nearly minimal algorithms for the case = , we can now determine the problem complexity and nd nearly optimal complexity algorithms. Theorem 4.1.2.2.
(1) For any " > 0, we have ?
comp("; ) = c (1=")d=(r?q) : (2) Suppose that the eigenvalues n are strictly monotonically decreasing in n. Then for any " > 0, we have ?
comp("; ) = c (1=")d=(r?q) : Let Nn; 2 n be the information given by (3.15) and let n; be the algorithm given by (3.16), with n = m("; ) + 1 and satisfying
1 0 < <
n ? 1 n + n n+1
(r?q)=d
? 1:
Then ?
c m("; ) comp("; ) cost( n; ; Nn; ) (c + 2) m("; ) + 1 ? 1: Hence for c 1, the algorithm n; using information Nn; is a nearly optimal complexity algorithm. Proof: This follows immediately from Theorems 3.4 and 4.1.2.1. Remark : Note that the assumption that the n are strictly monotonically decreasing in n
generally will not hold unless d = 1. For example, suppose that q = 0 and r = 0; then the n are eigenvalues of the operator (1 ? )?1 , where is the Laplacian. This operator has multiple eigenvalues, unless d = 1, and so Theorem 4.1.2.2 cannot be immediately applied. However, it is possible to modify this theorem to account for the case of multiple eigenvalues. Since we will not need this modi ed version of this theorem in what follows, we do not feel that it is important to state the modi ed theorem. 21
As in the previous section, the optimal information and algorithm of Theorem 4.1.2.2 are usually not available in closed form. Hence, we need to look at alternative nearly optimal information and algorithms. In particular, we now consider modi ed Galerkin methods for computing "-approximations to our problem. As in the previous section, we let Sn;k q ( ), whose degree is k , with the denote an n-dimensional nite element subspace of Hbd corresponding family fTng1 n=1 of triangulations being quasi-uniform. We let fg1 ; : : : ; gn g again be a basis for Sn;k . However, we require that this basis satisfy the condition that there exist a positive constant C , independent of n, such that X n
j =1
2 j
1=2
n
X Cn1=2
j gj
L2 ( ) j =1
8 1; : : : ; n 2 R:
(4.1.2.1)
This condition is satis ed by the usual nite element basis functions having small support; see [5] and [28, pp. 216 .] for details. Since the range of L is dense in X , there exist 1; : : : ; n 2 X such that (4.1.2.2) kgi ? L i kHbdq ( ) Cn?((r?q)=d+1):
We can now de ne our information and algorithm. For any n 2 N, let N~n;k = [1(f ); : : : ; n(f )] 8f 2 F: Note that N~n;k 2 n is an approximation of the nite element information Nn;k 2 Fn de ned in the previous section, and so we refer to Nn;k as modi ed nite element information, or modi ed FEI, for short. Then for f 2 F , we seek u~n;k 2 Sn;k satisfying hu~n;k ; giiHbdq ( ) = i(f ): (1 i n) (4.1.2.3) Since u~n;k depends on f only through the information N~n;k f , we may write
u~n;k = ~n;k (N~n;k f ): We call ~n;k an modi ed Galerkin method using modi ed FEI. Remark : It is easy to see that the Galerkin and modi ed Galerkin methods may be reduced to the solution of n n linear systems whose solution gives the (respective) coecients of un;k or u~n;k with respect to the basis fg1; : : : ; gng. The only di erence between the Galerkin and modi ed Galerkin methods is that the latter method uses i(f ) to approxq ( ) appearing in the de nition of the former method. However, the imate the hSf; giiHbd same coecient matrix is used for both algorithms. Our main error estimate is Theorem 4.1.2.3. Let k r ? 1. Then for any n 2 N, we have ? e( ~n;k ; N~n;k ) = (n?(r?q)=d ) = r(n; ) : Thus if k r ? 1, then the information N~n;k is nth nearly optimal information in , and the algorithm ~n;k is an nth nearly minimal error algorithm. 22
2 N. For fP2n F , let un;k = n;k (Nn;k f ) and let u~n;k = ~n;k (N~n;k f ). Writing en;k = un;k ?u~n;k = j=1 "j gj , we may use the discrete Cauchy-Schwarz inequality Proof: Let n
to see that
ken;k
k2 q
Hbd ( ) =
n X j =1
"j hen;k ; gj iHbdq ( )
X n
j =1
"2 j
1=2 X n
j =1
hen;k ; gj
i2 q
Hbd ( )
1=2
: (4.1.2.4)
By (4.1.2.1), there is a positive constant C such that
n X j =1
"2 j
1=2
Cn1=2ken;k kL2( ) Cn1=2ken;k kHbdq ( ):
(4.1.2.5)
Moreover, we may use (4.1.1.2), (4.1.2.3), and (4.1.2.2) to see that
jhen;k ; gj iHbdq ( )j = jhSf; gj iHbdq ( ) ? j (f )j = jhSf; gj ? L j iHbdq ( )j kSf kHbdq ( )kgj ? L j kHbdq ( ) Cn?((r?q)=d+1):
(4.1.2.6)
Using (4.1.2.5) and (4.1.2.6) in (4.1.2.4), we nd that
ken;k kHbdq ( ) Cn?(r?q)=d for some positive constant C . Hence
kSf ? ~n;k(N~n;k f )kH q ( ) kSf ? n;k(Nn;k f )kHbdq ( ) + ken;k kHbdq ( ) Cn?(r?q)=d for a positive constant C . Since f 2 F is arbitrary, we nd that
e( ~n;k ; N~n;k ) = O(n?(r?q)=d ): Using Theorem 4.1.2.1, we have
e( n;k ; Nn;k ) r(n; ) = (n?(r?q)=d ); completing the proof of the theorem. We now determine the "-complexity of the standard problem in the class , as well as the cost of using our modi ed Galerkin method to compute "-approximations. Let us denote this cost by costmod-Gal("; F ) = minf cost( ~n;k ; N~n;k ) : e( ~n;k ; N~n;k ) " g: We then have 23
Theorem 4.1.2.4. Let " > 0. Then ?
comp("; ) = c (1=")d=(r?q) and
?
costmod-Gal ("; ) = c (1=")d=(r?q) : Hence the modi ed Galerkin method using modi ed FEI is a nearly optimal complexity algorithm when = . ? Proof: From Theorem 4.1.2.3, we see that e( ~n;k ; N~n;k ) " holds i n = (1=")d=(r?q) . Thus ? comp("; ) = c O (1=")d=(r?q) Now use Theorem 4.1.2.2.
4.2. Elliptic boundary-value problems.
We now consider the complexity of solution-restricted elliptic boundary-value problems. For the sake of brevity, we provide neither de nitions of standard vocabulary nor lists of standard results in the study of elliptic problems. The interested reader should consult any standard reference (such as [1], [3], [7, Chapter V], [9], or [16]) for more details. Previous results on the complexity of elliptic problems may be be found in [28, Chapter 5], and the references found therein. To simplify the exposition, we will follow the approach taken in [25]. However, more complicated problems could have been handled as well. Let Rd be a smooth, bounded, simply-connected region. Recall that H0m ( ) is the space of all H m ( )-functions whose normal derivatives of order less than m vanish at the boundary of . Let X Z B(v; w) = a ; D vD w j j;j jm
be a symmetric H0m ( )-coercive bilinear form. Then B is an inner product on H0m ( ), yielding an energy norm k kB de ned by p
kvkB = B(v; v)
8 v 2 H0m ( )
that is equivalent to the usual norm k k on H0m ( ). Note that if we let
Lv =
X
(?1)j jD (a ; D v);
j j;j jm
then L is a uniformly strongly elliptic operator of order 2m. Moreover, we have
B(v; w) = hLv; wiL2 ( )
8 v; w 2 H0m ( );
where (as usual) the L2( )-inner product may be interpreted as the duality pairing of H0m ( ) and its dual space H ?m ( ). 24
We now show how our problem may be expressed in terms of the general framework that r ( ) = f v 2 H r ( ) : v 2 H m ( ) g. We we have developed. Let r m, and let W = Hbd 0 then take G = H0m ( ) and X = H ?m ( ). The class F of problem elements is now the unit ball of H r ( ). Note that the Lax-Milgram lemma implies that the range of L is H ?m ( ). So L : H0m ( ) ! H ?m ( ) is a bijection. Hence S = L?1 : H ?m ( ) ! H0m ( ). Hence our solution operator S : H ?m ( ) ! H0m ( ) is given as B(Sf; v) = hf; viL2 ( ) 8f 2 H ?m ( ); v 2 H0m ( ): (4.2.1) Again, the Lax-Milgram lemma tells us that S is well-de ned. For any f 2 H ?m ( ), we have that u = Sf is the variational solution of
Lu = f in
@j u = 0 on @ (0 j m ? 1); a 2mth-order elliptic boundary problem satisfying homogeneous Dirichlet boundary conditions. Note that our problem (4.2.1) is a standard problem (with q = m), and so we can apply the results in Section 4.1. We rst look at the case = F . For any n 2 N, let Sn;k be an n-dimensional nite element subspace of H0m ( ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by Nn;k f = [hf; g1 iL2 ( ); : : : ; hf; gn iL2 ( )]: For f 2 F , let un;k = n;k (Nn;k f ) be the Galerkin method given by B(un;k ; gi ) = hf; gi iL2 ( ) (1 i n): (4.2.2) Note that the Galerkin algorithm n;k is the standard nite element method (FEM). Theorem 4.2.1. Let = F . (1) For any n 2 N, we have r(n; F ) = (n?(r?m)=d ): (2) Let k r ? 1. For any n 2 N, we have e( n;k ; Nn;k ) = (n?(r?m)=d ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ? comp("; F ) = c (1=")d=(r?m) : ? (4) Let k r ? 1. For any " > 0, let n = (1=")d=(r?m) . Then ? costGal ("; F ) = c (1=")d=(r?m) ; and so the FEM n;k using FEI Nn;k is a nearly optimal complexity algorithm. 25
Proof: Recall that B (; ) is the inner product on H0m ( ). But B (Sf; gi ) = hf; gi iL2 ( )
by (4.2.1). Hence Nn;k and n;k as de ned in the statement of the theorem are the same as in (4.1.1.1) and (4.1.1.2). The result now follows from Theorems 4.1.1.3 and 4.1.1.4. So we see that the standard nite element method of degree k r ? 1 is nearly optimal for a 2mth-order elliptic problem, if the solution elements are constrained to lie in the unit ball of H r ( ). This result should not be too surprising, given the known results about optimality of FEMs for elliptic problems, see [28, Chapter 5] and the results cited therein. The novelty in this result lies in the fact that we did not need to use a shift theorem (i.e., a result saying that if f has r ? 2m derivatives, then Sf has r derivatives) to prove the optimality of the FEM. We now look at the case = , mainly for the sake of completeness. It will turn out that FEI for this problem (which is a priori only F -information) is really continuous linear information. Rather than use the standard H ?m ( ) inner product on H ?m ( ), it will be more convenient to consider H ?m ( ) as the dual of H0m ( ) under the energy norm, i.e., we use the norm (4.2.3) kvkB0 = supm hv;kwwikL2 ( ) 8 v 2 H ?m ( ): B w2H0 ( ) We let h; iB0 denote the corresponding inner product. Since the energy norm k kB is equivalent to the usual H0m ( )-norm, it follows that this norm k kB0 is equivalent to the usual H ?m ( )-norm. Lemma 4.2.1.
(1) For any v; w 2 H ?m ( ), we have
hv; wiB0 = B(Sv; Sw): (2) L = S . Proof: From (4.2.1) and (4.2.3), we nd
hv; wiL2 ( ) = sup B(Sv; w) = kSvk ; kvkB0 = sup B v2H0m( ) kwkB v2H0m( ) kwkB the last holding because B(; ) is the inner product corresponding to the norm k kB . Part (1) now follows immediately. To prove part (2), note that we must have
hLv; wiB0 = B(v; L w) But
8 v 2 H0m ( ); w 2 H ?m ( ):
hLv; wiB0 = B(SLv; Sw) = B(v; Sw)
Comparing these last two, we see that L = S . 26
8 v 2 H0m ( ); w 2 H ?m ( ):
Now suppose that Nn;k is FEI as in the statement of Theorem 4.2.1. If for 1 i n, we let vi = Lgi , then we have L vi = Svi = gi by Lemma 4.2.1. Thus we nd that
hf; giiL2 ( ) = hf; vi iB0
(1 i n):
We now see that
Nn;k f = [hf; v1 iB0 ; : : : ; hf; vn iB0 ]: For f 2 F , it now follows that un;k = n;k (Nn;k f ) satis es B(un;k ; gi) = hf; vi iB0
(1 i n):
(4.2.4)
So the information Nn;k de ned in Theorem 4.2.1 is continuous linear information. Theorem 4.2.1 now tells us that the FEM is a nearly optimal error algorithm and a nearly optimal complexity algorithm using continuous linear information. Of course, Theorems 4.1.1.3 and tell us that r(n; ) = r(n; F ) = (n?(r?m)=d ) and comp("; ) = c ? 4.1.1.4 (1=")d=(r?m) . Hence (4.2.2) and (4.2.4) are two formulations of the FEM algorithm using FEI. The rst formulation shows that this FEI is information from F , whereas the second shows that it actually is information from . However, the B0 -inner product is more complicated than the L2( )-inner product. So in practice, we would probably rather use (4.2.2) than (4.2.4).
4.3. Fredholm problems of the second kind.
We now look at the complexity of solution-restricted Fredholm integral equations of the second kind. For previous work on the complexity of Fredholm problems of the second kind, see [24] and [28, Section 6.3]. Let Rd be a smooth, bounded, simply-connected region. Suppose that k : ! R satis es Z Z jk(x; y)j2 dx dy < 1:
Then the operator K : L2( ) ! L2( ), de ned as (Kv)(x) =
Z
k(x; y)v(y) dy
8 v 2 L2( );
is an integral operator with a Hilbert-Schmidt kernel, and is thus compact. Assume that 1 is not an eigenvalue of (K K )1=2 . For f : ! R, we wish to nd (approximations to) u : ! R satisfying
u(x) ?
Z
k(x; y)u(y) dy = f (x)
8 x 2 ;
(4.3.1)
a Fredholm integral equation of the second kind. We formally describe our problem by taking X = G = L2( ) and W = H r ( ) for some r 0. Hence the class F of problem elements is the unit ball of H r ( ). Let L = I ? K . Then L is a bounded bijection of G onto X , and so S = L?1 is a bounded bijection of X 27
onto G. Note that our problem (4.3.1) is a standard problem (with q = 0), and so we can apply the results in Section 4.1. We rst look at the case = F . For any n 2 N, let Sn;k be an n-dimensional nite element subspace of L2( ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by
Nn;k f = [hSf; g1iL2 ( ); : : : ; hSf; gniL2 ( )]: For f 2 F , let un;k = n;k (Nn;k f ) be the Galerkin method given by
hun;k ; gi iL2 ( ) = hSf; giiL2 ( )
(1 i n):
(4.3.2)
We then have
Theorem 4.3.1. Let = F .
(1) For any n 2 N, we have
r(n; F ) = (n?r=d ): (2) Let k r ? 1. For any n 2 N, we have
e( n;k ; Nn;k ) = (n?r=d ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; F ) = c (1=")d=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")d=r . Then ?
costGal ("; F ) = c (1=")d=r ; and so the Galerkin method n;k using FEI Nn;k is a nearly optimal complexity algorithm for the Fredholm problem of the second kind. Proof: Immediate from Theorems 4.1.1.3 and 4.1.1.4. Remark : The algorithm n;k de ned by (4.3.2) is not a nite element method, a marked
contrast with what happened when we were looking at elliptic boundary-value problems. The only possibility we would have for using the approach in the previous section, in which the Galerkin method n;k turned out to be an FEM, would be to assume that K is symmetric and kK k < 1. If this were the case, then B(v; w) = hLv; wiL2 ( ) would be p an inner product on L2( ), with the norm k kB = B(; ) being equivalent to the usual L2( )-norm. Under these hypotheses, it then turns out that the Galerkin method (4.1.1.2) is the standard FEM with test and trial spaces Sn;k . However, we often need to solve our 28
problem (I ? K )u = f for non-symmetric K or kK k 1, so this approach is not generally applicable. Note that linear functionals hSf; giiL2 ( ) are hard to directly evaluate, because of their dependence on Sf . However, since S : X ! G is a bounded bijection, we can write
hSf; giiL2 ( ) = hf; S gi iL2 ( ); where S = (L )?1 = (I ? K )?1 . In principle, we can then consider the computation of S g1; : : : ; S gn as precomputation. We now see that
Nn;k f = [hf; S g1iL2 ( ); : : : ; hf; S gniL2 ( ) ]: For f 2 F , it now follows that un;k = n;k (Nn;k f ) satis es
hun;k ; giiL2 ( ) = hf; S giiL2 ( )
(1 i n):
(4.3.3)
So the information Nn;k de ned in Theorem 4.3.1 is continuous linear information. Moreover, the Galerkin algorithm (4.3.3) is a nearly optimal error algorithm and a nearly optimal complexity algorithm using continuous linear information. Of course, Theorems 4.1.1.3 ? F ? r=d and 4.1.1.4 tell us that r(n; ) = r(n; ) = (n ) and comp("; ) = c (1=")d=r . Hence (4.3.2) and (4.3.3) are two formulations of the Galerkin algorithm using FEI. The rst formulation shows that this FEI is information from F , whereas the second shows that it actually is information from . Moreover, the rst formulation uses functionals hSf; giiL2 ( ), whereas the second uses hf; S giiL2 ( ). Since the formulation (4.3.2) is simpler than the formulation (4.3.3), as well as making it clear that we are using continuous linear information. So in practice, we would probably rather use (4.3.3) than (4.3.2). Unfortunately, there is still one diculty with the Galerkin method, even if we decide to use the simpler formulation (4.3.2). Even though S g1; : : : ; S gn are well-de ned, they may not be easy to calculate. For this reason, we will look at modi ed Galerkin methods for the solution-restricted Fredholm problem of the second kind. Once again, we let Sn;k be a nite element subspace of L2( ), of dimension n and degree k. Let fg1; : : : ; gng be a basis of Sn;k satisfying (4.1.2.1). Next, we let r~ 2 f1; : : : ; rg, and let Sn~ ;k~ be a nite element subspace of H r~( ), of dimension n~ and degree k~. Of course, we must have k~ r~, since Sn~ ;k~ H r~( ). For any n 2 N, let n~ = (n(r?r~)=r~+d=(2~r)): (4.3.4) For 1 i n, let vi 2 Sn~ ;k~ satisfy
hL vi ; siL2 ( ) = hgi ; siL2 ( ): We then have 29
(4.3.5)
Lemma 4.3.1. For 1 i n, we have
kgi ? L vikL2 ( ) Cn?(r=d+1): Proof: Let 1 i n. We rst note that vi is an Sn~ ;k~ - nite element approximation (in the usual sense) of S gi . Using [28, Theorem 6.3.3.2] and (4.3.5), it follows that
kgi ? Lvi kL2 ( ) kL k kS gi ? vi kL2 ( ) C n~ ?r~=dkS gikH r~( ) C kS k n~?r~=d kgikH r~( ): (Here kL k and kS k respectively denote the L2( ) and H r~( ) operator norms. See [28, Theorem 6.3.1.1] for a proof that kS k is nite.) Recall that the spaces Sn;k are based on a quasi-uniform family of triangulations. For 0 j r, we may use [28, Lemma A.2.3.4] to see that the inverse inequalities
jgijH j ( ) Cn(~r?j)=d kgikL2 ( ) hold, where
jvjH j ( ) =
X
j j=j
kD vk2L2( )
1=2
8 v 2 H j ( )
is the H j ( )-seminorm. So
kgi kH r~( ) =
X r~
j =0
jg j2 j
i H ( )
1=2
Cnr~=d kgikL2 ( ) Cnr~=d?1=2
the latter since gi is a bounded function whose support has volume (n?1). Combining these results, we see that
kL vikL2 ( ) C n~ ?r~=d nr~=d?1=2 : Using this inequality along with (4.3.4), the lemma follows. We can now de ne our information and algorithm. For any n 2 N, let
N~n;k = [hf; v1 iL2 ( ); : : : ; hf; vn iL2 ( )]
8f 2 F:
Then for f 2 F , we seek u~n;k = ~n;k (N~n;k f ) satisfying
hu~n;k ; giiL2 ( ) = hf; vi iL2 ( ) We then have 30
(1 i n):
Theorem 4.3.2. Let = .
(1) For any n 2 N, we have
r(n; ) = (n?r=d ): (2) Let k r ? 1. For any n 2 N, we have
e( ~n;k ; N~n;k ) = (n?r=d ); so that ~n;k is an nth nearly minimal error algorithm, and N~n;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; ) = c (1=")d=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")d=r . Then ?
costmod-Gal("; ) = c (1=")d=r ; and so the modi ed Galerkin method ~n;k using modi ed FEI N~n;k is a nearly optimal complexity algorithm for the Fredholm problem of the second kind. Proof: Using Lemma 4.3.1, we see that inequality (4.1.2.2) holds. Now we can apply
Theorems 4.1.2.3 and 4.1.2.4. We close this section by discussing the optimal choice of r~ in (4.3.4), as well as the amount of preprocessing required in computing v1 ; : : : ; vn . Clearly, v1 ; : : : ; vn are independent of any problem element f , and so their calculation may be be considered precomputation. However, in practice, we would like to compute them as cheaply as possible. There are (at least) two con icting reasons why it may be dicult or expensive to calculate v1 ; : : : ; vn satisfying (4.3.5) with n~ given by (4.3.4). The rst is that we need Sn~ ;k~ to be a subspace of H r~( ). On the one hand, we want to simplify the task of designing the basis functions of the nite element space Sn~ ;k~ over a reference element; this tells us that we should choose r~ as small as possible, i.e., r~ = 1. On the other hand, we want to minimize amount of work required to calculate v1; : : : ; vn once we have designed these basis functions from (4.3.4), this criterion tells to choose r~ as large as possible, i.e., r~ = r. The question is now one of which criterion to use. To solve this conundrum, we note that we only design the reference element basis functions once, independent of n, whereas the calculation of v1; : : : ; vn depends on n. We are probably willing to expend the extra e ort involved in designing the basis functions (which only needs to be done once), thereby saving cost arising in the calculation of v1 ; : : : ; vn for various n. In other words, we feel that it would be preferable to choose r~ = r. Thus we choose a nite element subspace Sn~ ;k~ of H r ( ), where k~ is the degree of the subspace and the dimension n~ of the subspace satis es n~ = nd=(2r). 31
We now discuss the cost of computing v1; : : : ; vn. Assuming we choose r~ = r, the previous analysis implies that the cost of computing v1; : : : ; vn is (nd=(2r)+1). Since this cost grows faster than n, on-the- y calculation of v1 ; : : : ; vn will dwarf the remainder of the calculation of ~n;k(N~n;k f ) for a problem element f . However, it is possible to precompute v1; : : : ; vn, since they are independent of any f . If we decide to compute "-approximations for many f 2 F , with a xed value of ", then we may consider the cost of this precomputation as an overhead whose cost we can ignore.
4.4. Mixed elliptic-hyperbolic problems.
In this section, we look at the complexity of a solution-restricted Tricomi problem. This is a simple mixed hyperbolic-elliptic problem, which arises in the study of two-dimensional transonic ow across an airfoil; see [7, Chapter X] and [13] for further discussion. Let ?0 be a simple curve in the region y > 0 of the two-dimensional (x; y)-plane, intersecting the x-axis only at the points A(?1; 0) and B(1; 0). Let ?1 and ?2 be given by ( ?1 + 32 (?y)3=2 (x; y) 2 ?1 ; x= 1 ? 32 (?y)3=2 (x; y) 2 ?2 : Note that ?1 and ?2 intersect at the point C (0; yC ), where yC = ?( 23 )2=3 =: ?1:31037. Let
R2 be the region whose boundary is ?0 [ ?1 [ ?2 . De ne a partial di erential operator L by
Lu = yuxx + uyy : L is called the Tricomi operator. Note that (1) L is elliptic (but not strongly elliptic) in the elliptic region E = f (x; y) 2 : y > 0 g. This corresponds to subsonic ow. (2) L is parabolic on the parabolic line J = f (x; y) 2 : y = 0 g. This corresponds to sonic ow. (3) L is hyperbolic in the hyperbolic region E = f (x; y) 2 : y < 0 g. This corresponds to supersonic ow. We also note that ?1 and ?2 are the characteristic curves of L, respectively emanating from A and B. Let f : ! R. The Tricomi problem is to nd a function u : ! R satisfying Lu = f in ; (4.4.1) u = 0 on ?0 [ ?1 : Since the domain is divided by the sonic line into elliptic and hyperbolic regions, the Tricomi problem is an example of a mixed elliptic-hyperbolic problem. Remark : Note that we only prescribe boundary data for the Tricomi problem on part of @ . The simple explanation for this is that since the Tricomi problem is hyperbolic in H , prescribing boundary data on one of the characteristic lines that bounds H is sucient for solvability on H , whereas requiring the solution to satisfy given boundary 32
values on two characteristic lines would overly constrain the problem in H . Note that in principle, once we have solved the problem in H , we could solve the remaining degenerate elliptic problem in E . We now show how to express (4.4.1) as a standard problem. First, we let H 2 (bd) = f v 2 H 2( ) : v = 0 on ?0 [ ?1 g and H 2 (bd)+ = f v 2 H 2 ( ) : v = 0 on ?0 [ ?2 g: (The notations bd and bd+ respectively refer to boundary conditions and adjoint boundary conditions; see [4, pg. 71].) Then hLv; wiL2 ( ) = hv; LwiL2 ( ) 8 v 2 H 2 (bd); w 2 H 2(bd)+ ; (4.4.2) see [4, pg. 308]. Let H ?2 (bd) denote the dual space of H 2(bd), which is a Hilbert space under the norm hv; wiL2 ( ) : kvk?H2(bd) = sup w2H 2 (bd)+ kwkH 2( ) We can then de ne a linear transformation L : L2( ) ! H ?2 (bd) by letting Lv = hv; LiL2 ( ) 8 v 2 L2( ): Using (4.4.2), we see that there is a positive constant C for which kLvk?H2(bd) C kvkL2( ) 8 v 2 L2( ): Thus the bounded linear transformation L : L2( ) ! H ?2 (bd) is a weak extension of the original Tricomi operator L. Using [4, Theorem IV.3.1], we nd that L is a dense injection. Hence, we see that the problem (4.4.1) may be expressed as a standard problem, with X = H ?2 (bd), G = L2 ( ), and W = H r ( ) for some r 0. Note that q = 0. Once again, the class F of problem elements is the unit ball of H r ( ). Since L : L2 ( ) ! H ?2 (bd) is de ned weakly, we must also de ne S = L?1 weakly. That is, for f 2 F , we require that u = Sf satisfy hu; LviL2 ( ) = hf; viL2 ( ) 8 v 2 H 2(bd)+: Note that this weak de nition of S allows us to avoid the issue of boundary values for L2( )-functions. We now apply the results of Section 4.1. First, we look at the case = F . For any n 2 N, we once again let Sn;k be an n-dimensional nite element subspace of L2( ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by Nn;k f = [hSf; g1iL2 ( ); : : : ; hSf; gniL2 ( )]: For f 2 F , let un;k = n;k (Nn;k f ) 2 Sn;k be the Galerkin approximation given by hun;k ; gi iL2 ( ) = hSf; giiL2 ( ) (1 i n): (4.4.3) We then have 33
Theorem 4.4.1. Let = F .
(1) For any n 2 N, we have
r(n; F ) = (n?r=2 ): (2) Let k r ? 1. For any n 2 N, we have
e( n;k ; Nn;k ) = (n?r=2 ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; F ) = c (1=")2=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")2=r . Then ?
costGal("; F ) = c (1=")2=r ; and so the Galerkin method n;k using FEI Nn;k is a nearly optimal complexity algorithm for the Tricomi problem.
Proof: Immediate from Theorems 4.1.1.3 and 4.1.1.4.
Once again, we see that linear functionals hSf; giiL2 ( ) are hard to directly evaluate, because of their apparent dependence on Sf . To overcome this diculty, we describe an auxilliary adjoint problem. Even though (4.4.2) tells us that L is formally self-adjoint, we will now write L for the adjoint operator.5 For g : ! R, we wish to nd a function v : ! R satisfying Lv = g in ; (4.4.4) v = 0 on ?0 [ ?2 : This problem is called the adjoint Tricomi problem. We need to nd the proper weak formulation of this adjoint problem. To do this, let H ?2(bd)+ denote the dual space of H 2 (bd)+, which is a Hilbert space under the norm
kvk?H2(bd)+ = sup2 hkv;wwk iL22 ( ) : H ( ) w2H (bd) De ne a linear transformation L : L2( ) ! H ?2 (bd)+ by letting
L v = hL; viL2 ( ) 5
8 v 2 L2( ):
We do this because and , when considered as operators with domain 2 ( ), have di erent codomains. L
L
L
34
Using (4.4.2), we see that there is a positive constant C for which
kL vk?H2(bd)+ C kvkL2( )
8 v 2 L2( ): Thus the bounded linear transformation L : L2( ) ! H ?2 (bd)+ is a weak extension of
the adjoint Tricomi operator L . From [4, Theorem IV.3.1], it follows that L is a dense injection. Hence, we have de ned L weakly. This means that we can de ne S = (L )?1 weakly. For g 2 F , we require that v = S g satisfy
hLu; viL2 ( ) = hu; giL2 ( )
8 u 2 H 2 (bd):
This is the weak formulation of the adjoint Tricomi problem. Using (4.4.2), we nd that
hSf; giL2( ) = hf; S giL2( )
8f; g 2 L2( ):
In particular, we have
hSf; giiL2 ( ) = hf; S giiL2 ( )
(1 i n):
This means that we can once again consider the computation of S g1; : : : ; S gn as precomputation. Hence we have
Nn;k f = [hf; S g1iL2 ( ); : : : ; hf; S gniL2 ( ) ]: For f 2 F , it now follows that un;k = n;k (Nn;k f ) satis es hun;k ; giiL2 ( ) = hf; S giiL2 ( ) (1 i n): (4.4.5) So the information Nn;k de ned in (4.4.3) is continuous linear information. Moreover, the Galerkin algorithm (4.4.5) is a nearly optimal error algorithm and a nearly optimal complexity algorithm using continuous linear information. Of course, Theorems ? 4.1.1.3 F ? r= 2 and 4.1.1.4 tell us that r(n; ) = r(n; ) = (n ) and comp("; ) = c (1=")2=r . We now see that (4.4.3) and (4.4.5) are two formulations of the Galerkin algorithm using FEI, the rst showing that this FEI is information from F and the second showing that it actually is from . Moreover, the rst formulation uses functionals hSf; giiL2 ( ), whereas the second uses hf; S giiL2 ( ). As was the case in the previous section, we prefer to use (4.4.5) instead of (4.4.3). As in the previous section we see that S g1; : : : ; S gn may be hard to calculate, even though they are well-de ned. Hence we need to once again consider modi ed Galerkin methods for our problem. Of course, the only issue that we need to resolve is how to calculate v1 ; : : : ; vn such that L vi is suciently close to gi for 1 i n. One idea is to let Sn~ ;k+2 be an n~ -dimensional nite element subspace of L2( ) having degree k + 2, where n~ = (nr+2 ): (4.4.6) 35
Suppose that we choose vi solving the least squares problem
kgi ? Lvi kL2 ( ) = inf kgi ? L skL2 ( ); s2Sn~ ;k+2
i.e., vi 2 Sn~ ;k+2 satis es
8 s 2 Sn~ ;k+2:
hLvi ; LsiL2 ( ) = hgi ; LsiL2 ( )
We are now ready to de ne our information and algorithm. For any n 2 N, let
N~n;k = [hf; v1 iL2 ( ); : : : ; hf; vn iL2 ( )]
8f 2 F:
Then for f 2 F , we seek u~n;k = ~n;k (N~n;k f ) satisfying
hu~n;k ; giiL2 ( ) = hf; vi iL2 ( )
(1 i n):
We then have
Theorem 4.4.2. Let = .
(1) For any n 2 N, we have
r(n; ) = (n?r=2 ): (2) Let k r ? 1. For any n 2 N, we have
e( ~n;k ; N~n;k ) = (n?r=2 ); so that ~n;k is an nth nearly minimal error algorithm, and N~n;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; ) = c (1=")2=r : ?
(4) Let k r ? 1. For any " > 0, let n = (1=")2=r . Then ?
costmod-Gal ("; ) = c (1=")2=r ; and so the modi ed Galerkin method ~n;k using modi ed FEI N~n;k is a nearly optimal complexity algorithm for the Tricomi problem. 36
Proof: Since Lv 2 Pk for v 2 Pk+2 , one can exactly solve the problem vi = S gi , except possibly on the boundary elements. But these boundary elements have area (~n?1=2 ) = (n?(r=2+1)). Hence kgi ? Lvi kL ( ) Cn?(r=2+1) for 1 i n. Now use Theo2
rems 4.1.2.3 and 4.1.2.4. We once again note that on-the- y computation of v1 ; : : : ; vn may be expensive. Indeed, from (4.4.6), we see that v1; : : : ; vn may be calculated with cost (nr+3 ), which of course greatly outweighs the cost of calculating u~n;k . However, we once again point out that v1; : : : ; vn are independent of any f 2 F . So, if we precompute v1 ; : : : ; vn and if we do not charge for this precomputation (since it is independent of any problem element), we can ignore the cost of the precomputation. Remark : Note that we used a very weak error bound to show that (4.4.6) implies that v1; : : : ; vn are suciently accurate. This was motivated by our lack of a shift theorem for the Tricomi problem, so that we cannot assume enough global smoothness in S gi (where gi is piecewise polynomial) to use the error estimates in 2. It is quite possible that our estimate is overly-pessimistic, and that we can nd suciently accurate piecewise polynomial approximations to S gi using fewer degrees of freedom.
4.5. Inverse nite Laplace transform.
In this section, we look at the complexity of a solution-restricted inverse Laplace transform. This is an example of a Fredholm integral equation of the rst kind, and is thus an ill-posed problem. This problem arises in remote sensing problems of geomathematics; see [21] for discussion and further examples. Without loss of generality, we assume that our functions are de ned over the unit interval I = [0; 1]. De ne an operator L : L2(I ) ! L2 (I ) as (Lu)(s) =
Z 1
0
e?stu(t) dt
(0 s 1)
for u 2 L2(I ). Thus Lu is the nite Laplace transform of u. We are interested in the inverse nite Laplace transform problem : for f 2 L2 (I ), nd u 2 L2(I ) such that Lu = f , i.e., such that Z 1 e?st u(t) dt = f (s) (0 s 1): (4.5.1) 0
From [14], we see that L : L2(I ) ! L2 (I ) is an injection. Since L is self-adjoint as an operator on L2(I ), we see that the range of L is dense in L2 (I ). Hence the solution operator S = L?1 is densely de ned in L2(I ). Thus we can express our problem (4.5.1) as a standard problem if we choose G = X = L2(I ) and W = H r (I ) for some r 0. Note that L is compact. Thus the problem of nding u 2 L2(I ) satisfying Lu = f is ill-posed for f 2 L2(I ). Remark : The problem of inverting L is in fact very ill-posed. From [28, pp. 198-199], we nd that L : L2(I ) ! H r (I ) is a compact injection with dense range for any r 0. Thus the (densely de ned) solution operator S = L?1 : H r (I ) ! L2(I ) is unbounded, no matter how big we choose r to be. Hence the results of [26] imply that if we choose our problem elements to be the unit ball of a Hilbert Sobolev space H r (I ), then the error of 37
any nite-cost algorithm is in nite, no matter how large we choose r to be. Simply stated, this means that restricting the problem elements for (4.5.1) will not work. This explains why we are interested in a solution-restricted version of this problem. Having expressed (4.5.1) as a solution-restricted problem, we can now use the results of Section 4.1. First, we look at the case = F . For any n 2 N, let Sn;k be an ndimensional nite element subspace of L2(I ) having degree k. Let fg1; : : : ; gng be a basis for Sn;k . De ne nite element information Nn;k by
Nn;k f = [hSf; g1iL2 (I ); : : : ; hSf; gniL2 (I )]: For f 2 F , let un;k = n;k (Nn;k f ) be the Galerkin method given by
hun;k; gi iL2 (I ) = hSf; giiL2 (I )
(1 i n):
(4.5.2)
We then have
Theorem 4.5.1. Let = F .
(1) For any n 2 N, we have
r(n; F ) = (n?r ): (2) Let k r ? 1. For any n 2 N, we have e( n;k ; Nn;k ) = (n?r ); so that n;k is an nth nearly minimal error algorithm, and Nn;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; F ) = c (1=")1=r :
?
(4) Let k r ? 1. For any " > 0, let n = (1=")1=r . Then ?
costGal("; F ) = c (1=")1=r ; and so the Galerkin method n;k using FEI Nn;k is a nearly optimal complexity algorithm for the inverse nite Laplace transform problem. Proof: Immediate from Theorems 4.1.1.3 and 4.1.1.4.
As with problems that we have considered previously, we prefer to avoid using the functionals hSf; giiL2 (I ) making up the nite element information Nn;k . We would rather use functionals of the form hf; vi iL2 (I ). In the discussion that follows, we will take advantage of the self-adjointness of L as an operator on L2 (I ), which implies that S is self-adjoint in L2(I ). Thus we write L and S instead of L and S in what follows. Note that we cannot rewrite the functionals hSf; giiL2 (I ) in the form hf; SgiiL2 (I ). This is because functions belonging to the domain of S (i.e., the range of L) are in nitely 38
di erentiable, while the functions gi are only piecewise smooth. So the nearly optimal Galerkin method using FEI for the inverse nite Laplace transform, which (of course) uses F -information, cannot be cosmetically rewritten as a method using -information. This is an important di erence between this problem and the other problems that we have studied. This being the case, we now let = , and look for nearly optimal -information, using our general results in Section 4.1.2. The main idea is to choose v1; : : : ; vn 2 X such that Lv1 ; : : : ; Lvn are suciently good approximations of g1 ; : : : ; gn. To do this, we use ideas based on those in [28, pp. 225{226]. We assume that the nite element space Sn;k is de ned over a uniform partition of I . Suppose that Sn;k H p(I ) for some positive integer p, so that k p. Recall that fg1; : : : ; gng is a standard basis for Sn;k , having small supports. Our task is to nd functions fv1; : : : ; vng such that (4.1.2.2) holds. To do this, we let R and m be parameters depending on n, p, and r. We will give criteria for choosing R and m later. For any j 2 N, we let Gj (x)P= Pj (2x ? 1), where Pj is the usual Legendre polynomial of degree j . We write Gj (x) = jl=0 j;lxl . Next, for any l 2 N, we let (
wl =
? (?1)j jl Rl+1 if t 2 [j=R; (j + 1)=R) for some j 2 f0; : : : lg, 0 if t 2 [(l + 1)=R; 1]
We then let
vi =
m X
j X
j =0
l=0
(2j + 1)hgi ; Gj iL2 (I )
j;l wl
for 1 i n. Then we have Lemma 4.5.1. For any n and k , there exist R and m such that
kLvi ? gikL2 (I ) Cn?(r+1) Proof: Let i
(1 i n):
(4.5.3)
2 f1; : : : ; ng. Since the support of gi has length (n?1 ), there exists a
positive constant A such that
kgikL2 (I ) An?1=2
Since Sn;k is de ned over a uniform partition of I , we know that the inverse inequality [28, Lemma A.2.3.4] holds, and so there exists B > 0 such that
kgi(p)kL2(I ) BA?1 npkgikL2 (I ) Bnp+1=2: We claim that for (4.5.3) to hold, it suces to choose
R pA nr+1=2 3C
and
m 4e 2CB 39
1=p
n(r+p+1=2)=p:
(4.5.4)
Indeed, let
gi;m =
m X j =0
(2j + 1)hgi ; Gj iL2 (I )Gj
be the mth Legendre series approximation of gi . From [8, Theorem 1.3.2], we have e p kgi ? gi;m kL2 (I ) 4m kgi(p)kL2 (I ): Following [28, pp. 225{226], we nd that e p 1 A kLvi ? gikL2 (I ) 2Rp3 n1=2 + B 4m np?1=2: Using (4.5.4), we now nd that (4.5.3) holds, as claimed. We are now ready to de ne our information and algorithm. For any n 2 N, let v1; : : : ; vn be as de ned in Lemma 4.5.1. De ne information N~n;k as N~n;k = [hf; v1 iL2 ( ); : : : ; hf; vn iL2 ( )] 8f 2 F: The modi ed Galerkin algorithm ~n;k is as follows: For f 2 F , we seek u~n;k = ~n;k (N~n;k f ) satisfying hu~n;k ; giiL2 ( ) = hf; vi iL2 ( ) (1 i n): We then have Theorem 4.5.2. Let = . (1) For any n 2 N, we have r(n; ) = (n?r ): (2) Let k r ? 1. For any n 2 N, we have e( ~n;k ; N~n;k ) = (n?r ); so that ~n;k is an nth nearly minimal error algorithm, and N~n;k is nth nearly optimal information. (3) For any " > 0, we have ?
comp("; ) = c (1=")1=r :
?
(4) Let k r ? 1. For any " > 0, let n = (1=")1=r . Then ?
costmod-Gal ("; ) = c (1=")1=r ; and so the modi ed Galerkin method ~n;k using modi ed FEI N~n;k is a nearly optimal complexity algorithm for the inverse nite Laplace transform problem. 40
Proof: Immediate from Theorems 4.1.2.3 and 4.1.2.4, along with Lemma 4.5.1.
We brie y discuss the choice of p. Clearly the larger we make p, the smaller m needs to be. The optimal choice is then to make p = k. Since the best choice for k is to let k = r ? 1, we see that the best choice for p is p = r ? 1. It then follows that m = (n ), where = 2 + 3=(2r ? 2). Since m grows faster than n2, the cost of computing v1; : : : ; vn on-the- y grows faster than n3 , and is therefore impractical. However, since v1; : : : ; vn are independent of any f , they may be precomputed. If we wish to compute "-approximations for many f 2 F , with a xed value of ", then we can safely ignore the cost of this precomputation.
4.6. The backwards heat equation.
In our nal application, we look at the complexity of the heat equation running backwards in time. This is one of the most famous classical examples of an ill-posed problem. Further discussion and references may be found in [10], [11], [12], [17], and [19]. Let I = [0; 1]. For any positive integer j , let
p
8 x 2 I: (4.6.1) Then fsj g1 j =1 is an orthonormal basis for L2 (I ). For t 2 R, we de ne a positive-de nite sj (x) = 2 sin jx
self-adjoint operator Ht in L2(I ) by letting
Ht f =
1 X j =1
e?2 j2 t hf; sj iL2 (I )sj
8f 2 L2(I ):
(4.6.2)
Note that (1) If t > 0, then Ht is compact. (2) H0 is the identity map. (3) If t < 0, then Ht is a densely-de ned unbounded operator. (4) We have the semigroup properties
Ht H = Ht+ H?t = Ht?1
8 t; 2 R; 8 t 2 R:
(5) For any t 2 R, let u(; t) = Ht f . Then u(x; t) satis es
@u = @ 2 u @t @x2 u(; 0) = f u(0; ) = u(1; ) = 0
in I R; in I; in R:
Thus Ht f is the solution of the heat equation at time t, with u at time t = 0 being given by f . We will call these conditions at time t = 0 initial conditions if t > 0 and nal conditions if t < 0. 41
We wish to solve this problem at time t = ?t0 , where t0 > 0. Since the elapsed time is negative, we call this problem a backwards heat equation. The operator H?t0 is a unbounded in L2 (I ), and so this problem is ill-posed. We shall formulate our problem as a solution-restricted operator equation. Our restriction will be to assume that the solution is L2(I )-bounded at some time t = ?t1 in the past, where t0 < t1. It is well-known that the ill-posed backwards heat equation becomes well-posed under this hypothesis, see [10]. Note that the approach taken here is similar to that in [27]. To express this problem as a solution-restricted operator equation, we let X = G = L2(I ) and de ne L : G ! X be de ned as L = Ht0 . Then Lu = f i f is the solution of a heat equation at time t = t0 , with initial conditions given by u at time t = 0. From the previous comments, we see that L is a compact injection with dense range. Our solution operator is now S = L?1 : G ! X . From the semigroup properties, we see that S = H?t0 , i.e., u = Sf is the solution of a heat equation at time t = ?t0, with nal conditions given by f at time t = 0. Since S is an unbounded ? operator, we are trying to solve an ill-posed problem. Finally, we let W = Ht1 ?t0 L2(I ) , which is a Hilbert space under the norm
kwkW = kH?(t1?t0 )wkL2 (I ) 8 w 2 W: Since t1 > t0, we see that kkL2(I ) kkW , and so the identity embedding E : W ! L2(I ) is continuous, with kE k 1. Hence, the spaces X , G, and W , along with the operator L : G ! X , de ne a solution-restricted operator equation. It is straightforward to check that the class F of problem elements is the set of all f 2 L2(I ) for which H?t1 f belongs
to the unit ball of L2(I ). Hence this solution-restricted problem is the solution of the backwards heat equation at time t = ?t0 with prescribed nal data at time t = 0, under the constraint of a known bound on the solution at the earlier time t = ?t1. We will use the results contained in Theorem 3.3. This requires us to nd an orthonormal basis for W consisting of eigenvectors of E E , as well as the corresponding eigenvalues
12 22 > 0. For j = 1; 2; : : : , we will let
zj = Ht1 ?t0 sj = e?2 j2 (t1?t0 )sj :
(4.6.3)
Then fzj g1 j =1 is an orthonormal basis for W . Lemma 4.6.1. For any positive integer j , we have
E Ezj = j2zj ; where
j = e?2 j2 (t1?t0 )
(4.6.4)
and zj is given by (4.6.3). Proof: We rst claim that E E = H2(t1 ?t0 ) . Indeed, let v; w 2 W . Since H?(t1 ?t0 ) is self-adjoint on L2 (I ), we have
hv; wiL2 (I ) = hEv; EwiL2 (I ) = hv; E EwiW = hH?(t1 ?t0 )v; H?(t1 ?t0)E EwiL2 (I ) = hv; H?2(t1 ?t0 )E EwiL2 (I ): 42
Thus H?2(t1?t0 )E E is the identity operator on W , and so E E = H??2(1 t1?t0 ) = H2(t1?t0 ), as claimed. Next, we claim that for any w 2 W and any index j , we have hw; zj iW zj = hw; sj iL2 (I )sj : (4.6.5) Indeed, since H?(t1?t0 )sj is a scalar multiple of sj and H?(t1?t0 ) is self-adjoint, we have hw; zj iW zj = hH?(t1 ?t0 )w; H?(t1 ?t0)zj iL2 (I )zj = hH?(t1 ?t0 )w; sj iL2 (I )Ht1 ?t0 sj = hw; H?(t1 ?t0 )2sj iL2 (I )Ht1 ?t0 sj = hw; sj iL2 (I )sj ; as claimed. Now since E E = H2(t1?t0 ), we may use (4.6.2) and (4.6.5) to nd that 1 1 X X 2 j 2 (t1 ?t0 ) ? 2 E Ew = H2(t1?t0 )w = e hw; sj iL2 (I )sj = e?22 j2 (t1?t0)hw; zj iW zj : j =1
j =1
The result follows immediately. We are now ready to apply the results in Theorem 3.3. First, we consider the case = F . For any n 2 N, we de ne information Nn as Nnf = [hSf; z1iW ; : : : ; hSf; zniW ]; where z1 ; : : : ; zn are given by (4.6.3), and an algorithm n using Nn as
n(Nn f ) = We then have
n X j =1
hSf; zj iW zj
Theorem 4.6.1. Let = F .
8f 2 F:
(1) For any n 2 N, we have r(n; F ) = e( n; Nn ) = e?2 (n+1)(t1?t0 ): Hence Nn is nth optimal information in F , and n is an nth minimal error algorithm. (2) For any " > 0, the "-cardinality number m("; F ) = inf f integers n 0 : n+1 " g is given by (& ' ) r 1 1 F m("; ) = max pt ? t ln " ? 1; 0 : 1 0 (3) For " > 0, let n = m("; F ). Then c m("; F ) comp("; F ) cost( n; Nn) (c + 2)m("; F ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. 43
Proof: Immediate from Lemma 4.6.1 and Theorem 3.4.
?p Hence we see that comp("; F ) = c ln(1=") , giving the complexity of the solutionrestricted backwards heat equation when = F . We now look at the case = . For any n 2 N, we de ne information N~n 2 n as
N~nf = [hf; s1 iL2 (I ); : : : ; hf; sn iL2 (I )]; where s1 ; : : : ; sn are given by (4.6.1), and an algorithm ~n using N~n as n
X ~n(N~nf ) = e2 j2 t0 hf; sj iL2 (I )sj
j =1
8f 2 F:
Note that the algorithm ~n is the n-term truncation of the standard series representation H?t0 of the solution to the heat equation at time t = ?t0. We then have Theorem 4.6.2. Let = . (1) For any n 2 N, we have r(n; ) = e( n; Nn ) = e?2 (n+1)(t1?t0 ): Hence Nn is nth optimal information in , and n is an nth minimal error algorithm. (2) For any " > 0, the "-cardinality number m("; ) = inf f integers n 0 : n+1 " g is given by (& ' ) r 1 1 m("; ) = max pt ? t ln " ? 1; 0 : 1 0 (3) For " > 0, let n = m("; ). Then c m("; ) comp("; ) cost( n; Nn) (c + 2)m("; ) ? 1: Hence for c 1, the algorithm n using information Nn is a nearly optimal complexity algorithm. Proof: Let j 2 f1; : : : ; ng. Using (4.6.5) and the self-adjointness of S = H?t0 , we nd
hSf; zj iW zj = hH?t0 f; sj iL2 (I )sj = hf; H?t0 sj iL2 (I )sj = e2 j2 t0 hf; sj iL2 (I )sj : ~ ) = n(Nnf ) for any f 2 F . The theorem now follows immediately from Thus ~n(Nf Theorem 4.6.1.
Hence, we have shown that the optimal F -information is actually -information, and that we get a minimal error algorithm by truncating the standard series representation for ?p the solution of the backwards heat equation. Moreover, comp("; ) = c ln(1=") , giving the complexity of the solution-restricted backwards heat equation when = . 44
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