The complexity of indefinite elliptic problems with noisy data
The complexity of inde nite elliptic problems with noisy data Technical Report CUCS-004-97 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 March 25, 1997 Abstract. We study the complexity of second-order inde nite elliptic problems ? div(aru) + bu = f (with homogeneous Dirichlet boundary conditions) over a d-dimensional domain , the error being measured in the H 1( )-norm. The problem elements f belong to the unit ball of W ( ), where p 2 [2; 1] and r > d=p. Information consists of (possibly-adaptive) noisy evaluations of f , a, or b (or their derivatives). The absolute error in each noisy evaluation is at most . We nd that the nth minimal radius for this problem is proportional to n? + , and that a noisy nite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be eciently implemented using multigrid techniques. Using these results, we nd tight bounds on the "-complexity (minimal cost of calculating an "-approximation) for this problem, said bounds depending on the cost c() of calculating a -noisy information value. As an example, if the cost of a -noisy evaluation is c() = ? (for s > 0), then the complexity is proportional to (1=") + . r;p
r=d
s
d=r
s
1. Introduction
We study the complexity of elliptic partial di erential equations with noisy data. To ease notational burdens, we will consider a class of second-order elliptic problems, although the results in this paper may be extended to include elliptic problems of arbitrary order. To be speci c, we will consider (the variational form of) second-order elliptic problems
? div(aru) + bu = f u=0
in ; on @ :
(1.1)
Here, is a xed d-dimensional domain. We need a speci c way to measure the error of an approximation; in this paper, we choose to measure error in the Sobolev H 1 -norm, which is equivalent to the problem's natural energy norm. However, the results of this paper also hold if error is measured in the L2 -norm. Typically, we only know the values of a, b, and f at a nite number of points in . Hence, this is a problem of information-based complexity (IBC) [11]. We cannot obtain exact solutions with nite cost; we must settle for an "-approximation, i.e., an algorithm computing (for each choice of a, b, and f ) an approximation whose error is at most ". For any positive error level ", we want to know the "-complexity (the minimal cost of calculating an "-approximation), and we want to This research was supported in part by the National Science Foundation under Grant CCR-95-00850.
nd algorithms that are optimal, in the sense that they calculate an "-approximation at (nearly) minimal cost. Most earlier work on the complexity of elliptic problems has assumed that exact information is available, i.e., we calculate the values these coecient functions or of the right-hand side exactly. (An extensive treatment of the topic may be found in [12].) What happens if the information is contaminated by noise? Suppose that we can only guarantee that these calculated function values are accurate to within some positive noise level . What can we say about the complexity of such problems? The systematic study of complexity for problems with noisy information was initiated by [9], with noisy elliptic problems rst being investigated in [13]. Let us recall the main results of [13]. If the problem is a de nite elliptic problem (i.e., a is bounded away from zero and b is non-negative), then the nth minimal radius (the minimal error attained using n evaluations) is (n?r=d + ). This error is attained by a nite element method using quadrature (FEMQ) of degree r (or greater). Now let c() be the cost of obtaining a -noisy approximation of a function value. Then the "-complexity is (
comp(") =
1
c() C ?1 " ?
inf
0 0 for suciently small positive . (2) Real arithmetic operations and comparisons are done exactly, with unit cost. (3) We are not charged for Boolean operations. (4) Linear operations over H01 ( ) are done exactly, with cost g. For any noisy information N and any algorithm using N, we shall let cost( ; N) denote the ? worst case cost of calculating N ([f ; a; b]) over all [f ; a; b] 2 F . Now that we have de ned the error and cost of an algorithm, we can nally de ne the complexity of our problem. We shall say that comp(") = inf f cost( ; N) : N and such that e( ; N) " g is the "-complexity of our problem. An algorithm using noisy information N for which
e( ; N) "
and
?
cost( ; N) = comp(")
is said to be an optimal algorithm. 3. The Noisy FEMQ
In this section, we de ne the noisy nite element method with quadrature (FEMQ). This is an algorithm using standard information consisting only of function evaluations, i.e., no derivative evaluations are used. Our notation is the standard one found in, e.g., [5] and [12, Chapter 5]. This noisy FEMQ is the same as described in [13]. However, we will give a complete description of the noisy FEMQ, so that this paper can be reasonably self-contained. We rst establish some notation. Let K^ be a xed polyhedron in Rd. We call K^ a reference element. Select a xed value of k 2 Z++, and let Pk (K^ ) denote the space of polynomials having total degree at most k, considered as functions over K^ . We next let K be a (small) nite element, i.e., the ane image of K^ under a bijection FK , where ^ FK (^x) = BK x^ + xK 8 x^ 2 K; (3.1) where BK 2 Rdd is invertible and xK 2 Rd. Next, we let T be a triangulation of consisting of nite elements, where each K 2 T is the image of the reference element K^ under the ane bijection FK . Given this triangulation T , we de ne a nite element space
S (T ) = s 2 H01( ) : s K 2 Pk (K ) 8 K 2 T 5
of degree k. We will assume that the following conditions hold: (1) fTn g1 n=1 is a family of triangulations of such that Sn = S (Tn ) is a nite element space of dimension n. (2) fTn g1 n=1 is a quasi-uniform family of triangulations, i.e., lim sup sup hK < 1; n!1 K 2Tn K where hK is the diameter of K and K is the diameter of the largest sphere contained in K . (3) Let k k denote the `2 matrix norm on Rd. Then kBK k 1 for any element K 2 Tn and any triangulation Tn . We rst recall how the noise-free \pure" FEM is de ned. Let n 2 Z+, and let fs1 ; : : :; sn g be a basis for Sn . For [f ; a; b] 2 F , nd n X u n = j s j ; in Sn such that
j =1
Ba;b (un; si ) = hf; sii
(1 i n): (3.2) If we approximate the integrals appearing in the FEM by numerical quadrature, we get the (noisefree) FEMQ. The quadrature rule used to de ne the FEMQ is initially de ned on the reference element. This reference quadrature rule has the form
I^v^ =
J X j =1
!^j v^(^xj )
for functions v^ de ned on K^ . Said rule is said to be exact of degree q if Z
K^
v^ = I^v^
8 v^ 2 Pq (K^ ):
We de ne a local quadrature rule over a particular nite element K as
IK v =
J X j =1
!j;K v(xj;K );
where
!j;K = det BK !^j and xj;K = FK (^xj ) (1 j J ) for K = FK (K^ ), with FK given by (3.1). Next, for any ` 2 Z+, we let
N` =
J [ [ K 2T` j =1
(3.3)
fxj;K g
denote the set of all quadrature nodes in all the elements belonging to T` . This is usually not a disjoint union, since a quadrature node on the boundary of one element will be on the boundary of an adjacent element sharing a common face. 6
Let
= minfk + 1; rg:
We now assume that the following conditions hold: (1) The smoothness r of the problem elements F satis es r 1 (as well as our previous requirement r > d=p). (2) The degree k of the nite element subspaces Sn~ satis es k > d=p ? 1. (3) I^ is exact of degree 2k + ? 1 over the reference element K^ . We can now de ne the noise-free FEMQ. Given n 2 Z+, we de ne
n~ = maxf card N` : ` 2 Z+ and card N` 31 n g:
(3.4)
Roughly speaking, n~ = b 13 nc, allowing for the fact that n~ must be the cardinality of the set N` of quadrature nodes for some triangulation T` . Let fs1 ; : : :; sn~ g denote a basis for the nite element space Sn~ . For [f ; a; b] 2 F , we de ne a new bilinear form Ba;b;n~ on Sn~ by X
d X
IK (a @i v @j w) + IK (bvw) K 2Tn~ i;j =1 J X d X X = a(xj;K ) (@i v)(xj;K ) (@iw)(xj;K ) + b(xj;K ) v(xj;K ) w(xj;K ) K 2Tn~ j =1 i=1
Ba;b;n~ (v; w) =
and a linear functional fn~ on Sn~ by X
J X X
!j;K f (xj;K ) v(xj;K ) K 2Tn~ j =1 Then the noise-free FEMQ consists of nding uQn~ 2 H01 ( ) such that fn~ (v) =
K 2Tn~
IK (fv) =
Ba;b;n~ (uQn~ ; si) = fn~ (si)
(1 i n):
(3.5)
We are ready to de ne the noisy FEMQ. Given n 2 Z+, we again choose the largest n~ 2 Z+ satisfying (3.4), and a basis fs1 ; : : :; sn~ g for the nite element space Sn~ . We now calculate a noisy version of the information that would be used by the noise-free FEMQ. That is, for each element K 2 Tn~ and each index j 2 f1; : : :; J g, we obtain real numbers a~j;K; , ~bj;K; and f~j;K; satisfying
ja~j;K; ? a(xj;K )j ; j~bj;K; ? b(xj;K )j ; jf~j;K; ? f (xj;K )j : Let where
N~ n; ([f ; a; b]) = fN~ n;(f ); N~ n;(a); N~ n;(b)g N~ n;(a) = fa~j;K; satisfying (3.6) : 1 j J and K 2 Tn~ g; N~ n;(b) = f~bj;K; satisfying (3.7) : 1 j J and K 2 Tn~ g; N~ n;(f ) = ff~j;K; satisfying (3.8) : 1 j J and K 2 Tn~ g; 7
(3.6) (3.7) (3.8)
Clearly, N~ n; is noisy information of cardinality at most n. For [f ; a; b] 2 F , we de ne a new bilinear form B~a;b;n~ ; on Sn~ by
B~a;b;n~ ; (v; w) =
J X d X X K 2Tn~ j =1 i=1
~aj;K; (@i v )(xj;K ) (@i w)(xj;K ) + ~bj;K; v (xj;K ) w(xj;K )
and a linear functional f~n~ ; on Sn~ by
f~n~ ; (v) =
J X X K 2Tn~ j =1
Then we seek
u~Qn~ = such that The coecient vector satis es where and
!j;K f~j;K; v(xj;K ): n~ X j =1
j sj
B~a;b;n~ ; (~uQn~ ; si) = f~n~ ; (si )
(1 i n):
(3.9)
a = [ 1 ; : : :; n]T Ga = b; G = [B~a;b;n~ ; (si; sj )]1i;jn~
b = [f~n~ ; (s1 ); : : :; f~n~; (sn~ )]T : We see that u~Qn~ depends on [f ; a; b] only through the noisy information N~ n; ([f ; a; b]), and so we write u~Qn~ = ~n; (N~ n;([f ; a; b])), with ~n; an algorithm using our noisy standard information N~ n; . 4. The Noisy FEMQ is a Minimal Error Algorithm
In this section, we prove that the noisy FEMQ is well-de ned. We also establish an error bound for the noisy FEMQ, which allows us to show that the FEMQ is a minimal error algorithm whenever k r. First, we note that the conditions de ning A imply several important inequalities, which hold independently of (a; b) 2 A : Lemma 4.1.
(1) Garding's inequality: There exists a constant 0 = 0;m > 0 such that
Ba;b(v; v) 0 kvk2H 1 ( ) ? 0 kvk2L2 ( )
8 v 2 H01( );
for all (a; b) 2 A . (2) Weak coercivity of Ba;b : There exists a constant 1 = 1;m;M;0 ; > 0 such that
8 v 2 H01( ); 9 nonzero w 2 H01( ) : jBa;b(v; w)j 1kvkH1( )kwkH1 ( ); 8
for all (a; b) 2 A . (3) Shift theorem: If f 2 H r ( ), then there exists a constant = m;M;0 ;;r such that
?1 kS ([f ; a; b])kHr+2 ( ) kf kH r ( ) kS ([f ; a; b])kH r+2 ( ) ;
(4.1)
for all (a; b) 2 A . (4) Uniform weak coercivity of Ba;b : There exists a constant = m;M;0; > 0 and an index n 2 Z+ such that if n n, then
8 v 2 Sn~ ; 9 w 2 Sn~ : jBa;b(v; w)j kvkH1( ) kwkH1( ) ; for any (a; b) 2 A .
(4.2)
Proof: If we use the inequality
Ba;b(v; v) mjvj2H 1 ( ) ? 0 kvk2L2 ( );
(4.3)
and the Poincare inequality j jH 1 ( ) C k kH 1 ( ) , we see that Garding's inequality follows, with
0 = C m. We next consider weak coercivity. Let v 2 H01 ( ). Since 0 is not an eigenvalue of La;b , there exists a unique z 2 H 3 ( ) satisfying
La;bz = 0v in ; z = 0 on @ :
(4.4)
Ba;b (v; z) = 0 kvk2L2 ( ):
(4.5)
Ba;b (v; w) mjvj2H 1 ( ) 0 kvk2H 1 ( ) :
(4.6)
Then
Now take w = v + z . Using (4.3), (4.5), and the Poincare inequality, we have Let fzi gi2Z ++ be an orthonormal H01( ) basis satisfying
La;b0 zi = izi in ; z = 0 on @ : If we expand
v= we nd that It now follows that Thus
z=
kzkH1 ( )
sup i2Z++
1 X i=1
i zi;
1 X
i 0 z : i i=1 i ?
0 kvk 1 0 kvk 1 : i ? H ( ) H ( )
0 kwkH1( ) kvkH1( ) + kzkH1 ( ) 1 + kvkH1( ) :
9
Using this inequality along with (4.6), we have
0
Ba;b(v; w) + kvkH 1 ( ) kwkH1 ( ) ; 0
as required. We next consider the uniform shift theorem. Let [f ; a; b] 2 F and let u = S ([f ; a; b]). Using the weak coercivity of Ba;b , we nd that there exists nonzero w 2 H01 ( ) such that Ba;b (u; w) 1 kukH 1 ( ) kwkH 1 ( ): Since Ba;b (u; w) = hf; wiL2( ) , we nd that
1kukH 1( ) kwkH 1 ( ) Ba;b (u; w) kf kH ?1 ( ) kwkH 1 ( ); and so
kukL2( ) kukH1( ) 1?1kf kH?1( ) :
From [7, Theorem 8.8], we have the a priori inequality kukHr+2 ( ) C (kf kHr ( ) + kukL2( ) ); the constant C depending only on m, M , 0 , , and r. Using these last two inequalities, we see 0 that there exists 0 = m;M; 0 ;;r such that The reverse inequality
kukHr+2( ) 0kf kHr ( ) :
kf kHr ( ) 00 kukHr+2( ) ;
00 with 00 = m;M; 0 ;;r , follows easily from the conditions de ning the class A . Combining these last two inequalities, we obtain the uniform shift theorem. To check uniform weak coercivity, we let : H01 ( ) ! Sn~ denote the Sn~ -interpolation operator given by
(v )(x) =
n~ X j =1
v(xj )sj
8 v 2 H01( ):
Here x1 ; : : :; xn~ are the interior nodes of the triangulation Tn~ , and fs1 ; : : :; sn~ g is the usual dual nite element basis de ned by si (xj ) = i;j (1 i; j n~): From the usual nite element approximation theory, there is a positive constant C1 such that kz ? zkH1 ( ) C1n~?=d kzkH3( ) for any z 2 H01 ( ) \ H 3 ( ). Now choose (a; b) 2 A . From the conditions on A , there is another positive constant C2, depending only on M and 0 , such that jBa;b(v; w)j C2kvkH1( ) kwkH1( ) 8 v; w 2 H01( ): Let m l n = (C1 C2 1 0?1)?d= : For v 2 Sn~ , we let w = v + z , where z 2 H 3 ( ) satis es (4.4). Following [12, Lemmas 5.2.1 and 5.4.4], we nd that w is nonzero and that Ba;b (v; w) 1 + + C 2 (n )?=d kvkH 1 ( )kwkH 1 ( ) : 0 1 0 Hence the problem is uniform weakly coercive. Our main tool is Strang's Lemma (see [12, pp. 310{312] for a proof of a version having slightly more restrictive hypotheses). 10
Lemma 4.2. Suppose that there exists 0 2 (0; 1] and n 2 Z++ such that for any 2 [0; 0], any n n and any (a; b) 2 A , we have
(4.7) jBa;b(v; w) ? B~a;b;n~ ;(v; w)j 21 kvkH1( )kwkH1 ( ) 8 v; w 2 Sn~ ; where is as in (4.2). Then for any n n , any 2 [0; 0], and any [f ; a; b] 2 F , there is a unique u~Qn~ 2 Sn~ such that (3.9) holds. Moreover, there exists a positive constant C , such that if u = S ([f ; a; b]) is the solution to (2.2), then ~a;b;n~ ; (v; w)j jf (w) ? f~n~ ; (w)j j B ( v; w ) ? B a;b + kwk 1 C inf ku ? vkH1 ( ) + sup ; kwkH1( ) v2Sn~ w2Sn~ H ( ) the constant C being independent of n, , and [f ; a; b].
ku ? u~Qn~ kH1 ( )
We now recall some preliminary error estimates, whose proofs may be found in [13]. Lemma 4.3. There exists a positive constant C , depending only on m, M , 0 , and r, such that jBa;b;n~ (v; w) ? B~a;b;n~; (v; w)j CkvkH1( ) kwkH1( ) 8 v; w 2 Sn~ and
jfn~ (v) ? f~n~ ; (v)j CkvkL2( ) 8 v 2 Sn~ ; for any > 0, any [f ; a; b] 2 F , and any n n~ , where n~ satis es (3.4).
We are now ready to prove the main result of this section. Here, and in the remainder of this paper, C will denote a generic constant that depends only on m, M , 0 , , and r, but whose value may change from place to place. Theorem 4.1. There exist n 2 Z++ and 0 > 0 such that ~n; is well-de ned for all n n and all 2 [0; 0]. Furthermore, there exists a constant C such that e( ~n; ; N~ n;) C (n?=d + );
where
= minfk; rg:
Proof: We rst show that ~n; is well-de ned. As in [12, pg. 106], there exists a positive constant C
such that
jBa;b(v; w) ? Ba;b;n~ (v; w)j Cn?=d kvkH1( ) kwkH1( ) 8 v; w 2 Sn~ for any n 2 Z++. Using the rst inequality in Lemma 4.3, we have jBa;b(v; w) ? B~a;b;n~; (v; w)j C (n?=d + )kvkH1( )kwkH1 ( ) 8 v; w 2 Sn~ (4.8) for any n 2 Z++ and any 2 [0; 1]. It now follows that there exists 0 2 (0; 1] and n 2 Z++ such that (4.7) holds for any 2 [0; 0], any n n and any (a; b) 2 A . Thus Strang's Lemma implies that the noisy FEMQ ~n; is well-de ned for any such and n. We now turn to the error of the FEMQ. Let 2 [0; 0] and n n . For [f ; a; b] 2 F , let u = S ([f ; a; b]). From [12, pg. 107], there exists v 2 Sn~ such that ku ? vkH1( ) Cn?=dkf kHr ( ) Cn?=d; 11
(4.9)
the latter since there exists a positive constant C for which
kf kHr ( ) C kf kW r;p( ) C:
(4.10)
Now for any w 2 Sn~ , we nd from [12, pg. 106] that
jf (w) ? fn~ (w)j Cn?=d kf kHr ( )kwkH1 ( ) Cn?=dkwkH1 ( ); where we have again used (4.10). Using this inequality and the second inequality in Lemma 4.3, we have jf (x) ? f~n~ ;(w)j C (n?=d + )kwkH1( ) : (4.11): Use (4.8), (4.11), and (4.9) in Strang's Lemma. Since , we nd
ku ? u~Qn~ kH1( ) C (n?=d + );
as required. Using Theorem 4.1, we nd Corollary 4.1.
(1) rn ( ) = (n?r=d + ). (2) The noisy FEMQ, using a quadrature rule that is exact of degree at least 2k + r ? 1, is a minimal error algorithm if k r. (3) Adaption is no stronger than non-adaption.
Proof: We only need to prove the lower bound
rn () = (n?r=d + ): But the proof of this bound is exactly the same as the analogous lower bound in [13].
5. Multigrid Implementation of the Noisy FEMQ
We have shown that the the FEMQ algorithm ~n; is an nth minimal error algorithm. Said algorithm consists of three steps. First, we evaluate n information samples (values of f , a, or b). Next, we use these values to construct the n~ n~ linear system Ga = b. So far, the cost of the algorithm is (n). The nal step is solving the linear system Ga = b. Unfortunately, we do not know how to solve the linear system in time (n). For this reason, we will pursue a multigrid implementation of the noisy FEMQ for inde nite ellipitc problems, which is an nth minimal error algorithm whose running time is (n). This multigrid method is similar to that in our previous paper [13], the di erence being that we use the inner multigrid step of [3] (which has been crafted to work with inde nite problems). Nonetheless, we give a complete description of the method, to help keep this paper reasonably self-contained. As in [13], we use the notation of [4, Chapter 6]. Recall that fTn g1 n=1 is a quasi-uniform grid sequence. Let us write
hj = Kmax h 2T K j
for the meshsize of Tj . Recall (from Theorem 4.1) that the noisy FEMQ ~n; is well-de ned if n n. Let n1 = n < n2 < < nl?1 < nl 12
be a sequence of integers, chosen so that
Tnj?1 Tnj , and thus Snj?1 Snj and
(5.1) hnj 21 hnj?1 (2 j l): We let j be xed, but arbitrary, index in f1; : : :; lg. If p1 ; : : :; pnj are the interior nodes of the triangulation Tnj , then we get the standard nite element basis fs1 ; : : :; snj g for Snj by requiring that si (pi0 ) = i;i0 for 1 i; i0 nj (see, e.g., the discussion in [12, Sections 5.7 and A.2.3]). We de ne a mesh-dependent inner product h; ij on Snj by nj X d hv; wij = hnj v(pi)w(pi) i=1
8 v; w 2 Snj :
Then the operator Aj on Snj is de ned by
hAj v; wij = B~a;b;nj ; (v; w)
8 v; w 2 Snj :
We also de ne the operator A^j on Snj by
hA^j v; wij = B^a;b;nj ; (v; w)
8 v; w 2 Snj :
Here, the bilinear form B^a;b;nj ; on Snj is de ned to be
B^a;b;nj ; (v; w) =
J X d X X K 2Tnj j =1 i=1
a~j;K; (@iv)(xj;K ) (@iw)(xj;K) + v(xj;K ) w(xj;K ) :
That is, B^a;b;nj ; (v; w) approximates
B^a (v; w) =
Z
[arv rw + vw]
using numerical quadrature and noisy information about a. Note that B^a is a uniformly strongly coercive bilinear form on H01 ( ), as are the forms B^a;b;nj ; for 1 j l. Following the proof of [4, Lemma 6.2.8], we obtain an upper bound
(A^j ) j = Ch?nj2
(5.2)
on the spectral radius of A^j , where the constant C is independent of the index j and the coecient vector (a; b). Let us de ne fj 2 Snj by requiring that
hfj ; sij = f~nj (s) 8 s 2 Snj and let us write u~j for the solution u~j = u~Qnj of the noisy FEMQ for Snj , so that Aj u~j = fj : 13
We then let Ijj?1 : Snj?1 ! Snj be the natural embedding, and let Ijj ?1 : Snj ! Snj?1 be its adjoint, i.e., hIjj?1w; vij?1 = hw; Ijj?1vij = hw; vij 8 v 2 Snj?1 ; w 2 Snj : Recalling that j is an upper bound on (A^j ), we now de ne the j th-level multigrid iteration recursively, in terms of the multigrid iterations at lower levels: function MG(j : Z+; z0; g : Snj ): Snj ;
begin if k = 1 then
MG := A?1 1 g
else begin
z1 := z0 + ?j 1(g ? Aj z0 ); f pre-smoothing g g := Ijj?1 (g ? Aj z1); f ne-to-coarse intergrid transfer g q1 := MG(j ? 1; 0; g); f error correcting g z2 := z1 + Ijj?1 q1 ; f coarse-to- ne intergrid transfer g z3 := z2 + j?1(g ? Aj z2 ); f post-smoothing g
end; MG := z3
end
Then for any index t, the t-fold full multigrid scheme produces an approximation u^j to u~j as follows: function FMG(j; t : Z+): Snj ;
begin if j = 1 then ?1
u^j := A1 f1
else begin j
u0 := Ijj?1 u^j?1 ; for ji := 1 to t do ui := MG(j; uji?1 ; fj ); u^j := ujr
end; FMG := u^j
end Let
Nn; = [N~ n1; ; N~ n2; ; : : :; N~ nl; ]; with l the maximal index for which card Nn; n. Then we may write
?
u^l = n; Nn; ([f ; a; b]) ; where n; is the full multigrid algorithm. Note that the main di erence between this multigrid scheme and the one used in [13] is that we use di erent smoothing operators. The main result for this section is 14
Theorem 5.1.
(1) The full multigrid algorithm is well-de ned. (2) There exits an index t such that the error of the full multigrid algorithm is
e( n; ; Nn; ) = O(n?=d ); where (as in Theorem 4.1)
= minfk; rg:
(3) The combinatory cost of the full multigrid scheme FMG(l; t) is (n). Here, the O- and - constants depend only on m, M , 0 , , and r. Proof: The well-de nedness follows from Theorem 4.1. To prove the desired error estimate, let us rst consider the j th-level multigrid iteration. Let k kEj be the energy norm de ned by
kvkEj = B^a;b;nj ;(v; v)1=2; this energy norm being equivalent to the usual H01 ( )-norm, the equivalency constants depending only on m, M , 0 , , and r. Since the pre-smoothing and post-smoothing operators for the j th-level multigrid step satisfy the hypotheses of [3], it follows that there exists a constant 2 (0; 1) such that (5.3) kz ? MG(j; z0; g)kEj kz ? z0 kEj ; the constant being independent of g; z; z0 2 Snj , j 2 Z+, and [f ; a; b] 2 F . Since a j th-level multigrid iteration reduces the error by a constant factor , the error estimate for the full multigrid algorithm follows, exactly as shown in [13, Theorem 5.1]. To determine the combinatory cost of the full multigrid scheme, we rst determine the cost of the j th-level scheme. As in [13, Theorem 5.1], we nd that the cost of the j th-level scheme is O(nj ), and that the cost of the full multigrid scheme is O(n). 6. Complexity
In this Section, we determine the complexity of the noisy inde nite elliptic problem. It will be useful to explicitly specify some of the order-of-magnitude constants in some of the estimates in the previous sections. Thus, Corollary 4.1 tells us that there exists a positive constant C1 such that rn () C1 (n?r=d + ): (6.1) Moreover, let ~n; be the noisy FEMQ of degree k r, using a quadrature rule that is exact of degree at least 2k + r ? 1. Then by Theorem 5.1, there exist positive constants C2 and C3 = C3 (g) such that e( n; ; Nn; ) C2 (n?r=d + ) (6.2) and
cost( n; ; Nn; ) C3 c( )n:
As in [13, Theorem 7.1], we have 15
(6.3)
Theorem 6.1. The problem complexity is bounded from below by (
comp(") inf c( )
&
C1?1 " ?
>0
and from above by
1
(
comp(") C3 inf c( )
&
d=r ')
;
(6.4)
d=r ')
1
: C2?1 " ? The upper bound is attained by using the noisy FEMQ n; described above, with >0
n=
&
1
C2?1 " ?
d=r '
;
(6.5)
(6.6)
and with chosen minimizing (6.5). Hence ( d=r )! 1 comp(") = inf c( ) C ?1 " ? ;
(6.7)
>0
for some constant C . This allows us to determine the complexity for various cost functions c(), as well as to nd the optimal noise level for a particular value of ". For example, suppose that c( ) = ?s , where s > 0. Then for any " > 0, the optimal is = C (rsrs"+ d) = ("); so that
comp(") =
d sr
s
C (rs + d) d"
d=r+s !
Bibliography
=
d=r+s!
1
"
:
1. Babuska, I. and Aziz, A. K., Survey lectures on the mathematical foundations of the nite element method, The Mathematical Foundations of the Finite Element Method With Applications to Partial Di erential Equations, (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 3{359. 2. Bramble, J., Multigrid Methods, Pitman Research Notes in Mathematics, Vol. 294, Wiley, New York, 1993. 3. Bramble, J. H., Kwak, D. Y., and Pasciak, J. E., Uniform convergence of multigrid V -cycle iterations for inde nite and nonsymmetric problems, SIAM J. Numer. Anal. 31 (1994), 1746{1764. 4. Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer, New York, 1994. 5. Ciarlet, P. G., The Finite Element Method For Elliptic Problems, North-Holland, Amsterdam, 1978. 6. Coulton, D. and Monk, P., On a class of integral equations of the rst kind in inverse scattering theory, SIAM J. Appl. Math. 53 (1993 3), 847{860. 7. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Di erential Equations of Second Order, Springer-Verlag, Berlin, 1983. 8. Oden, J. T. and Carey, G. F., Finite elements: Mathematical aspects, Prentice-Hall, Englewood Cli s, 1983. 9. Plaskota, L., Noisy Information and Computational Complexity, Cambridge University Press, Cambridge, 1996. 10. Plaskota, L., Worst case complexity of problems with random information noise, J. Complexity (1996) (to appear). 11. Traub, J. F., Wasilkowski, G. W., and Wozniakowski, H., Information-Based Complexity, Academic Press, New York, 1988. 12. Werschulz, A. G., The Computational Complexity of Di erential and Integral Equations: An InformationBased Approach, Oxford University Press, Oxford, 1991. 13. , The complexity of de nite elliptic problems with noisy data, J. Complexity 12 (1996), 440{473. 16
r=d
s
d=r
s
1. Introduction
We study the complexity of elliptic partial di erential equations with noisy data. To ease notational burdens, we will consider a class of second-order elliptic problems, although the results in this paper may be extended to include elliptic problems of arbitrary order. To be speci c, we will consider (the variational form of) second-order elliptic problems
? div(aru) + bu = f u=0
in ; on @ :
(1.1)
Here, is a xed d-dimensional domain. We need a speci c way to measure the error of an approximation; in this paper, we choose to measure error in the Sobolev H 1 -norm, which is equivalent to the problem's natural energy norm. However, the results of this paper also hold if error is measured in the L2 -norm. Typically, we only know the values of a, b, and f at a nite number of points in . Hence, this is a problem of information-based complexity (IBC) [11]. We cannot obtain exact solutions with nite cost; we must settle for an "-approximation, i.e., an algorithm computing (for each choice of a, b, and f ) an approximation whose error is at most ". For any positive error level ", we want to know the "-complexity (the minimal cost of calculating an "-approximation), and we want to This research was supported in part by the National Science Foundation under Grant CCR-95-00850.
nd algorithms that are optimal, in the sense that they calculate an "-approximation at (nearly) minimal cost. Most earlier work on the complexity of elliptic problems has assumed that exact information is available, i.e., we calculate the values these coecient functions or of the right-hand side exactly. (An extensive treatment of the topic may be found in [12].) What happens if the information is contaminated by noise? Suppose that we can only guarantee that these calculated function values are accurate to within some positive noise level . What can we say about the complexity of such problems? The systematic study of complexity for problems with noisy information was initiated by [9], with noisy elliptic problems rst being investigated in [13]. Let us recall the main results of [13]. If the problem is a de nite elliptic problem (i.e., a is bounded away from zero and b is non-negative), then the nth minimal radius (the minimal error attained using n evaluations) is (n?r=d + ). This error is attained by a nite element method using quadrature (FEMQ) of degree r (or greater). Now let c() be the cost of obtaining a -noisy approximation of a function value. Then the "-complexity is (
comp(") =
1
c() C ?1 " ?
inf
0 0 for suciently small positive . (2) Real arithmetic operations and comparisons are done exactly, with unit cost. (3) We are not charged for Boolean operations. (4) Linear operations over H01 ( ) are done exactly, with cost g. For any noisy information N and any algorithm using N, we shall let cost( ; N) denote the ? worst case cost of calculating N ([f ; a; b]) over all [f ; a; b] 2 F . Now that we have de ned the error and cost of an algorithm, we can nally de ne the complexity of our problem. We shall say that comp(") = inf f cost( ; N) : N and such that e( ; N) " g is the "-complexity of our problem. An algorithm using noisy information N for which
e( ; N) "
and
?
cost( ; N) = comp(")
is said to be an optimal algorithm. 3. The Noisy FEMQ
In this section, we de ne the noisy nite element method with quadrature (FEMQ). This is an algorithm using standard information consisting only of function evaluations, i.e., no derivative evaluations are used. Our notation is the standard one found in, e.g., [5] and [12, Chapter 5]. This noisy FEMQ is the same as described in [13]. However, we will give a complete description of the noisy FEMQ, so that this paper can be reasonably self-contained. We rst establish some notation. Let K^ be a xed polyhedron in Rd. We call K^ a reference element. Select a xed value of k 2 Z++, and let Pk (K^ ) denote the space of polynomials having total degree at most k, considered as functions over K^ . We next let K be a (small) nite element, i.e., the ane image of K^ under a bijection FK , where ^ FK (^x) = BK x^ + xK 8 x^ 2 K; (3.1) where BK 2 Rdd is invertible and xK 2 Rd. Next, we let T be a triangulation of consisting of nite elements, where each K 2 T is the image of the reference element K^ under the ane bijection FK . Given this triangulation T , we de ne a nite element space
S (T ) = s 2 H01( ) : s K 2 Pk (K ) 8 K 2 T 5
of degree k. We will assume that the following conditions hold: (1) fTn g1 n=1 is a family of triangulations of such that Sn = S (Tn ) is a nite element space of dimension n. (2) fTn g1 n=1 is a quasi-uniform family of triangulations, i.e., lim sup sup hK < 1; n!1 K 2Tn K where hK is the diameter of K and K is the diameter of the largest sphere contained in K . (3) Let k k denote the `2 matrix norm on Rd. Then kBK k 1 for any element K 2 Tn and any triangulation Tn . We rst recall how the noise-free \pure" FEM is de ned. Let n 2 Z+, and let fs1 ; : : :; sn g be a basis for Sn . For [f ; a; b] 2 F , nd n X u n = j s j ; in Sn such that
j =1
Ba;b (un; si ) = hf; sii
(1 i n): (3.2) If we approximate the integrals appearing in the FEM by numerical quadrature, we get the (noisefree) FEMQ. The quadrature rule used to de ne the FEMQ is initially de ned on the reference element. This reference quadrature rule has the form
I^v^ =
J X j =1
!^j v^(^xj )
for functions v^ de ned on K^ . Said rule is said to be exact of degree q if Z
K^
v^ = I^v^
8 v^ 2 Pq (K^ ):
We de ne a local quadrature rule over a particular nite element K as
IK v =
J X j =1
!j;K v(xj;K );
where
!j;K = det BK !^j and xj;K = FK (^xj ) (1 j J ) for K = FK (K^ ), with FK given by (3.1). Next, for any ` 2 Z+, we let
N` =
J [ [ K 2T` j =1
(3.3)
fxj;K g
denote the set of all quadrature nodes in all the elements belonging to T` . This is usually not a disjoint union, since a quadrature node on the boundary of one element will be on the boundary of an adjacent element sharing a common face. 6
Let
= minfk + 1; rg:
We now assume that the following conditions hold: (1) The smoothness r of the problem elements F satis es r 1 (as well as our previous requirement r > d=p). (2) The degree k of the nite element subspaces Sn~ satis es k > d=p ? 1. (3) I^ is exact of degree 2k + ? 1 over the reference element K^ . We can now de ne the noise-free FEMQ. Given n 2 Z+, we de ne
n~ = maxf card N` : ` 2 Z+ and card N` 31 n g:
(3.4)
Roughly speaking, n~ = b 13 nc, allowing for the fact that n~ must be the cardinality of the set N` of quadrature nodes for some triangulation T` . Let fs1 ; : : :; sn~ g denote a basis for the nite element space Sn~ . For [f ; a; b] 2 F , we de ne a new bilinear form Ba;b;n~ on Sn~ by X
d X
IK (a @i v @j w) + IK (bvw) K 2Tn~ i;j =1 J X d X X = a(xj;K ) (@i v)(xj;K ) (@iw)(xj;K ) + b(xj;K ) v(xj;K ) w(xj;K ) K 2Tn~ j =1 i=1
Ba;b;n~ (v; w) =
and a linear functional fn~ on Sn~ by X
J X X
!j;K f (xj;K ) v(xj;K ) K 2Tn~ j =1 Then the noise-free FEMQ consists of nding uQn~ 2 H01 ( ) such that fn~ (v) =
K 2Tn~
IK (fv) =
Ba;b;n~ (uQn~ ; si) = fn~ (si)
(1 i n):
(3.5)
We are ready to de ne the noisy FEMQ. Given n 2 Z+, we again choose the largest n~ 2 Z+ satisfying (3.4), and a basis fs1 ; : : :; sn~ g for the nite element space Sn~ . We now calculate a noisy version of the information that would be used by the noise-free FEMQ. That is, for each element K 2 Tn~ and each index j 2 f1; : : :; J g, we obtain real numbers a~j;K; , ~bj;K; and f~j;K; satisfying
ja~j;K; ? a(xj;K )j ; j~bj;K; ? b(xj;K )j ; jf~j;K; ? f (xj;K )j : Let where
N~ n; ([f ; a; b]) = fN~ n;(f ); N~ n;(a); N~ n;(b)g N~ n;(a) = fa~j;K; satisfying (3.6) : 1 j J and K 2 Tn~ g; N~ n;(b) = f~bj;K; satisfying (3.7) : 1 j J and K 2 Tn~ g; N~ n;(f ) = ff~j;K; satisfying (3.8) : 1 j J and K 2 Tn~ g; 7
(3.6) (3.7) (3.8)
Clearly, N~ n; is noisy information of cardinality at most n. For [f ; a; b] 2 F , we de ne a new bilinear form B~a;b;n~ ; on Sn~ by
B~a;b;n~ ; (v; w) =
J X d X X K 2Tn~ j =1 i=1
~aj;K; (@i v )(xj;K ) (@i w)(xj;K ) + ~bj;K; v (xj;K ) w(xj;K )
and a linear functional f~n~ ; on Sn~ by
f~n~ ; (v) =
J X X K 2Tn~ j =1
Then we seek
u~Qn~ = such that The coecient vector satis es where and
!j;K f~j;K; v(xj;K ): n~ X j =1
j sj
B~a;b;n~ ; (~uQn~ ; si) = f~n~ ; (si )
(1 i n):
(3.9)
a = [ 1 ; : : :; n]T Ga = b; G = [B~a;b;n~ ; (si; sj )]1i;jn~
b = [f~n~ ; (s1 ); : : :; f~n~; (sn~ )]T : We see that u~Qn~ depends on [f ; a; b] only through the noisy information N~ n; ([f ; a; b]), and so we write u~Qn~ = ~n; (N~ n;([f ; a; b])), with ~n; an algorithm using our noisy standard information N~ n; . 4. The Noisy FEMQ is a Minimal Error Algorithm
In this section, we prove that the noisy FEMQ is well-de ned. We also establish an error bound for the noisy FEMQ, which allows us to show that the FEMQ is a minimal error algorithm whenever k r. First, we note that the conditions de ning A imply several important inequalities, which hold independently of (a; b) 2 A : Lemma 4.1.
(1) Garding's inequality: There exists a constant 0 = 0;m > 0 such that
Ba;b(v; v) 0 kvk2H 1 ( ) ? 0 kvk2L2 ( )
8 v 2 H01( );
for all (a; b) 2 A . (2) Weak coercivity of Ba;b : There exists a constant 1 = 1;m;M;0 ; > 0 such that
8 v 2 H01( ); 9 nonzero w 2 H01( ) : jBa;b(v; w)j 1kvkH1( )kwkH1 ( ); 8
for all (a; b) 2 A . (3) Shift theorem: If f 2 H r ( ), then there exists a constant = m;M;0 ;;r such that
?1 kS ([f ; a; b])kHr+2 ( ) kf kH r ( ) kS ([f ; a; b])kH r+2 ( ) ;
(4.1)
for all (a; b) 2 A . (4) Uniform weak coercivity of Ba;b : There exists a constant = m;M;0; > 0 and an index n 2 Z+ such that if n n, then
8 v 2 Sn~ ; 9 w 2 Sn~ : jBa;b(v; w)j kvkH1( ) kwkH1( ) ; for any (a; b) 2 A .
(4.2)
Proof: If we use the inequality
Ba;b(v; v) mjvj2H 1 ( ) ? 0 kvk2L2 ( );
(4.3)
and the Poincare inequality j jH 1 ( ) C k kH 1 ( ) , we see that Garding's inequality follows, with
0 = C m. We next consider weak coercivity. Let v 2 H01 ( ). Since 0 is not an eigenvalue of La;b , there exists a unique z 2 H 3 ( ) satisfying
La;bz = 0v in ; z = 0 on @ :
(4.4)
Ba;b (v; z) = 0 kvk2L2 ( ):
(4.5)
Ba;b (v; w) mjvj2H 1 ( ) 0 kvk2H 1 ( ) :
(4.6)
Then
Now take w = v + z . Using (4.3), (4.5), and the Poincare inequality, we have Let fzi gi2Z ++ be an orthonormal H01( ) basis satisfying
La;b0 zi = izi in ; z = 0 on @ : If we expand
v= we nd that It now follows that Thus
z=
kzkH1 ( )
sup i2Z++
1 X i=1
i zi;
1 X
i 0 z : i i=1 i ?
0 kvk 1 0 kvk 1 : i ? H ( ) H ( )
0 kwkH1( ) kvkH1( ) + kzkH1 ( ) 1 + kvkH1( ) :
9
Using this inequality along with (4.6), we have
0
Ba;b(v; w) + kvkH 1 ( ) kwkH1 ( ) ; 0
as required. We next consider the uniform shift theorem. Let [f ; a; b] 2 F and let u = S ([f ; a; b]). Using the weak coercivity of Ba;b , we nd that there exists nonzero w 2 H01 ( ) such that Ba;b (u; w) 1 kukH 1 ( ) kwkH 1 ( ): Since Ba;b (u; w) = hf; wiL2( ) , we nd that
1kukH 1( ) kwkH 1 ( ) Ba;b (u; w) kf kH ?1 ( ) kwkH 1 ( ); and so
kukL2( ) kukH1( ) 1?1kf kH?1( ) :
From [7, Theorem 8.8], we have the a priori inequality kukHr+2 ( ) C (kf kHr ( ) + kukL2( ) ); the constant C depending only on m, M , 0 , , and r. Using these last two inequalities, we see 0 that there exists 0 = m;M; 0 ;;r such that The reverse inequality
kukHr+2( ) 0kf kHr ( ) :
kf kHr ( ) 00 kukHr+2( ) ;
00 with 00 = m;M; 0 ;;r , follows easily from the conditions de ning the class A . Combining these last two inequalities, we obtain the uniform shift theorem. To check uniform weak coercivity, we let : H01 ( ) ! Sn~ denote the Sn~ -interpolation operator given by
(v )(x) =
n~ X j =1
v(xj )sj
8 v 2 H01( ):
Here x1 ; : : :; xn~ are the interior nodes of the triangulation Tn~ , and fs1 ; : : :; sn~ g is the usual dual nite element basis de ned by si (xj ) = i;j (1 i; j n~): From the usual nite element approximation theory, there is a positive constant C1 such that kz ? zkH1 ( ) C1n~?=d kzkH3( ) for any z 2 H01 ( ) \ H 3 ( ). Now choose (a; b) 2 A . From the conditions on A , there is another positive constant C2, depending only on M and 0 , such that jBa;b(v; w)j C2kvkH1( ) kwkH1( ) 8 v; w 2 H01( ): Let m l n = (C1 C2 1 0?1)?d= : For v 2 Sn~ , we let w = v + z , where z 2 H 3 ( ) satis es (4.4). Following [12, Lemmas 5.2.1 and 5.4.4], we nd that w is nonzero and that Ba;b (v; w) 1 + + C 2 (n )?=d kvkH 1 ( )kwkH 1 ( ) : 0 1 0 Hence the problem is uniform weakly coercive. Our main tool is Strang's Lemma (see [12, pp. 310{312] for a proof of a version having slightly more restrictive hypotheses). 10
Lemma 4.2. Suppose that there exists 0 2 (0; 1] and n 2 Z++ such that for any 2 [0; 0], any n n and any (a; b) 2 A , we have
(4.7) jBa;b(v; w) ? B~a;b;n~ ;(v; w)j 21 kvkH1( )kwkH1 ( ) 8 v; w 2 Sn~ ; where is as in (4.2). Then for any n n , any 2 [0; 0], and any [f ; a; b] 2 F , there is a unique u~Qn~ 2 Sn~ such that (3.9) holds. Moreover, there exists a positive constant C , such that if u = S ([f ; a; b]) is the solution to (2.2), then ~a;b;n~ ; (v; w)j jf (w) ? f~n~ ; (w)j j B ( v; w ) ? B a;b + kwk 1 C inf ku ? vkH1 ( ) + sup ; kwkH1( ) v2Sn~ w2Sn~ H ( ) the constant C being independent of n, , and [f ; a; b].
ku ? u~Qn~ kH1 ( )
We now recall some preliminary error estimates, whose proofs may be found in [13]. Lemma 4.3. There exists a positive constant C , depending only on m, M , 0 , and r, such that jBa;b;n~ (v; w) ? B~a;b;n~; (v; w)j CkvkH1( ) kwkH1( ) 8 v; w 2 Sn~ and
jfn~ (v) ? f~n~ ; (v)j CkvkL2( ) 8 v 2 Sn~ ; for any > 0, any [f ; a; b] 2 F , and any n n~ , where n~ satis es (3.4).
We are now ready to prove the main result of this section. Here, and in the remainder of this paper, C will denote a generic constant that depends only on m, M , 0 , , and r, but whose value may change from place to place. Theorem 4.1. There exist n 2 Z++ and 0 > 0 such that ~n; is well-de ned for all n n and all 2 [0; 0]. Furthermore, there exists a constant C such that e( ~n; ; N~ n;) C (n?=d + );
where
= minfk; rg:
Proof: We rst show that ~n; is well-de ned. As in [12, pg. 106], there exists a positive constant C
such that
jBa;b(v; w) ? Ba;b;n~ (v; w)j Cn?=d kvkH1( ) kwkH1( ) 8 v; w 2 Sn~ for any n 2 Z++. Using the rst inequality in Lemma 4.3, we have jBa;b(v; w) ? B~a;b;n~; (v; w)j C (n?=d + )kvkH1( )kwkH1 ( ) 8 v; w 2 Sn~ (4.8) for any n 2 Z++ and any 2 [0; 1]. It now follows that there exists 0 2 (0; 1] and n 2 Z++ such that (4.7) holds for any 2 [0; 0], any n n and any (a; b) 2 A . Thus Strang's Lemma implies that the noisy FEMQ ~n; is well-de ned for any such and n. We now turn to the error of the FEMQ. Let 2 [0; 0] and n n . For [f ; a; b] 2 F , let u = S ([f ; a; b]). From [12, pg. 107], there exists v 2 Sn~ such that ku ? vkH1( ) Cn?=dkf kHr ( ) Cn?=d; 11
(4.9)
the latter since there exists a positive constant C for which
kf kHr ( ) C kf kW r;p( ) C:
(4.10)
Now for any w 2 Sn~ , we nd from [12, pg. 106] that
jf (w) ? fn~ (w)j Cn?=d kf kHr ( )kwkH1 ( ) Cn?=dkwkH1 ( ); where we have again used (4.10). Using this inequality and the second inequality in Lemma 4.3, we have jf (x) ? f~n~ ;(w)j C (n?=d + )kwkH1( ) : (4.11): Use (4.8), (4.11), and (4.9) in Strang's Lemma. Since , we nd
ku ? u~Qn~ kH1( ) C (n?=d + );
as required. Using Theorem 4.1, we nd Corollary 4.1.
(1) rn ( ) = (n?r=d + ). (2) The noisy FEMQ, using a quadrature rule that is exact of degree at least 2k + r ? 1, is a minimal error algorithm if k r. (3) Adaption is no stronger than non-adaption.
Proof: We only need to prove the lower bound
rn () = (n?r=d + ): But the proof of this bound is exactly the same as the analogous lower bound in [13].
5. Multigrid Implementation of the Noisy FEMQ
We have shown that the the FEMQ algorithm ~n; is an nth minimal error algorithm. Said algorithm consists of three steps. First, we evaluate n information samples (values of f , a, or b). Next, we use these values to construct the n~ n~ linear system Ga = b. So far, the cost of the algorithm is (n). The nal step is solving the linear system Ga = b. Unfortunately, we do not know how to solve the linear system in time (n). For this reason, we will pursue a multigrid implementation of the noisy FEMQ for inde nite ellipitc problems, which is an nth minimal error algorithm whose running time is (n). This multigrid method is similar to that in our previous paper [13], the di erence being that we use the inner multigrid step of [3] (which has been crafted to work with inde nite problems). Nonetheless, we give a complete description of the method, to help keep this paper reasonably self-contained. As in [13], we use the notation of [4, Chapter 6]. Recall that fTn g1 n=1 is a quasi-uniform grid sequence. Let us write
hj = Kmax h 2T K j
for the meshsize of Tj . Recall (from Theorem 4.1) that the noisy FEMQ ~n; is well-de ned if n n. Let n1 = n < n2 < < nl?1 < nl 12
be a sequence of integers, chosen so that
Tnj?1 Tnj , and thus Snj?1 Snj and
(5.1) hnj 21 hnj?1 (2 j l): We let j be xed, but arbitrary, index in f1; : : :; lg. If p1 ; : : :; pnj are the interior nodes of the triangulation Tnj , then we get the standard nite element basis fs1 ; : : :; snj g for Snj by requiring that si (pi0 ) = i;i0 for 1 i; i0 nj (see, e.g., the discussion in [12, Sections 5.7 and A.2.3]). We de ne a mesh-dependent inner product h; ij on Snj by nj X d hv; wij = hnj v(pi)w(pi) i=1
8 v; w 2 Snj :
Then the operator Aj on Snj is de ned by
hAj v; wij = B~a;b;nj ; (v; w)
8 v; w 2 Snj :
We also de ne the operator A^j on Snj by
hA^j v; wij = B^a;b;nj ; (v; w)
8 v; w 2 Snj :
Here, the bilinear form B^a;b;nj ; on Snj is de ned to be
B^a;b;nj ; (v; w) =
J X d X X K 2Tnj j =1 i=1
a~j;K; (@iv)(xj;K ) (@iw)(xj;K) + v(xj;K ) w(xj;K ) :
That is, B^a;b;nj ; (v; w) approximates
B^a (v; w) =
Z
[arv rw + vw]
using numerical quadrature and noisy information about a. Note that B^a is a uniformly strongly coercive bilinear form on H01 ( ), as are the forms B^a;b;nj ; for 1 j l. Following the proof of [4, Lemma 6.2.8], we obtain an upper bound
(A^j ) j = Ch?nj2
(5.2)
on the spectral radius of A^j , where the constant C is independent of the index j and the coecient vector (a; b). Let us de ne fj 2 Snj by requiring that
hfj ; sij = f~nj (s) 8 s 2 Snj and let us write u~j for the solution u~j = u~Qnj of the noisy FEMQ for Snj , so that Aj u~j = fj : 13
We then let Ijj?1 : Snj?1 ! Snj be the natural embedding, and let Ijj ?1 : Snj ! Snj?1 be its adjoint, i.e., hIjj?1w; vij?1 = hw; Ijj?1vij = hw; vij 8 v 2 Snj?1 ; w 2 Snj : Recalling that j is an upper bound on (A^j ), we now de ne the j th-level multigrid iteration recursively, in terms of the multigrid iterations at lower levels: function MG(j : Z+; z0; g : Snj ): Snj ;
begin if k = 1 then
MG := A?1 1 g
else begin
z1 := z0 + ?j 1(g ? Aj z0 ); f pre-smoothing g g := Ijj?1 (g ? Aj z1); f ne-to-coarse intergrid transfer g q1 := MG(j ? 1; 0; g); f error correcting g z2 := z1 + Ijj?1 q1 ; f coarse-to- ne intergrid transfer g z3 := z2 + j?1(g ? Aj z2 ); f post-smoothing g
end; MG := z3
end
Then for any index t, the t-fold full multigrid scheme produces an approximation u^j to u~j as follows: function FMG(j; t : Z+): Snj ;
begin if j = 1 then ?1
u^j := A1 f1
else begin j
u0 := Ijj?1 u^j?1 ; for ji := 1 to t do ui := MG(j; uji?1 ; fj ); u^j := ujr
end; FMG := u^j
end Let
Nn; = [N~ n1; ; N~ n2; ; : : :; N~ nl; ]; with l the maximal index for which card Nn; n. Then we may write
?
u^l = n; Nn; ([f ; a; b]) ; where n; is the full multigrid algorithm. Note that the main di erence between this multigrid scheme and the one used in [13] is that we use di erent smoothing operators. The main result for this section is 14
Theorem 5.1.
(1) The full multigrid algorithm is well-de ned. (2) There exits an index t such that the error of the full multigrid algorithm is
e( n; ; Nn; ) = O(n?=d ); where (as in Theorem 4.1)
= minfk; rg:
(3) The combinatory cost of the full multigrid scheme FMG(l; t) is (n). Here, the O- and - constants depend only on m, M , 0 , , and r. Proof: The well-de nedness follows from Theorem 4.1. To prove the desired error estimate, let us rst consider the j th-level multigrid iteration. Let k kEj be the energy norm de ned by
kvkEj = B^a;b;nj ;(v; v)1=2; this energy norm being equivalent to the usual H01 ( )-norm, the equivalency constants depending only on m, M , 0 , , and r. Since the pre-smoothing and post-smoothing operators for the j th-level multigrid step satisfy the hypotheses of [3], it follows that there exists a constant 2 (0; 1) such that (5.3) kz ? MG(j; z0; g)kEj kz ? z0 kEj ; the constant being independent of g; z; z0 2 Snj , j 2 Z+, and [f ; a; b] 2 F . Since a j th-level multigrid iteration reduces the error by a constant factor , the error estimate for the full multigrid algorithm follows, exactly as shown in [13, Theorem 5.1]. To determine the combinatory cost of the full multigrid scheme, we rst determine the cost of the j th-level scheme. As in [13, Theorem 5.1], we nd that the cost of the j th-level scheme is O(nj ), and that the cost of the full multigrid scheme is O(n). 6. Complexity
In this Section, we determine the complexity of the noisy inde nite elliptic problem. It will be useful to explicitly specify some of the order-of-magnitude constants in some of the estimates in the previous sections. Thus, Corollary 4.1 tells us that there exists a positive constant C1 such that rn () C1 (n?r=d + ): (6.1) Moreover, let ~n; be the noisy FEMQ of degree k r, using a quadrature rule that is exact of degree at least 2k + r ? 1. Then by Theorem 5.1, there exist positive constants C2 and C3 = C3 (g) such that e( n; ; Nn; ) C2 (n?r=d + ) (6.2) and
cost( n; ; Nn; ) C3 c( )n:
As in [13, Theorem 7.1], we have 15
(6.3)
Theorem 6.1. The problem complexity is bounded from below by (
comp(") inf c( )
&
C1?1 " ?
>0
and from above by
1
(
comp(") C3 inf c( )
&
d=r ')
;
(6.4)
d=r ')
1
: C2?1 " ? The upper bound is attained by using the noisy FEMQ n; described above, with >0
n=
&
1
C2?1 " ?
d=r '
;
(6.5)
(6.6)
and with chosen minimizing (6.5). Hence ( d=r )! 1 comp(") = inf c( ) C ?1 " ? ;
(6.7)
>0
for some constant C . This allows us to determine the complexity for various cost functions c(), as well as to nd the optimal noise level for a particular value of ". For example, suppose that c( ) = ?s , where s > 0. Then for any " > 0, the optimal is = C (rsrs"+ d) = ("); so that
comp(") =
d sr
s
C (rs + d) d"
d=r+s !
Bibliography
=
d=r+s!
1
"
:
1. Babuska, I. and Aziz, A. K., Survey lectures on the mathematical foundations of the nite element method, The Mathematical Foundations of the Finite Element Method With Applications to Partial Di erential Equations, (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 3{359. 2. Bramble, J., Multigrid Methods, Pitman Research Notes in Mathematics, Vol. 294, Wiley, New York, 1993. 3. Bramble, J. H., Kwak, D. Y., and Pasciak, J. E., Uniform convergence of multigrid V -cycle iterations for inde nite and nonsymmetric problems, SIAM J. Numer. Anal. 31 (1994), 1746{1764. 4. Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer, New York, 1994. 5. Ciarlet, P. G., The Finite Element Method For Elliptic Problems, North-Holland, Amsterdam, 1978. 6. Coulton, D. and Monk, P., On a class of integral equations of the rst kind in inverse scattering theory, SIAM J. Appl. Math. 53 (1993 3), 847{860. 7. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Di erential Equations of Second Order, Springer-Verlag, Berlin, 1983. 8. Oden, J. T. and Carey, G. F., Finite elements: Mathematical aspects, Prentice-Hall, Englewood Cli s, 1983. 9. Plaskota, L., Noisy Information and Computational Complexity, Cambridge University Press, Cambridge, 1996. 10. Plaskota, L., Worst case complexity of problems with random information noise, J. Complexity (1996) (to appear). 11. Traub, J. F., Wasilkowski, G. W., and Wozniakowski, H., Information-Based Complexity, Academic Press, New York, 1988. 12. Werschulz, A. G., The Computational Complexity of Di erential and Integral Equations: An InformationBased Approach, Oxford University Press, Oxford, 1991. 13. , The complexity of de nite elliptic problems with noisy data, J. Complexity 12 (1996), 440{473. 16
