quadrature error bounds with applications to lattice rules - CiteSeerX

QUADRATURE ERROR BOUNDS WITH APPLICATIONS TO LATTICE RULES FRED J. HICKERNELL



Abstract. Reproducing kernel Hilbert spaces are used to derive error bounds and worst-case integrands for a large family of quadrature rules. In the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature. However, the theory developed here does not require periodicity and is not restricted to lattice rules. An ANOVA decompositionis employed in de ning the inner product. It is shown that imbedded rules are superior when integrating functions with large high order ANOVA eects. Key words. ANOVA decomposition, good lattice points, imbedded rules, multidimensional integration, Monte Carlo, number-theoretic, quasirandom, periodic functions, reproducing kernel Hilbert spaces AMS subject classi cations. 65D30, 65D32

1. Introduction. Many multidimensional integrals can be written as I(f) =

Z

Cd

f(x) dx;

where C d = [0; 1)d is the d-dimensional unit cube. A simple approximation to I(f) is the arithmetic mean of the values of f on a set, S, of N points: X (1.1) Q(f) = N1 f(z): z2S For example, if S is a rectangular grid, (1.1) is the multidimensional rectangle rule. The Monte Carlo method corresponds to choosing the points in S randomly. Several dierent error bounds for quadrature rules of the form (1.1) have been derived. There is a widely applicable error bound based on the discrepancy of S. For lattice rules a more convenient, but specialized, error bound exists. Both of these error bounds are reviewed below. This article derives new quadrature error bounds using reproducing kernel Hilbert spaces. The new error bound coecients and worstcase integrands are relatively simple to compute. As a special case we obtain an error bound similar to the usual one for lattice rules. However, most of the new results are applicable to arbitrary quadrature rules of the form (1.1). Lattice rules were introduced by Sloan and his collaborators [22, 25]. For a review of the literature on lattice rules see [16, 23, 24]. An integration lattice, L, is a discrete additive subgroup of Rd that contains Zd . Lattice quadrature rules evaluate the integrand on the set S = L \ C d . A rectangular grid is one example of a lattice. Another example is the rank-1 lattice, fmg=N : m 2 Zg, where g 2 Zd is a generating vector normally chosen to be relatively prime to N. Quadrature rules based on rank-1 lattices (also called good lattice points) were proposed independently by Korobov [14] and Hlawka [10]. They have been studied by Hua and Wang [11], Fang and Wang [6] and others. Because integration lattices have period 1 in each coordinate direction, it is convenient to de ne f on Rd by a periodic extension of its de nition on C d . If f is  Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, Email: [email protected]. This research was supported by a Hong Kong UPGC-RGC grant. 1

continuous on Rd , it may usually be expressed as an absolutely convergent Fourier series: f(x) =

X

k2Zd

F(k)e2ikx;

where F(k) =

Z

Cd

f(x)e?2ikx dx:

In this case Sloan and Kachoyan [25] showed that the error of a lattice rule is (1.2)

I(f) ? Q(f) = ?

X 0

k2L?

F(k);

where L? is the dual of the integration lattice, and the notation 0 means summation excluding zero. To bound this error one may consider classes of functions whose Fourier coecients decay suciently rapidly. For any  > 1 and K > 0 let   K  E (K) = f : jF(k)j  (k


k ) ; (1.3) 1 d where kj = max(jkj j; 1). Greater  and K correspond to greater regularity and function size, respectively. The following error bound is due to Sloan and Kachoyan [25]: Theorem 1.1. The lattice rule quadrature error for any integrand f 2 E (K) satis es the inequality jI(f) ? Q(f)j  KP , where X 0 1 (1.4) P =  ( k


kd ) : 1 k2L? Equality holds for integrands that are constant multiples of

(1.5)

 (x) =

e2ikx :    k2Zd (k1


kd ) X

Whereas K indicates the size of the integrand, the error bound coecient P depends solely on the lattice. Therefore, much of the literature on lattice rules is devoted to deriving theoretical bounds on P or generating lattices with small P . The above error analysis for lattice rules is particularly convenient for periodic integrands. There is a dierent kind of error analysis assuming only bounded variation of the integrand in the sense of Hardy and Krause. Let D (S), the discrepancy of S, be de ned as follows: jS \ [0; x)j  (1.6) D (S) = supd N ? Vol([0; x)) ; x2C where jS \ [0; x)j denotes the number of points in the set S \ [0; x), and Vol([0; x)) denotes the volume of the rectangular solid [0; x). The Koksma-Hlawka inequality [16, Theorem 2.11], holds for any quadrature rule of the form (1.1), not only lattice rules. Theorem 1.2 (Koksma-Hlawka). The quadrature error for any integrand, f , with bounded variation V (f) on [0; 1]d in the sense of Hardy and Krause satis es the inequality jI(f) ? Q(f)j  V (f)D (S): 2

Although asymptotic bounds on the discrepancy of integration lattices have been derived [17], D (S) itself is not as simple to calculate as P. Furthermore the worstcase integrand, as constructed in [16, x2.2] does not have as simple a form as (1.5). Other error bounds, including those based on an L2 discrepancy, may be found in [15]. The approach we use to derive new quadrature error bounds diers from those outlined above in that it is based on the theory of reproducing kernel Hilbert spaces. Unlike the standard error analysis for lattice rules the integrands need not be periodic and Fourier analysis is not used. The advantage of working in a Hilbert space (rather than the space of functions of bounded variation) is that one can derive computationally simple formulas for the error bounds and worst-case integrands. Reproducing kernel Hilbert spaces have proven to be a valuable tool in the analysis of multivariate splines [27]. Thus, it is not surprising that they might also be useful in the analysis of multidimensional quadrature rules. The paragraphs below outline the main ideas underlying reproducing kernel Hilbert spaces. More details are contained in [2, 21]. Let (X; h; i) be some Hilbert space of real-valued functions on Rd , where h; i denotes the inner product. For any x 2 Rd let Tx denote the linear functional de ned as

Tx(f) = f(x) 8f 2 X: If Tx is bounded, then by the Riesz Representation Theorem there exists a reproducing kernel (; x) 2 X such that f(x) = Tx(f) = h(; x); f i 8f 2 X: The rst argument of  is the variable, and the second argument is a parameter. However,  is symmetric in its two arguments since (x; y) = h(; x); (; y)i = h(; y); (; x)i = (y; x). Once (x; y) is known, it is possible to express any other bounded, linear functional T as follows: (1.7) T (f) = h; f i; where (x) = h(; x);  i = h; (; x)i = T ((; x)): The Cauchy-Schwarz inequality then implies that (1.8)

jT (f)j = jh; f ij  k k kf k;

equality holding when f is a constant multiple of . In this article T is chosen to be I ? Q, the error of quadrature rule (1.1). Then (1.8) provides an error bound in terms of the sizes of the integrand, f, and the worstcase integrand, . By (1.7) this worst-case integrand is simply the error in integrating the reproducing kernel and depends only on the quadrature rule (and the de nition of the inner product). Therefore, quadrature rules that yield smaller values of k k are expected to be more accurate. In the next section a convenient reproducing kernel Hilbert space is de ned and the explicit error bound is given. Section 3 considers the case of periodic integrands and gives a result comparable to Theorem 1.1. Section 4 discusses the class of imbedded quadrature rules, and Monte Carlo rules are considered brie y in Section 5. Some numerical examples are presented in Section 6. 3

2. Quadrature Error Bound. The space of integrands considered in this article, X, is a subset of L2(C d ), the space of Lebesque square-integrable functions on C d . In de ning the inner product on X it is convenient to consider analysis of variance (ANOVA) decompositions of functions in X. These decompositions have been used in the statistical literature by Efron and Stein [5], Owen [20] and others. Let D = f1; : : :; dg be the set of coordinate indices. For any index set u  D, let juj denote its cardinality. Let xu denote Qthe juj-vector containing the components of x indexed by u. The quantity dxu = j 2u dxj is the uniform measure on the cube C juj = [0; 1)juj. This cube is denoted C u to distinguish cubes of the same dimension in dierent coordinate directions. Any function f 2 L2(C d ) may be written as the sum of its ANOVA eects, fu , de ned recursively as follows: fu (x) =

Z

C

[f ? D?u

fD (x) = f ?

X

X

vu

fv (x)] dxD?u (u 6= D);

fv (x);

vD

f(x) =

X

uD

fu (x):

Every function has a total of 2d ANOVA eects. The ANOVA eect fu is the part of the function depending only on the xj with j 2 u. The constant f; is the average value of the function: f; =

Z

Cd

f(x) dx = I(f):

The integral of f 2 and the variance of f can both be written in terms of integrals of the ANOVA eects: Z

Cd

[f(x)]2 dx =

(2.1)

Var(f) = =

XZ

u2D C u

Z

C

[fu(xu )]2 dxu;

[f(x) ? I(f)]2 dx = d

XZ

u6=; C u

[fu(xu )]2 dxu:

Z

Cd

[f(x)]2 dx ? [I(f)]2

The main eect along the j th axis is ffj g , while ffj;mg is the interaction along the axes j and m. The ANOVA eect fu can be considered as either a function on C u or a function on C D which is constant along the axes in D ? u. Note from the de nition R1 that 0 fu (xu ) dxj = 0 for any j 2 u. The order of an eect fu is juj. One would expect that it is easier to approximate the integrals of low order ANOVA eects because they are essentially functions on a low-dimensional cube. To illustrate the idea of the ANOVA decomposition consider the function f(x) = e?0:5jx1?0:5j?jx2?0:9j; There is one constant ANOVA eect, f; =

Z

0

1Z 1 0

?


?




f(x) dx1 dx2 = 4 ? 4e?0:25 2 ? e?0:1 ? e?0:9 ; 4

two rst order ANOVA eects, ff1g (x1) =

1

Z

0

[f(x) ? f; ] dx2

?
= e?0:5jx1?0:5j ? 4 + 4e?0:25 2 ? e?0:1 ? e?0:9 ;

ff2g (x2) =

Z ?

0

1

[f(x) ? f; ] dx1






= 4 ? 4e?0:25 e?jx2 ?0:9j ? 2 + e?0:1 + e?0:9 ; and one second order ANOVA eect, ff1;2g(xf1;2g) = f(x) ? f; ? ff1g (x1 ) ? ff2g (x2)    = e?0:5jx1?0:5j ? 4 + 4e?0:25 e?jx2 ?0:9j ? 2 + e?0:1 + e?0:9 : A reproducing kernel does not exist for L2(C d ) because Tx is unbounded on this space. Requiring certain partial derivatives of the ANOVA eects to be square integrable guarantees sucient regularity to insure that Tx is bounded. Let X; be de ned as the space of constant functions: X; = ff : f is a constantg; hf; f 0 i; = ff 0 : R For any nonempty u 2 D, Xu is a space of functions f for which C v f dxv = 0 for all v  u. Furthermore, all mixed partial derivatives @ jvj f=@xv (v  u) are required to exist as Lebesque square integrable distributions (generalized functions). For example, the (xf1g ?1=2)3=5 2 Xf1g since its derivative is square integrable on [0; 1), even though the derivative is not nite at xf1g = 1=2. On the other hand sign(xf1g ? 1=2) 2= Xf1g since its derivative, the Dirac delta function centered at 0.5, is not square integrable. The de nition of Xu is:   Z jvj f @ 2 u 2 u Xu = f 2 L (C ) : @x 2 L (C ) and f dxv = 0 8v  u ; v Cv Z

@ juj f @ juj f 0 dx : u C u @xu @xu We consider integrands whose ANOVA eects lie in Xu : X X X = ff : fu 2 Xu g = Xu ; hf; f 0 i = (2.2a)

?juj hfu ; fu0 iu ;

hf; f 0 iu =

uD

uD

where fu and fu0 are the ANOVA eects of f and f 0 , respectively. The norms induced by h; i and h; iu are related as follows: ?juj kfuk2 8fu 2 Xu ; (2.2b) kfu k2 = X u X 2 2 kf k = kfu k = ?juj kfu k2u 8f 2 X: (2.2c) uD

uD

The arbitrary constant in the de nition of h; i might naturally be taken to be one. However, it is shown in the next section that for lattice rules applied to periodic integrands there is an error bound similar to Theorem 1.1 for = 42. 5

To derive the reproducing kernel for X we rst nd the reproducing kernel for Xf1g . Let 1(x1; y1 ) = 21 (x21 + y12 ) + 13 ? max(x1 ; y1)      1 1 1 1 = 2 jx1 ? y1 j(jx1 ? y1 j ? 1) + 6 + x1 ? 2 y1 ? 2 : R Since 01 1(x1; y1 ) dx1 = 0 for all y1 and @1 1 @x1 = x1 ? 2 [1 + sign(x1 ? y1 )] is square integrable, it follows that 1 (; y1) 2 Xf1g . Furthermore, for all f 2 Xf1g Z 1 Z 1 1 @1 @f @f @f dx h1(; y1 ); f if1g = @x @x dx1 = x1 @x dx1 ? @x 1 1 1 1 0 0 y1 1 1 Z 1 = x1f(x1 ) 0 ? f(x1 ) dx1 + f(y1 ) ? f(1) = f(y1 ); 0 Z

since 01 f(x1 ) dx1 = 0. Thus, 1(; y1 ) is the reproducing kernel for Xf1g . The reproducing kernel for the subspace Xu is formedQ by taking the product of one-dimensional reproducing kernels. Let ^u(xu ; yu ) = j 2u 1 (xj ; yj ). Using the properties of 1 it is straightforward to show that R

Z

C

@ jvj ^u 2 L2(C u ) 8v  u; and @xv

^u(xu ; yu) dxj = 0 8j 2 u; u

h^u (; yu); f iu = f(yu ) 8f 2 Xu : So, it follows that ^u is the reproducing kernel for Xu . The reproducing kernel for X has ANOVA eects u, which are closely related to the ^u . Note that u(; yu ) belongs to the space Xu , whose reproducing kernel is ^u. On the other hand ^u(; yu ) belongs to the space X and has a single nonzero ANOVA eect identical to itself: ^u;u(; yu ) = ^u (; yu) 2 Xu . These two observations are used to derive the formula for u : u(xu ; yu ) = h^u(; xu); u(; yu )iu = hu(; yu ); ^u(; xu)iu = juj h(; yu ); ^u(; xu)i by (2.2) j u j j u j = ^u (yu ; xu) = ^u (xu; yu ): Therefore,  can be written in terms of 1 as follows: (2.3a) (2.3b)

; = 1;

u (xu; yu ) = juj ^u (xu; yu ) = juj

(x; y) =

X

uD

u(xu ; yu) = 6

d Y

Y

j 2u

1(xj ; yj );

[1 + 1 (xj ; yj )]:

j =1

From the reproducing kernel it is possible to compute the worst-case integrand, R (x), as given by (1.7). Since 01 1(x1; y1 ) dx1 = 0 for all y1 , it follows that I((; x)) = 1, and X (2.4a) (x) = I((; x)) ? Q((; x)) = 1 ? N1 (z; x): z2S The ANOVA eects of the worst-case integrand, u (x), are the errors in integrating u(; xu). Since quadrature rule (1.1) integrates constants exactly, (2.4b) ; = I(; ) ? Q(; ) = 0; and since I(u (; xu)) = 0 for u 6= ;, X u (x) = ?Q(u (; xu)) = ? N1 u(zu ; xu) (u 6= ;): (2.4c) z2S Given an explicit formula for the worst-case integrand, it is now possible to derive an explicit quadrature error bound by (1.8). To show how each ANOVA eect contributes to the total quadrature error, h; f i is expressed in terms of the hu ; fu iu using (2.2): X = ?juj uD

jI(f) ? Q(f)j = jh; f ij 

X

uD

u ; fu u X

h

i

?juj ku ku kfu ku =

uD

ku k kfu k

 k k kf k: The quantities cu  ?juj ku ku and c  k k that appear in the above inequality

depend only on the quadrature rule, i.e. the set S, and not on the integrand. They are therefore called error bound coecients. Formulas for these error bound coecients follow immediately from (2.2)-(2.4): *

(2.5a)

(2.5b)

+

X X c2u  ?2juj ku k2u = ?2juj N1 u(; zu); N1 u (; zu0 ) z2S z0 2S u ? 2 j u j X

0 hu(; zu); u(; zu )iu = N2 z;z0 2S X X h^u (; zu); ^u(; zu0 )iu = N12 ^u (zu ; zu0 ) = N12 z;z0 2S z;z0 2S X Y 1(zj ; zj0 ) (u 6= ;); = N12 z;z0 2S j 2u

c2  k k2 =

X

uD

= ?1 + N12

?juj ku k2u =

X

d Y

X

uD

juj c2u

[1 + 1 (zj ; zj0 )]:

z;z0 2S j =1 7

The following theorem summarizes the main results of this section. Theorem 2.1. For any integrand f 2 X P the error of quadrature rule (1.1) satis es the inequality jI(f) ? Q(f)j = jh; f ij  u6=; cu kfuku  ckf k, where the error bound coecients cu and c are given by (2.5). Equality holds when f is proportional to  as given in (2.4). Obviously, quadrature rules with small error bound coecients are more desirable. If one knows the relative sizes of the ANOVA eects of f, then one can reduce the error by choosing a quadrature rule that has small cu for kfu ku large. In practice, however, the relative magnitudes of the ANOVA eects of the integrand are rarely known. Thus, one might choose S so that c2 is as small as possible. By (2.5), P 2 c = uD juj c2u is a weighted average of the c2u, each of which depends on S but not on the weight . It is important to clarify the meaning of . This is not a parameter to be adjusted in order to obtain a smaller c. For any given S and any given integrand, reducing causes c to decrease and kf k to increase monotonically. Thus, choosing small does not necessarily give a good error bound ckf k. If one is able to minimize ckf k with respect to , then one probably knows enough P about the norms of the ANOVA eects to use the tighter and -free error bound u6=; cu kfu ku. The role of is to re ect the user's preference for accurate quadrature of low order or high order ANOVA eects. A small value of implies that c depends mainly on the low order cu . Thus, choosing S to minimize c will insure small cu for smaller juj. Consequently the quadrature error bounds cu kfuk for low order ANOVA eects will also be small. On the other hand a large value of implies a preference for accurately integrating the high order ANOVA eects. 3. Periodic Integrands. The analysis in the previous section makes no assumptions on the periodicity of the integrands. Because much of the theory for lattice rules applies to periodic integrands we now consider this case. For any u  D let P (C u ) be the set of all continuous, periodic functions on Ru with period interval C u = [0; 1]u. ~ h; i) are Let X~u = Xu \ P (C u ) and X~ = X \ P (C d ). The spaces (X~u ; h; iu) and (X; subspaces of the Hilbert spaces de ned in the previous section. Their reproducing kernels must be periodic, in contrast to ^u and , which are not. It can be shown that the reproducing kernel for X~f1g is (3.1)

~1 (x1; y1 ) = 1(fx1 g; fy1 g) ? (fx1g ? 1=2)(fy1 g ? 1=2) = 21 fx1 ? y1 g(fx1 ? y1 g ? 1) + 61 = 12 B2 (fx1 ? y1 g);

where fxg denotes the fractional part of a real number or vector x, and B2 (t) is the Bernoulli polynomial of degree two [1, Chapter 23]. The reproducing kernel for for X~ has a form analogous to (2.3): (3.2a)

~; = ; = 1;

(3.2b)

~(x; y) =

X

uD

~u (xu; yu ) = juj ~u(xu ; yu) =

d Y

Y

j 2u

~1(xj ; yj );

[1 + ~1 (xj ; yj )]:

j =1

The formulas for the worst-case integrand, ~(y), and the corresponding error 8

bound coecients are analogous to (2.4) and (2.5): X ~; = 0; ~u(x) = ? N1 ~u(zu ; xu) (u 6= ;); (3.3a) z2S X ~(x) = 1 ? N1 ~(z; x);

(3.3b) (3.4a) (3.4b)

z2S

c~2u  ?2jujk~u k2u = N12 c~2  k~k2 =

X

u6=;

X Y

z;z0 2S j 2u

juj c~2u = ?1 + N12

~1(zj ; zj0 ) (u 6= ;);

d X Y z;z0 2S j =1

[1 + ~1 (zj ; zj0 )]:

Because they apply to a smaller space of integrands these error bound coecients are no larger than their counterparts for general integrands. For any particular S c~2u = ?2juj k~u k2u = ?2jujh~u ; ~u iu = ?juj h~u ; ~ui = ?juj [I(~u ) ? Q(~u )] = ?juj hu ; ~u i = ?2juj hu ; ~uiu  ?2juj k~u ku ku ku = c~u cu : So, c~u  cu and c~  c.

2 X~ the error of quadrature rule (1.1) P ~ satis es the inequality jI(f) ? Q(f)j = jh; f ij  u6=; c~u kfu ku  c~ kf k, where the error bound coecients are given in (3.4). Equality holds when f is proportional to ~ Theorem 3.1. For any integrand f

as de ned in (3.3).

For lattice rules formulas (3.4) can be reduced to sums of only N terms, which is a signi cant computational advantage over the original sums of N 2 terms. Recall that integration lattices are invariant under translations that shift the origin to any lattice point. Furthermore, note that ~(x; y) is periodic and a function of x ? y only. Thus, it follows that X X c~2u = N12 (3.5a) ~u(zu ; zu0 ) = N12 ~u (zu + zu0 ; zu0 ) 0 0 z;z 2S z;z 2S X XY 1 1 ~1(zj ; 0) (u 6= ;); = N ~u (zu ; 0) = N z2S z2S j 2u

(3.5b)

c~2 =

X

u6=;

d XY [1 + ~1 (zj ; 0)]

juj c~2u = ?1 + N1

X = ?1 + N1 ~(z; 0): z2S

z2S j =1

In order to compare Theorem 3.1 to Theorem 1.1 consider the Fourier series representations of the functions in L2(C d ). For any function f 2 L2 (C d ) with ANOVA eects fu let F and Fu denote their respective Fourier coecients. The coecients 9

Fu depend only on the wave numbers ku 2 Zu . For such a wave number let k~u denote its extension to Zd where all the components outside u are set to zero. Then by the de nition of the ANOVA decomposition F(k~u) = Fu(ku ). The norm on X~ can be written in terms of the Fourier coecients as follows: X (3.6) kf k2 = (42= )jkj 6=0j (k1


kd )2jF(k)j2: k2Zd

where jkj 6= 0j denotes the number of nonzero components of the wavenumber k. Comparing this expression with (1.3) we see that E 2 (K)  X~ for any nite K. The Fourier series of the reproducing kernel follows from the Fourier series for the Bernoulli polynomial B2 (t) [1, Equation 23.1.18] and is X ~h(k; y)e2ikx ; ~(x; y) = k2Zd

where h~ 1 (k1; y1 ) = ~h(k; y) =

Z

C

Z

0

1

~1(x1 ; y1)e?2ik1 x1 dx1 =

~(x; y)e?2ikx dx = d



e?2ik1y1 =(42k12 ) (k1 6= 0) 0 (k1 = 0);

d Y

2 jkj 6=0j ?2iky [kj + h~ 1 (kj ; yj )] = ( =4 ) e 2 ; (k1


kd ) j =1

and k denotes the Kroneker delta function. The notation jkj 6= 0j means the number of nonzero components of the wave number k. If S = L \ C d for some integration lattice L, then [25, Theorem 1] can be used to express ~(y) and c~2 in terms of the above Fourier coecients: X X X ~h(k; y)e2ikz ~(y) = 1 ? N1 ~(z; y) = 1 ? N1 d d z2S k2Z z2L\C X 0 ( =42 )jkj 6=0j e?2iky =? ; (k1


kd )2 k2L? X X X c~2 = ?1 + N1 ~(z; 0) = ?1 + N1 h~ (k; 0)e2ikz z2S k2Zd z2L\C d X 0 ( =42 )jkj 6=0j =   2 : k2L? (k1


kd ) In this paragraph we consider speci cally the case = 42, which implies c~2 is identical to P2 as de ned in (1.4). Both Theorems 1.1 and 3.1 imply that lattices with small P2 give more accurate quadrature rules p for periodic integrands. However, the error bound is KP2 in Theorem 1.1 versus P2kf k in Theorem 3.1. This is due to the dierent measures for the size of an integrand | K for E 2(K) versus k k for ~ Recall from (1.2) that the error for lattice rules is the sum of only those Fourier X. coecients with nonzero wave numbers k 2 L? . Therefore, the worst-case integrand in Theorem 1.1 is not unique. Reducing any of the Fourier coecients of 2 with k 62 L? leaves the error unchanged. For example, ~ has the same quadrature error and decay rate of its Fourier coecients as 2, so ~ is also a worst-case integrand for 10

Theorem 1.1. However, k2 k > k~k, so 2 is not a worst-case integrand for Theorem 3.1. The ANOVA decomposition and reproducing kernel Hilbert space approach can, in fact, produce error bounds similar to Theorem 1.1 for any even integer   2. For any positive integer t let tu denote an index set consisting of t copies of u. Let Pt (C u) be the set of functions on Rd such that @ jvjf=@xv is continuous and periodic with period interval C u for all v  tu. By de nition P0(C u ) = P (C u). Let the Hilbert spaces (X~t;u ; h; it;u) and (X~t ; h; it) be de ned as follows for any positive integer t: X~t;; = ff : f is a constantg; hf; f 0 it;; = ff 0 

jvj

X~ t;u = f 2 Pt?1(C u) : @@x f 2 L2(C u) and v

hf; f 0 it;u = X~t = ff : fu 2 Xt;ug =

Z

C

f dxv = 0 8v  tu v



(u 6= ;);

@ tjujf @ tjujf 0 dx (u 6= ;); u C u @xtu @xtu

Z

X

uD

X~t;u ;

hf; f 0 it =

X

uD

?juj hfu ; fu0 it;u:

Here t = 1 corresponds to the case treated in Theorem 3.1. The norm k kt can be written in terms of the Fourier coecients in an analogous way to (3.6): X kf k2t = [(42)t= ]jkj 6=0j (k1


kd )2tjF(k)j2; k2Zd

implying E 2t(K)p X~t . For = (42 )t the quadrature error for lattice rules becomes jI(f) ? Q(f)j  P2tkf kt . 4. Imbedded Rules. For any positive integer vector n let Gn be de ned as the rectangular grid lattice:  (4.1) Gn = (m1 =n1; : : :; md =nd ) : m 2 Zd : For any S0  C d the set S = (S0 +Gn )\C d consists of n1


nd translated copies of the set S0 . Quadrature rules based on S are called imbedded rules. Sloan and Walsh [26], Disney and Sloan [4], Joe and Sloan [13] and Joe and Disney [12] studied imbedded lattice rules. They recommended choosing S0 = L0 \ C d for some rank-1 lattice L0 and choosing nj = 1 or 2. Their computations and theoretical results suggest that imbedded lattice rules may be superior to rank-1 lattice rules with the same number of points. In this section we explore the properties of imbedded rules using the reproducing kernel Hilbert space methodology adopted in the previous sections. First we establish a useful property of the reproducing kernels  and ~ de ned in (2.3) and (3.2) respectively. The following lemma is similar to [4, Lemma 1]. Lemma 4.1. For any real x1 and y1 (4.2a) (4.2b)

nX ?1

m;m0 =0

1 (fx1 + m=ng; fy1 + m0 =ng) = 1(fnx1g; fny1g); nX ?1

m;m0 =0

~1(x1 + m=n; y1 + m0 =n) = ~1(nx1 ; ny1): 11

Proof. The proof follows from some properties of Bernoulli polynomials. Recall from (3.1) that ~1(x1 + m=n; y1 + m0 =n) = B2 (fx1 ? y1 + (m ? m0 )=ng)=2. For m; m0 = 0;


; n ? 1 the quantity fx1 ? y1 +(m ? m0 )=ng takes on the same values (in perhaps a dierent order) as [fn(x1 ? y1 )g + m]=n. The proof of (4.2b) then follows by straightforward calculation: nX ?1 nX ?1 1 ~1 (x1 + m=n; y1 + m0 =n) = B2 (fx1 ? y1 + (m ? m0 )=ng) 2 m;m0 =0 m;m0 =0

=

nX ?1

1 B ([fn(x ? y )g + m]=n) 2 1 1 2 m;m0 =0

=n

nX ?1

1 B ([fn(x ? y )g + m]=n) 2 1 1 2 m=0

= 21 B2 (fn(x1 ? y1 )g) = ~1(nx1; ny1 );

where the previous line follows from [1, Equation 23.1.10]. The proof of (4.2a) is similar to that of (4.2b). First note from from (3.1) that 1(fx1 + m=ng; fy1 + m0 =ng) = 21 B2 (fx1 ? y1 + (m ? m0 )=ng) + B1 (fx1 + m=ng)B1 (fy1 + m0 =ng); where B1 is the rst degree Bernoulli polynomial. Following the argument above: nX ?1

1(fx1 + m=ng; fy1 + m0 =ng)

m;m0 =0 nX ?1



1 B (fx ? y + (m ? m0 )=ng) + B (fx + m=ng)B (fy + m0 =ng) 2 1 1 1 1 1 1 m;m0 =0 2 nX ?1 B1 ([fnx1g + m]=n)B1 ([fny1 g + m0 ]=n) = 21 B2 (fn(x1 ? y1 )g) + m;m0 =0 1 = 2 B2 (fn(x1 ? y1 )g) + B1 (fnx1g)B1 (fny1 g) = 1(fnx1g; fny1 g); =

where again [1, Equation 23.1.10] is used. Lemma 4.1 can be used to simplify the formulas for the error bound coecients of Theorems 2.1 and 3.1 for the case of imbedded rules. This result is contained in the following theorem. Theorem 4.2. Suppose S0  C d has N0 points, and let c20;u and c~20;u denote the the error bound coecients corresponding to the quadrature rule using the points S^0 = ff(n1 z1 ;


; nd zd )g : z 2 S0 g, that is: X Y 1 (fnj zj g; fnj zj0 g) (u 6= ;); c20;u = N12 0 z;z0 2S0 j 2u X Y 1 c~20;u = N 2 ~1 (fnj zj g; fnj zj0 g) (u 6= ;): 0 z;z0 2S0 j 2u 12

Let Gn be the grid de ned in (4.1). The error bound coecients (2.5a) and (3.4a) for the imbedded quadrature rule based on S = (S0 +Gn) \ C d can be expressed as follows for any u 6= ;:

c2u = c20;u

(4.3)

1; 2 n j 2u j Y

c~2u = c~20;u

1: 2 n j 2u j Y

Proof. The proof for c2u proceeds by rearranging the sum in (2.5a) and applying

Lemma 4.1. For any u 6= ; X Y c2u = N12 1(zj ; zj0 ) 0 z;z 2S j 2u

j ?1 1 nX 1(fzj + m=nj g; fzj0 + m0 =nj g) 0 z;z0 2S0 j 2u n2j m;m0 =0 X Y 1 Y 1 = N12  ( fnj zj g; fnj zj0 g) = c20;u j 2 2: 0 z;z0 2S0 j 2u nj j 2u nj

= N12

X Y

The argument for c~2u is the same. A special case of imbedded rules is the shifted rectangle rule. If S0 consists of a single point  2 C d , then the quadrature rule is nX nX 1 ?1 d ?1 Q(f) = n
1

n


f(f + (m1 =n1;


; md =nd )g): 1 d m1 =0 md =0

Since the sum over S0 is trivial, the error bound coecients in Theorem 4.2 can be simpli ed as follows: Y (4.4a) c2u = n12 1(fnj j g; fnj j g) j 2u j Y = n12 (fnj j g2 ? fnj j g + 1=3) (u 6= ;); j 2u j Y 1 c~2u = 12n (4.4b) 2 (u 6= ;): j j 2u The expression fnj j g2 ?fnj j g+1=3 has a minimumof 1=12 for fnj j g = 1=2, which corresponds to the midpoint rule. A maximum of 1=3 occurs for fnj j g = 0, which corresponds to the left-rectangle rule. For the periodic case c~u and c~ are independent of the shift. Notice that in (4.4) the low order error bound coecients are larger than the high order ones. For example, c~fj g = 1=(12nj ) whereas c~D = 1=(12dn1


nd ). This principle is true for imbedded rules in general according to Theorem 4.2. By using n1


nd times as many points an imbedded rule reduces cu by a factor of Q ?1 j 2u nj in comparison to a quadrature rule based on S^0 . The reduction is greatest for the dth order coecient, cD , and least for the rst order coecients, cfj g . Therefore, an imbedded rule performs much better than the original rule when integrating functions consisting mainly of high order ANOVA eects. Theorem 4.2 helps explain why several authors [4, 12, 13, 26] found imbedded lattice rules to have smaller P2 than rank-1 rules. Recall that for lattice rules P2 = 13

P

c~2 = u6=; juj c~2u where = 42  39:5. In this case the high order error bound coecients, c~2u , contribute heavily to P2 because of the large weight juj . Since these high order coecients are signi cantly reduced for imbedded rules, P2 is naturally much smaller. On the other hand, if our measuring stick for quadrature rules is c~2 with a smaller value of , then the low order error bound coecients are more important and imbedded rules should not do as well. 5. Monte Carlo Quadrature Rules. In the next section various lattice rules are generated based on minimizing c2 and c~2 . In this section we compute the expected values of these quantities when S is a simple random sample of N points uniformly distributed on C d . Since Monte Carlo rules are basic, their error bound coecients serve as a benchmark which we expect to improve upon by using lattice rules or other quasirandom rules. Let E denote the expected value of a random variable. From (2.5) one can see that the calculation of E[c2 ] depends mainly on E[1(zj ; zj0 )] for z; z 0 2 S. There are two cases to consider: ( R R 1 1 0 0 0 0 E[1(zj ; zj )] = R01 0 1 (x1; x1) dx1 dx1 = 0 for 0z 6= z ; 0 (x1 ; x1) dx1 = 1=6 for z = z : Similar expressions hold for the periodic case. This leads to the following theorem. Theorem 5.1. If S is a simple random sample of N points uniformly distributed on C d , then the expected values of the squared error bound coecients given in (2.5) and (3.4) can be expressed as follows:

E[c2u ] = 1=(6jujN);

E[c2] = [(1 + =6)d ? 1]=N;

E[~c2u] = 1=(12jujN); E[~c2] = [(1 + =12)d ? 1]=N: This theorem implies that the expected error bound coecients for Monte Carlo rules are O(N p ?1=2). This is asymptotically greater than the asymptotic bounds on P2 (or even P 2 ) for lattice rules. However, as pointed out by Disney and Sloan [3] an asymptotically larger bound may not be larger in practice. Applying Theorem 5.1 to Theorems 2.1 and 3.1 implies (5.1a) E[I(f) ? Q(f)]2 = [(1 + =6)d ? 1] kf k2=N; for general integrands, and (5.1b) E[I(f) ? Q(f)]2 = [(1 + =12)d ? 1] kf k2 =N; for periodic integrands. These error bounds depend on the dimension d, which is in contrast the typical error bound for Monte Carlo quadrature where there is no inherent dimensional eect: (5.2) E[I(f) ? Q(f)]2 = Var(f)=N; where Var(f) was de ned in (2.1). One reason for the dierence between the two cases is the dierent measures of the size of the integrand: k k as de ned in (2.2), which is an inherently dimension dependent measure, versus the variance, which is inherently dimension independent. Another reason may be that (5.1) arises by considering worstcase integrands, whereas (5.2) does not. It is worth noting that the expected value of the L2 discrepancy for a random sample also depends on the dimension [15]. 14

Generating vectors determined by minimizing c ( = 1) by the greedy algorithm. d N

8 13 21 34 55 89 144 233 377 610 987 K s

1

g1

2

g2

3

4

g3

g4

5

g5

6

g6

7

g7

1 5 3 7 6 2 4 1 5 11 4 3 6 7 1 13 16 19 4 10 9 1 15 25 21 20 22 29 1 34 32 43 26 18 7 1 63 69 73 28 72 83 1 89 65 25 43 82 53 1 147 179 99 191 202 90 1 233 218 242 311 337 315 1 233 439 129 452 519 334 1 610 722 764 782 677 857 0.2887 0.5101 0.8074 0.9857 1.2654 1.4697 1.8564 1.0000 0.9441 0.9029 0.8442 0.8065 0.7659 0.7544

6. Numerical Computations. This section presents three kinds of computations. First, we calculate a table of generating vectors for rank-1 lattice rules based on the criteria of minimizing c and c~. Next, these rules are applied to a sample integrand. Finally, we compute some imbedded rules and compare their error bound coecients to those of rank-1 rules. When computing c we consider lattice rules shifted by 1=(2N) in each direction. This corresponds to a midpoint rule for the single coordinate ANOVA eects and should be more accurate than an unshifted rule. For the periodic case it is not necessary to consider shifts since, c~ is invariant under shifts. Minimizing c for a rank-1 rule with respect to its generating vector g is equivalent to minimizing the function H(g1 ;


; gp) =

NX ?1

p Y

m;m0 =0 j =1

[1 + 1 (f(2mgj + 1)=(2N)g; f(2m0 gj + 1)=(2N)g)]

for the case p = d. The algorithm adopted here is a greedy one. We begin by setting g1 = 1. Then we iteratively choose gp for p = 2;


; d such that H(g1 ;


; gp ) = r=1min H(g1 ;


; gp?1; r): ;


;N An analogous approach is used for the periodic case. The disadvantage of this greedy algorithm is that the resulting g may not give a true minimum of c or c~. On the other hand, the resulting g is useful for all dimensions p  d, and the amount of work required to extend g to the next higher dimension is nearly independent of d. Tables 6.1-6.4 give the generating vectors g for dimensions 1-7 for N equal to the Fibonacci numbers 8; 13; 21;


; 987. The four tables correspond to minimizing c and c~ for = 1; 42. Figures 6.1-6.4 are log-log plots of the error bound coecients versus N for these rank-1 lattice rules for d = 1(); 2(); 3(+); 4(); 5(); 6() and 7(+). For any particular N the error bound coecients increase with increasing dimension. The p p seven solid lines in each gure show E[c2 ] or E[~c2] versus N for Monte Carlo rules for d = 1;


; 7 as given by Theorem 5.1. Several observations can be made from these tables and gures. Since c increases as increases, it is not surprising that the best values of c are much smaller for = 1 15

Generating vectors determined by minimizing c ( = 42 ) by the greedy algorithm. d N

8 13 21 34 55 89 144 233 377 610 987 K s

1

g1

2

g2

3

4

g3

g4

5

g5

6

g6

7

g7

1 6 2 4 5 3 7 1 5 10 2 9 6 7 1 9 3 15 11 6 12 1 20 30 28 12 18 26 1 34 5 2 47 39 13 1 63 43 39 50 37 86 1 89 105 62 124 34 2 1 147 16 25 50 100 99 1 233 170 95 187 285 97 1 354 112 390 384 170 280 1 610 279 69 282 420 702 1.8138 12.774 33.702 85.713 273.94 904.53 3276.8 1.0000 0.9011 0.7301 0.6428 0.6338 0.6437 0.6693

Generating vectors determined by minimizing c~ ( = 1) by the greedy algorithm. d N

8 13 21 34 55 89 144 233 377 610 987 K s

1

2

3

4

5

6

7

g1

g2

g3

g4

g5

g6

g7

1 3 5 7 2 6 4 1 5 3 2 4 6 7 1 8 5 2 4 10 9 1 13 9 15 6 5 11 1 21 12 23 7 18 4 1 34 25 40 37 33 35 1 55 61 43 19 37 31 1 89 64 101 75 112 52 1 144 159 135 167 69 57 1 233 171 129 209 276 99 1 377 265 886 172 223 416 0.2887 0.4270 0.5769 0.6700 0.8258 0.9535 1.1609 1.0000 0.9681 0.9365 0.8944 0.8690 0.8445 0.8350

Generating vectors determined by minimizing c~ ( = 42 ) by the greedy algorithm. d N

8 13 21 34 55 89 144 233 377 610 987 K s

1

g1

2

g2

3

4

g3

g4

5

g5

6

g6

7

g7

1 3 5 2 6 4 7 1 5 3 8 10 2 11 1 8 3 13 5 16 18 1 10 16 32 2 18 14 1 21 39 29 26 13 42 1 34 64 19 17 72 55 1 55 128 46 32 80 64 1 89 105 79 2 158 154 1 144 170 249 123 121 256 1 233 272 151 473 304 2 1 377 547 738 261 231 2 1.8138 7.0187 14.549 25.216 44.253 84.465 169.08 1.0000 0.8996 0.7398 0.6308 0.5608 0.5301 0.5177 16

0

10

10-1

c

-2

10

-3

10

-4

10

1

10

2

10 N

3

10

. c ( = 1) for rank-1 rules in Table 6.1 and (E [c2 ])1=2 ( = 1) for Monte Carlo rules for dimensions d = 1;


; 7. 3

10

102

1

10

c

0

10

-1

10

-2

10

-3

10

1

10

2

10 N

3

10

. c ( = 42 ) for rank-1 rules in Table 6.2 and (E [c2 ])1=2 ( = 42 ) for Monte Carlo rules for dimensions d = 1;


; 7.

than for = 42 , particularly in higher dimensions. For N up to 1000 the error bound coecients are approximately KN ?s , where K and s depend on the dimension d. Tables 6.1-6.4 list the values of K and s found by using linear least squares regression to approximate log(c) and log(~c) by log(K) ? s log(N). It follows from (4.4) that p c = c~ = =12N ?1 for d = 1. As d increases, s decreases and K increases. The eect of dimension is greater for = 42 , since in this case c and c~ depend more strongly on the high order cu and c~u . The superiority of rank-1 lattice rules over Monte Carlo methods decreases with increasing dimension. For higher dimensions the error bound 17

0

10

10-1

c˜

-2

10

-3

10

-4

10

1

10

2

10 N

3

10

. c~ ( = 1) for rank-1 rules in Table 6.3 and (E [~c2 ])1=2 ( = 1) for Monte Carlo rules for dimensions d = 1;


; 7. 3

10

102

1

10

0

c˜

10

-1

10

-2

10

-3

10

1

10

2

10 N

3

10

. c~ ( = 42 ) for rank-1 rules in Table 6.4 and (E [~c2 ])1=2 ( = 42 ) for Monte Carlo rules for dimensions d = 1;


; 7.

coecients for rank-1 lattice rules and Monte Carlo methods are similar, especially for larger . This means that advantages of rank-1 lattice rules are greatest for integrands with relatively small high order ANOVA eects. The error bound coecients c and c~ ( = 1; 42) were computed for all four sets of rank-1 lattice rules in Tables 6.1-6.4. It was found that the generating vectors that yield small c~ tend to yield small c as well. This is useful to know since the eort required to compute a row of Table 6.3 or 6.4 is O(N) less than that required to compute the corresponding row of Table 6.1 or 6.2. It is also worth noting that the 18

Quadrature errors for (6.1) for rank-1 rules. N

8 13 21 34 55 89 144 233 377 610 987

Table 6.1 2.9234E-1 9.3135E-2 1.1393E-1 7.0900E-3 -9.7565E-2 2.1320E-2 -4.0653E-3 2.2734E-3 -5.4076E-3 -5.8700E-4 -2.7689E-4

Table 6.2 2.2599E-2 -9.9472E-2 5.7981E-2 -3.0130E-2 -7.4248E-3 1.2820E-1 -2.0095E-2 -4.8914E-2 1.3907E-2 2.8393E-2 9.0740E-2

Table 6.3 6.0916E-2 7.1400E-2 7.5975E-2 -1.3621E-2 7.9934E-3 2.5874E-2 6.9101E-3 -2.3148E-4 2.8916E-3 -9.2905E-4 -2.3386E-4

Table 6.4 -3.0094E-1 3.2030E-1 1.5805E-1 -1.2536E-1 -1.7670E-1 6.4856E-2 1.1262E-1 2.8008E-2 9.4192E-4 4.1885E-3 -1.6847E-3

Quadrature errors for ANOVA eects of (6.1) of various orders, N = 987.

Table 6.1 Table 6.2 Table 6.3 Table 6.4 1 -2.6547E-7 -9.3594E-2 -9.9018E-7 -1.6153E-5 2 3.2945E-4 1.2851E-3 -3.1012E-5 -1.8268E-4 3 -4.3829E-4 1.6268E-3 4.0142E-6 1.1077E-3 4 4.1870E-4 -4.2478E-5 2.6813E-4 8.3665E-4 5 -3.2711E-5 -1.5936E-5 -6.2869E-6 -6.0770E-5

u

j j

rank-1 lattice rules generated by the greedy algorithm with = 42 do not necessarily give small error bound coecients for = 1, and to a lesser extent visa versa. Although it is beyond the scope of this paper to perform an exhaustive test of various lattice rules, we do consider an example from the test suite of Genz [7, 8] (6.1) f(x) = exp(?0:5jx1 ? 0:5j ? jx2 ? 0:9j ? 1:5jx3 ? 0:3j ?2jx4 ? 0:7j ? 2:5jx5 ? 0:1j); R where  9:0461 is chosen to make C 5 f(x) dx = 1. Table 6.5 contains the quadrature errors using the rank-1 rules in Tables 6.1-6.4. For Tables 6.3 and 6.4 the quadrature rules are applied to the periodic integrand f(j2x1 ? 1j;


; j2x5 ? 1j) rather than f(x). Because this integrand is a product of functions of a single variable, the ANOVA eects can be calculatedR in a straightforward manner. For any function f(x) = f1 (x1)


fd (xd ) let Ij = 01 fj (xj ) dxj . Then the ANOVA eects are fu (xu) =

Y

[fj (xj ) ? Ij ]

j 2u

Y

j 62u

Ij :

The quadrature error for f is the sum of ?Q(fu ) over all u 6= ;. The sums of ?Q(fu ) for constant juj are given in Table 6.6 for the case N = 987. For this example the errors in integrating low order ANOVA eects are generally less for the rules from Tables 6.1 and 6.3 (those generated using = 1). For juj = 1 it is easy to see why. Along each axis j the rank-1 lattice points take on N=gcd(gj ; N) distinct values in the interval [0,1), where gcd denotes the greatest common divisor. For N = 987 in Tables 6.1 and 6.3 gcd(gj ; N) = 1 for all j, so the lattice points take on 987 distinct values along each axis. In Tables 6.2 and 6.4 gcd(gj ; N) > 1 for some j. Along such axes the lattice points take on fewer than 987 distinct values, which results in a less accurate integration of the rst order ANOVA eects. 19

Selected imbedded rules from [25]. d

g0

3 (1, 125, 108) 4 (1, 50, 133, 75) 5 (1, 90, 210, 227, 179) (1, 98, 136, 178, 86, 12) 6 8 (1, 60, 81, 77, 149, 261, 143, 164)

N0

263 263 263 263 263

Error bound coecients c~ ( = 1) for imbedded and rank-1 rules. d

2 3 4 5 6 7 8

Table 6.7 | 1.7614E-3 3.8114E-3 7.9255E-3 7.5883E-3 | 3.3667E-2

Imbedded Rules (6.3a) (6.3b) 8.3239E-4 8.3239E-4 1.7023E-3 1.7518E-3 2.7757E-3 3.0066E-3 4.2406E-3 6.0065E-3 5.9881E-3 1.0398E-2 7.9445E-3 1.7305E-2 1.0202E-2 2.7871E-2

Rank-1 Rules (6.4a) (6.4b) 5.1191E-4 5.1191E-4 8.6714E-4 1.0416E-3 1.3896E-3 2.0051E-3 2.0443E-3 3.4360E-3 2.7953E-3 8.1903E-3 3.6878E-3 2.1028E-2 4.7414E-3 4.7088E-2

In Section 4 it was suggested that the superiority of imbedded rules depends on the value of used to de ne the error bound coecients. This assertion is now supported by some computations. Based on a numerical search Sloan and Walsh [26] found 2  2 copies of rank-1 rules to have smaller P2 than rank-1 rules with the same number of points. Speci cally, they recommend rules of the form (6.2a) S = (L0 + Gn) \ C d ; where (6.2b) L0 = fmg0 =N0 : m 2 Zg; Gn = f(m1 =2; m2=2; m3;


; md ) : m 2 Zd g: Some of the better rules of their Table 4 are reproduced in Table 6.7. The aforementioned greedy algorithm, modi ed to search for rules of the form (6.2), yields the following generating vectors with small c~: (6.3a) g0 = (1; 109; 112; 115; 71; 72; 59; 31); N0 = 263 ( = 1); (6.3b) g0 = (1; 109; 125; 27;88; 83; 81; 45); N0 = 263 ( = 42): The following rank-1 rules with the same number of points are generated by the greedy algorithm: (6.4a) g = (1; 311; 471; 205;458;241; 870;269); N = 1052 ( = 1); (6.4b) g = (1; 311; 87; 184; 625;250;368; 684); N = 1052 ( = 42 ): The above generating vectors (6.3) and (6.4) can be used for any dimension p  8 by taking the rst p coordinates. Tables 6.8 and 6.9 give the values of c~ with = 1; 42 for the two rank-1 rules (6.4a,b) and the three rank-2 imbedded rules (6.2) corresponding to Table 6.7 and (6.3a,b).p All ve of these rules use 1052 points. For = 42 , which corresponds to ~c = P2 , the rules of Sloan and Walsh are superior in nearly every dimension, although c~ for the imbedded rules (6.3b) and rank-1 rules (6.4b) are no more than 20

Error bound coecients c~ ( = 42 ) for imbedded and rank-1 rules. d

2 3 4 5 6 7 8

Table 6.7 | 6.8109E-2 2.5429E-1 7.1781E-1 1.7730E0 | 8.6167E0

Imbedded Rules (6.3a) (6.3b) 1.2837E-2 1.2837E-2 8.6812E-2 7.3906E-2 2.8405E-1 2.4857E-1 8.1597E-1 7.4441E-1 2.0429E0 1.8200E0 4.0635E0 4.0478E0 8.8028E0 8.6130E0

Rank-1 Rules (6.4a) (6.4b) 1.3404E-2 1.3404E-2 8.2603E-2 7.6494E-2 3.5167E-1 2.8550E-1 9.0622E-1 8.5341E-1 2.3757E0 2.0823E0 5.1299E0 4.6581E0 1.0409E1 9.9161E0

10% and 20% worse, respectively. On the other hand for = 1 the rank-1 lattice rules (6.4a) are superior in every dimension, having c~ as small as half that of the best imbedded rules. 7. Discussion and Conclusion. Easily computable error bounds and worstcase integrands have been established for general quadrature rules of the form (1.1) under some regularity conditions. These error bounds apply both to general integrands (Theorem 2.1) and periodic integrands (Theorem 3.1). For lattice quadrature rules the new error bounds are closely related to error bounds in the literature (Theorem 1.1). On the other hand Theorems 2.1 and 3.1 are not restricted to lattice rules. Theorem 5.1 gives the expected error bound coecients for Monte Carlo rules. Recently error bound coecients for a variety of other random and quasirandom sets S have been computed by the author [9]. The use of the ANOVA decomposition in de ning the inner product makes it possible to investigate the eciency of quadrature rules in integrating the high and low order eects. Imbedded rules and Monte Carlo rules tend to be better for integrating functions with large high order eects, while rank-1 lattice rules tend to be better for integrating functions with large low order eects. The error bound coecients derived here bear some resemblance to the L2 discrepancy, T  (S), which according to Warnock [28] can be written as 2 Z  j S \ [0; x) j  2 ? Vol([0; x)) dx [T (S)] = d N C d d ?d+1 X Y X Y = N12 (1 ? zj2) + 3?d : [1 ? max(zj ; zj0 )] ? 2 N 0 z2S j =1 z;z 2S j =1

Moroko and Ca isch [15] have studied the L2 discrepancy and computed its values for several quasirandom sequences. They note that the L2 discrepancy \suers as a means of comparing sequences and predicting performance because of its strong emphasis on putting points near 0." Both the de nition of D (S) in 1.6 and the de nition of T  (S) above attach particular signi cance to the origin. Thus re ecting S about the plane xj = 1=2 through the center of the cube will in general change its discrepancy and L2 discrepancy. Replacing xj by 1 ? xj also changes the variation of the function. In contrast the error bounds in Theorems 2.1 and 3.1 do not have this de ciency. Both c and c~ are invariant under a re ection about any xj = 1=2 since (1 ? xj ; 1 ? yj ) = (xj ; yj ); ~(1 ? xj ; 1 ? yj ) = (xj ; yj ): Likewise the norm used to measure the size of the integrand is also unchanged when any xj is replaced by 1 ? xj . 21

Because of the generality of the reproducing kernel Hilbert space approach, we suspect that it may lead to further results for multidimensional quadrature. Perhaps for certain problems an inner product dierent than the one chosen here would be more suitable. Given any other inner product space it would be straightforward to derive the corresponding quadrature error bound, provided that the reproducing kernel is known. Although this article deals with quadrature rules having uniform weights, the theory presented here may be extended to the case of quadrature rules with non-uniform weights, provided that the weights depend only on the points in S and not on the integrand. Some rules of this type have recently been studied by Niederreiter and Sloan [18, 19]. Acknowledgments. The author wishes to thank Kai-Tai Fang, James Lyness, Art Owen, Ian Sloan and two anonymous referees for valuable discussions and suggestions regarding this manuscript. The author is also grateful for the hospitality extended to him by the Department of Statistics, Stanford University while much of this research was being conducted. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, U. S. Government Printing Oce, Washington, DC, 1964. , Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), pp. 337{ 404. , Error bounds for the method of good lattice points, Math. Comp., 56 (1991), pp. 257{266. , Lattice integration rules of maximal rank formed by copying rank 1 rules, SIAM J. Numer. Anal., 29 (1992), pp. 566{577. , The jackknife estimate of variance, Ann. Stat., 9 (1981), pp. 586{596. , Number Theoretic Methods in Statistics, Chapman and Hall, New York, 1994. , Testing multidimensional integration routines, in Tools, Methods and Languages for Scienti c and Engineering Computation, B. Ford, J. C. Rault, and F. Thomasset, eds., Amsterdam, 1984, North-Holland, pp. 81{94. , A package for testing multiple integration subroutines, in Numerical Integration: Recent Developments, Software and Applications, P. Keast and G. Fairweather, eds., Dordrecht, 1987, D. Reidel Publishing, pp. 337{340. , Comparison of random and quasirandom points for multidimensional quadrature, in A Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scienti c Computing, 1994. , Zur angenaherten berechnung mehrfacher integrale, Monatsh. Math., 66 (1962), pp. 140{151. , Applications of Number Theory to Numerical Analysis, SpringerVerlag and Science Press, Berlin and Beijing, 1981. , Intermediate rank lattice rules for multidimensional integration, SIAM J. Numer. Anal., 30 (1993), pp. 569{582. , Imbedded lattice rules for multidimensional integration, SIAM J. Numer. Anal., 29 (1992), pp. 1119{1135. , The approximate computation of multiple integrals, Dokl. Adad. Nauk. SSR, 124 (1959), pp. 1207{1210. (Russian). , Quasi-random sequences and their discrepancies, SIAM J. Sci. Comput., 15 (1994). in press. , Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992. , Lattice rules for multiple integration and discrepancy, Math. Comp., 54 (1990), pp. 303{312. , Quasi-Monte Carlo methods with modi ed vertex weights, in Numerical Integration IV, vol. 112 of Internat. Series Numerical Math., Basel, 1993, Birkhauser, pp. 253 { 265. 22

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

, Integration of nonperiodic functions of two variables by Fibonacci lattice rules, J. Comput. Appl. Math., 51 (1994), pp. 57{70. , Orthogonal arrays for computer experiments, integration and visualization, Statist. Sinica, 2 (1992), pp. 439{452. , Theory of Reproducing Kernels and Its Applications, Longman Scienti c & Technical, Essex, England, 1988. , Lattice methods for multiple integration, J. Comput. Appl. Math., 12 & 13 (1985), pp. 131{143. , Numerical integration in high dimensions | the lattice rule approach, in Numerical Integration: Recent Developments, Software and Applications, T. O. Espelid and A. Genz, eds., Dordrecht, 1992, Kluwer Academic Publishers, pp. 55{69. , Lattice Methods for Multiple Integration, Oxford University Press, New York, 1994. , Lattice methods for multiple integration: Theory, error analysis and examples, SIAM J. Numer. Anal., 24 (1987), pp. 116{128. , A computer search of rank-2 lattice rules for multidimensional quadrature, Math. Comp., 54 (1990), pp. 281{302. , Spline Models for Observational Data, SIAM, Philadelphia, 1990. , Computational investigtions of low discrepancy point sets, in Applications of Number Theory to Numerical Analysis, S. K. Zaremba, ed., New York, 1972, Academic Press, pp. 319{343.

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