OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
MATHEMATICS OF COMPUTATION Volume 00, Number 0, Xxxx XXXX, Pages 000{000 S 0025-5718(XX)0000-0
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE Abstract. This article considers the error of the scrambled equidistribution
quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, Hwav . The asymptotic orders of the errors are derived for the case of the scrambled (; t; m; s)-nets and (t; s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands Hwav .
1. Introduction Base b scrambling quadrature, proposed by Owen [Owe95], is a hybrid method of Monte Carlo and quasi-Monte Carlo methods. One starts with a low discrepancy set or sequence and randomly permutes the digits of the points. When applied to a (t; m; s)-net or a (t; s)-sequence [Nie92, Lar98] the result is a new net or sequence with the same t-value. Since each scrambled point is uniformly distributed over the unit cube, the resulting scrambled quadrature rule is unbiased. Moreover, the quadrature error may be estimated by performing a number of replications of the scrambled rule. Furthermore, there is a central limit theorem for (0; m; s)-nets as m tends to in nity [Loh01]. The purpose of this article is to show that scrambled net quadrature rules are optimal for spaces of functions that are multidimensional Haar wavelet series. This means that the asymptotic convergence rate for scrambled net quadrature is the highest possible for any quadrature rule. This result holds for worst-case, randomcase and average-case error analyses, and the asymptotic convergence rates are obtained explicitly. The integration problem studied here is integration over the s-dimensional unit cube:
I (f ) =
Z
[0;1)s
f (x)dx:
Received by the editor October 8, 2001. 1991 Mathematics Subject Classi cation. 65C05, 65D30. Key words and phrases. Quasi-Monte Carlo methods, Monte Carlo methods, high dimensional integration, lower bounds. This work was partially supported by a Hong Kong Research Grants Council grant HKBU/2030/99P, by Hong Kong Baptist University grant FRG/97-98/II-99, and by Shanghai NSF Grant 00JC14057. 1
c 1997 American Mathematical Society
2
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
Quadrature rules to approximate this integral take the form:
Q(f ; P; fwi g) =
(1)
n X i=1
wi f (xi )
for some set of nodes P = fx1 ; : : : ; xn g [0; 1)s and some set of weights fwi g = fw1 ; : : : ; wn g. Quasi-Monte Carlo quadrature methods choose P to be a set of points evenly distributed over the integration domain and wi = n?1 for all i. The quality of a quadrature rule can be assessed by a worst-case, random-case, or average-case error analysis [HW01]. Let H denote some separable Hilbert space of measurable functions, and let B be the unit ball in H, i.e., B = ff 2 H : kf kH 1g. The quadrature error for a speci c integrand in H and a speci c quadrature rule Q is given by Err(f ; Q) = I (f ) ? Q(f ; P; fwi g). Suppose that Q is random, i.e., the nodes, weights, and number of function evaluations are all chosen randomly. Speci cally, let Q be chosen from some sample space, Qn , equipped with some probability distribution, , where the average number of function evaluations is n. (Deterministic quadrature rules are the case where Qn has a single element.) In the average case we assume that H is equipped with a probability measure such that E kf k2H = 1. The worst-case, random-case, and average-case error criteria for a given Hilbert space of integrands and type of rules are: worst-case ew (H; Qn ) := Qrms (2a) sup jErr(f ; Q)j ; 2Q (2b) (2c)
n f 2B
er (H; Qn ) := sup Qrms jErr(f ; Q)j ; 2Q
random-case:
ea (H; Q
average-case:
f 2B
n
rms jErr(f ; Q)j : n ) := Qrms 2Qn f 2H
The operator rms means root mean square (or the respective expression with the upper integral if the function to which rms is applied is not measurable). The optimal error criteria for a Hilbert space are de ned as the in ma of the above with respect to all possible quadrature rules: ex (H; n) := inf (3) ex(H; Qn ); x 2 fw; r; ag: Q n
A sequence of random quadrature rules (Qnm )m=0;1;2;::: is said to be optimal if it has the same asymptotic order as the best possible quadrature rules. Speci cally, one has worst-case, random-case, and average-case optimality, respectively, if there exists some nonzero constant C independent of n such that for all n = 1; 2; : : : min ex (H; Qnm ) Cex (H; n); n n m
where x = w; r; a, respectively. It is possible for a sequence of quadrature rules to be optimal for one of the above criteria and not for the others. This article is concerned with Haar wavelet spaces, Hwav , de ned in Section 2, and quadrature rules based on scrambled (; t; m; s)-nets and (t; s)-sequences in base b and weights all chosen to be wi = 1=n. It is shown in Corollary 11 that scrambled net rules, Qsc;n , are optimal for integrating functions in Hwav in the worst, random, and average cases. This is done by showing that for some constants Cup and Clo and some function hx (n), (4) min ex (Hwav ; Qsc;nm ) Cup hx (n); n = 1; 2; : : : ; n n m
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
3
in Theorem 8, and
Clo hx (n) ex(Hwav ; n); n = 1; 2; : : : ; in Theorems 9 and 10. Here x 2 fw; r; ag again. The formula for hx (n) is found
(5)
explicitly, so the asymptotic convergence rate is known. Equation (4) can be derived by applying the results of [Hic96, HH99, HY01] for worst-case and average-case error and the results of [Owe97a, Owe97b, Owe98, Yue99, YM99] for random-case error. The problem of tractability (unbounded dimension) is not considered here but has been studied by [YH01b]. Obtaining lower bounds relies on constructing test functions (or systems of such, with a measure on them, in the random-case) that cannot be integrated too well by any quadrature rule. This approach is explained in [Nov88] and [Hei93]. There are a couple of reasons for considering the Hilbert space Hwav . First, the technical details of computing the upper bounds in (4) for scrambled nets are relatively easy for this space compared to other spaces. In other words, Hwav seems to be the natural space for studying scrambled net quadrature. Second, it is known that for many other spaces, H, computing ew (H; Qsc;n ) is equivalent to computing ew (Hwav ; Qsc;n ) [HY01]. A similar relationship holds for average-case error analysis, but unfortunately the situation is not so simple for random-case error analysis. 2. Function Spaces Spanned by Haar Wavelets This section de nes the multidimensional spaces of Haar wavelets that are used in [Owe97a] and elsewhere. Let b be an integer greater than one that will denote the base of the Haar wavelets and the (t; s)-sequences. De ne the univariate basic wavelet functions
(x) = b1=2 1bbxc= ? b?1=2 1bxc=0 ;
= 0; 1; : : : ; b ? 1; where 1fg denotes the characteristic function, and bxc denotes the oor function of x or greatest integer less than or equal to x. For integers 0 and 0 < b the
dilated and translated versions of the above functions are the following univariate wavelets: (x) = bk=2 (b x ? ) = b(k?1)=2 (b1bbk+1 xc=b + ? 1bbk xc= ):
For each subset u of the coordinate axes f1; : : : ; sg, let juj denote the cardinality of u, and let u denote the complement, f1; : : : ; sg ? u. For each r 2 u let r , r and r be integers with r 0, 0 r < br and 0 r < b. De ne the vectors = (r )r2u , = ( r )r2u , and = ( r )r2u . Let u be a product over r 2 u of the dilated and translated wavelets, i.e., u (x) :=
Y
r2u
P
r r r (xr )
= b(jj?juj)=2
Y
r2u
(b1bbr +1 xr c=b r + r ? 1bbr xr c= r );
where jj = r2u r . For u = ; we take by convention u (x) = ; (x) = 1. The wavelets de ned above are not orthogonal nor linearly independent, but they
4
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
are nearly so. As observed in [Owe97a], b?1 X
(6) (7)
r =0
Z
u (x) = 0;
8r 2 u; 8u 6= ;; ; ; u?frg Y
u (x) u0 0 0 0 (x) dx = uu0 0 0 ( r r0 ? b?1 ); s [0;1) r2u
where is the Kronecker delta function. From [Owe97a] any function f 2 L2 ([0; 1)s ) can be represented as X f= (8) f^u u ; u;; ;
where the coecients f^u are given by Z (9) f (x) u (x) dx: f^u := s [0;1)
The coecients satisfy
X
u;; ;
f^u2 < 1
and the series (8) converges in L2 ([0; 1)s ). Note that f^; is simply I (f ), the integral of the function. Because the wavelets are not linearly independent, they do not form a basis, and the series expression in (8) for f is not unique. By (6) and (9) it follows that (10)
b?1 X
r =0
f^u = 0; 8r 2 u; 8u 6= ;; ; ; u?frg :
This condition ensures uniqueness of the representation in (8) in the sense that whenever X f= cu u ; u;; ;
where the cu satisfy and it follows that
X
u;; ; b?1 X
r =0
cu = 0
c2u < 1
8r 2 u; 8u 6= ;; ; ; u?frg ;
cu = f^u
8u; ; ; :
The space of integrands, Hwav , considered in this article consists of all wavelet series whose coecients converge to zero quickly enough. Let !u denote some positive scalars de ned for all u and , satisfying (11) sup !u < 1: De ne (12)
u
(
Hwav = f
2 L2 ([0; 1)s ) :
X
u;; ;
!?1 f^2
u u
)
0 one can only obtain upper bounds on the gain coecients [Owe97a, HY01]. In the case of digital nets and sequences, precise formulae for the gain coecients can be obtained [NP01, YH01a], but these o er no improvement in the asymptotic decay rates over the upper bounds on the gain coecients for general nets and sequences. The results mentioned in this paragraph are summarized in the following two lemmas. Lemma 6. Let P be a (; t; m; s)-net in base b. Then the gain coecients of P have the following upper bounds: 8 > = 0; jj < m ? t ? juj; > > ijuj < h b +1 ?u (P ) > bt h b?1 ; i m ? t ? juj < jj m ? t; > > :
bt
bjuj +(b?2)juj 2(b?1)juj
; jj > m ? t:
For the case of t = 0, the gain coecients can be computed exactly as
?u
j+juj?1 juj (?1) hbjuj? ? bm?jj i 1 ?m+jX (b ? 1)juj =1 ( = 0; jj m ? juj; 1; jj m + 1:
(P ) = ?0?net (; m) = u
Let P contain the rst n terms of a (t; s)-sequence in base b. Let n be expressed as n = 0 + 1 b + + q bq = 0q q?1 1 0 (base b); where q = blogb (n)c and where m 2 f0; 1; : : : ; b ? 1g. Then the gain coecients of P have the following upper bounds:
juj+1 juj+1 b + 1 b + 1 1 t +1 t min( t + j j + j u j ;q +1) b : ?u (P ) n b b b?1 b?1 For t = 0, these gain coecients can be computed precisely as
q+1 X ?u (P ) = nb(b1? 1) bm Cm ?0u?net (1; m);
where C0 = b0 (b ? 0 ),
m=0
Cm = m?1 (m?1 ? 1) + bm (b ? m ) ? 2m?1 m + 2(m?1 ? m )b?m+1 and
Cq+1 = q (q ? 1) + 2q b?q
mX ?2 h=0
q?1 X h=0
h bh; m = 1; : : : ; q;
h bh:
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
11
Lemma 7. Let a > 0 be a scalar, and let u be a subset of f1; : : : ; sg. If P is a (; t; m; s)-net in base b, then
n?1
where n then
= bm . If
X
b?ajj ?u (P ) = O(n?1?a [log n]s?1 );
P consists of the rst n points of a (t; s)-sequence in base b, 8 >
O(n?2 [log n]s ); a = 1; : ? 2 s ? 1 O(n [log n] ); a > 1;
with constants in the O-notation not depending on n and P . According to Theorem 1 the asymptotic convergence rates of the three error criteria depend, not only on the decay rates of the gain coecients, which are given in the lemma above, but also on the decay rates of the the !u (and u for the average case). From here on it is assumed that ?s ( b ? 1) j u j ? 2 j j (24) !u = 0 1 b juj b?2 jj ; ; u = 1 + 1 ? b1?2 where 0 ; 1 ; are positive numbers, and is non-negative. The summability condition in (14) for worst-case analysis is equivalent to > 1=2, while condition (11) is equivalent to 0. Condition (17) for the average-case analysis is equivalent to > 1=2. The constants and determine the asymptotic rates of decay of the three error criteria. Let Qsc;n denote Owen's scrambled quasi-Monte Carlo quadrature using a (; t; m; s)-net with n = bm points as the node set, where = 0; 1; : : : ; b ? 1 and m = 0; 1; : : : . Theorem 8. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Then for scrambled (; t; m; s)-nets in base b the three error criteria have the following asymptotic orders: (25a) ew (Hwav ; Qsc;n ) = O(n? [log n](s?1)=2 ); > 1=2; (25b) er (Hwav ; Qsc;n ) = O(n? ?1=2 ); 0; (25c) ea (Hwav ; Qsc;n ) = O(n? ? [log n](s?1)=2 ); 0; > 1=2: Proof. This theorem is obtained by applying the formulas in Lemma 7 to those of Theorem 1 under the assumption of (24). The key step is to note that the value of a in the term b?ajj determines the asymptotic rate of decay. The formulae given in Theorem 4 have been evaluated numerically for scrambled (; 0; m; s)-nets and (0; s)-sequences in base b, where s = 1; : : : ; 9, and the base b equal to all prime powers between s and 9 inclusively. For the (; 0; m; s)-net, the sample size is n = bm with m 0, 1 < b, while for the (0; s)-sequence, n can be any integer. The !u and u are assumed to be as follows: (26) !u = (6b)?juj b?2jj ; u = (bs ? 1)?1 (1 ? b?1)juj b?2jj : Conditions (14) and (17) are automatically satis ed. Some of these calculations are shown in Figures 1 and 2. The three error criteria for simple Monte Carlo quadrature as given in Theorem are also shown for comparison. The error criteria
12
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
for scrambled net or sequence quadrature are never larger than the corresponding errors of the simple Monte Carlo quadrature rules. Moreover, as the dimension s increases, ew (Hwav ; Qsc;n ) increases, er (Hwav ; Qsc;n ) stays about the same, and ea (Hwav ; Qsc;n ) decreases. 5. Lower Bounds for the Error Criteria In the previous section the asymptotic convergence rates of worst-case, randomcase and average-case quadrature errors were determined for scrambled net quadrature applied to integrands in Hwav . In this section it is shown that these convergence rates are optimal, i.e., they are the best possible for any random quadrature rule. Theorem 9. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Then the worst-case error and average-case criteria for this space as de ned in (3) have the lower bounds (27) ew (Hwav ; n) Cw n? [log n](s?1)=2 ; > 1=2; a ? ? ( s ? 1) = 2 (28) e (Hwav ; n) Ca n [log n] ; 0; > 1=2; for some constants Cw and Ca that are independent of n. Proof. This proof proceeds by constructing a function in Hwav that cannot be integrated very well by any quadrature rule. Let n be any given positive integer. Choose an integer m such that (29) bm?1 < 2n bm : For each s-vector = (1 ; : : : ; s ) of non-negative integers and each s-vector ` = (`1 ; : : : ; `s ) of integers 0 `r < bj , de ne the elementary interval in base b as s ` ` + 1 Y r; r (30) : B = `
r=1
br br
It is clear that the set of all B` is a partition of [0; 1)s into bjj elementary intervals of volume b?jj . Let P = fx1 ; : : : ; xn g [0; 1)s be any given set of points. We de ne the function f for each with j j = m by (
f (x) = 1; for all x 2 B` with B` \ P = ;; 0; otherwise:
Then f 2 Hwav and
I (f ) =
Z
[0;1)
1 ?m m f (x) dx b (b ? n) 2 s P
since 2n bm from (29). Now we de ne f0 = j j=m f . Then f0 2 Hwav , f0(xi ) = 0, for i = 1; : : : ; n, and X I (f ) 12 m s+?s 1? 1 c1 ms?1 ; (31) Err(f0 ; Q) = I (f0 ) = jj=m where c1 is a positive constant.
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
13
1
10
0
10
−1
10
−2
10
−3
10
eworst sc−wav
−4
10
−5
10
erand sc−wav
−6
10
eavg sc−wav
−7
10
s=4, b=11 −8
10
−9
10
0
1
10
10
2
3
10
10
4
5
10
10
n=λ bm 1
10
0
10
−1
10
−2
10
−3
worst
10
esc−wav
−4
10
−5
10
erand sc−wav
−6
10
−7
10
eavg sc−wav
s=7, b=11 −8
10
−9
10
0
10
1
10
2
3
10
10
4
10
5
10
n=λ bm
Figure 1. Performance of scrambled (; 0; m; s)-net quadrature for the space Hwav with the !u and u speci ed in (26). The three reference lines are for ew (Hwav ; Qmc;n ) (solid), er (Hwav ; Qmc;n ) (dashed) and ea (Hwav ; Qmc;n ) (dot-dashed), re-
spectively.
14
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE 1
10
0
10
−1
10
eworst sc−wav
−2
10
−3
10
erand sc−wav eavg sc−wav
−4
10
−5
s=4, b=4
10
−6
10
0
1
10
2
10
3
10
10
n 1
10
0
10
−1
10
eworst sc−wav −2
10
erand
−3
sc−wav
10
−4
10
−5
eavg sc−wav
s=7, b=7
10
−6
10
0
10
1
2
10
10
3
10
n
Figure 2. Performance of scrambled (0; s)-sequence quadrature for the space Hwav with the !u and u speci ed in (26). The three reference lines are for ew (Hwav ; Qmc;n ) (solid), er (Hwav ; Qmc;n ) (dashed) and ea (Hwav ; Qmc;n ) (dot-dashed), re-
spectively.
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
15
To complete the proof the norm of f0 must be estimated. For any juj-vector and any s-vector , the notation means that r r for all r 2 u. The wavelet series expansion for the function f de ned above is a nite sum:
f (x) =
X
;
f^;u u (x); where f^ ;u =
Z
[0;1)s
f (x) u (x) dx:
Note that the set of support for u is B(;0) by de nition (30), where (; 0) is the s-vector obtained by padding with zeros for all r 2= u. Since kf k1 = 1 and k u k1 = b(jj?juj)=2 (b ? 1)juj , it follows that ^ ;u
f
=
Z ( B(;0)
f x) u (x) dx
b?jj b(jj?juj)=2 (b ? 1)juj = (b ? 1)juj b?(jj+juj)=2 :
Based on the above upper bound, the square norm of f0 can now be written as follows:
kf0 k2Hwav = hf0; f0 iHwav =
X
X
j j=j0 j=m
X
X
hf ; f0 iHwav
!u?1 f^;u f^ 0 ;u
j j=j0 j=m uf1;::: ;sg0 min( ; ) X X 0?1 1?juj b2 jj(b ? 1)2juj uf1;::: ;sg ; 0 0 0jjm j j=j j=m
0?1 max(1; 1?s )(b ? 1)2s ?
m X p=0
3
2
X
6
b2 p 64
X
uf1;::: ;sg ; 0 jj=p j?(;0)j=j0 ?(;0)j=m?p
7
175 :
Given u, and , there are m?sp?+1s?1 (m ? p + s ? 1)s?1 ways to choose a non?s negative integer s-vector ? (; 0) with cardinality m ? p . Moreover, there are q ? subsets u with juj = q, and for each of them p+q?q?1 1 ways to construct a with jj = p. Since s X q=0
s q
p + q ? 1 2s (p + s ? 1)s?1 2s (m + s ? 1)s?1 ; q?1
16
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
the upper bound on kf0 k2Hwav may be written as
kf0 k2Hwav 0?1 max(1; 1?s )(b ? 1)2s 2s (m + s ? 1)s?1
(32)
m X p=0
(m ? p + s ? 1)2(s?1) b2 p
0 max(1; 1?s )(b ? 1)2s 2s (m + s ? 1)s?1 b2 m
m X
(q + s ? 1)2(s?1) b?2 q
q=0 s ? 1 2 c2 m b m :
for some c2 depending on b, s, and , but not on m. Letting f = f0 = kf0kHwav , it follows that f lies in the unit ball, Bwav . Moreover, from (29), (31) and (32) it follows that s?1
jErr(f ; Q)j pc cm1 ms?1 b2 m = pcc1 m(s?1)=2 b? m Cw n? [log n](s?1)=2 2
2
for some constant Cw . This implies (27). Formula (28) then follows from Lemma (2).
Theorem 10. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Then the random-case error criterion for this space as de ned in (3) has the lower bound
(33) er (Hwav ; n) Cr n? ?1=2 ; 0; for some constant Cr that is independent of n. Proof. It is sucient to consider the case with s = 1, since integration in the space Hwav with s = 1 is no harder than integration in the respective Hwav with s > 1 (the one dimensional Hwav can be identi ed with the subspace of the multidimensional Hwav consisting of functions depending only on the rst variable, or equivalently, of functions f with f^u = 0 for all u except u = ; and u = f1g). By convention the weights !u in (24) for the one dimensional case are written as !; = 0 and ! = 0 b?2 , for = 0; 1; 2; : : : . The square norm of the function in Hwav with s = 1 is ?1 b?1 1 bX X X ? 1 2 ! ?1 ; 2 2 f^ kf k = f^; !; + =0 =0 =0
where
f^; =
Z
1
0
f (x) dx and f^ =
Z
0
1
f (x) (x) dx:
Let n; m be taken as in (29) in the proof of the previous theorem. Again de ne basic intervals B = ` ; ` + 1 ; ` = 0; 1; : : : ; bm ? 1; m`
bm bm
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
17
and let g` = 1Bm` (x) be the indicator function of Bm` . It follows that I (g` ) = b?m. De ne
g=
m ?1 bX
`=0
"`g` ; where "` 2 f1; ?1g:
The norm of this function can be bounded in a similar manner as was done in the proof of the previous theorem. First the series coecients of g are bounded as follows: jg^;j 1 and
jg^ j =
Z ( +1)b? b?
g(x) (x) dx b(?1)=2(b ? 1)b? = b?(+1)=2(b ? 1):
This gives an upper bound on the norm of g: ?1 b?1 m bX X
?1 X Hwav = !; +
kgk2
=0 =0 =0
2 ! ?1 ?1 + ?1 (b ? 1)2 g^ 0 0
m X =0
b2 c23 b2 m ;
for some constant c3 that is independent of m. Now consider the following functions f` = g` =(c3 b m ). These f` have disjoint supports and satisfy
I (f` ) c3 b?( +1)m ;
m ?1
bX
`=0
`
"` f 1 for any "` 2 f1; ?1g:
From [Nov88, Section 2.2.4, Proposition 1] it follows that er(Hwav ; n) c4 b?( +1)m(bm )1=2 = c4 b?( +1=2)m Cr n? ?1=2 : This completes the proof of (33). Theorems 8, 9, and 10 may be combined to summarize the main results of this article, namely that scrambled net quadrature is optimal for Haar wavelet series in all three settings. We use the standard notation an bn for non-negative sequences an and bn meaning that there are constants clo ; cup > 0 such that clo bn an cupbn . Corollary 11. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Let Qsc;n denote Owen's scrambled quasi-Monte Carlo quadrature using a (; t; m; s)-net with bm points as the node set, where = 0; 1; : : : ; b ? 1 and m = 0; 1; : : : . Then, > 1=2; min ew (Hwav ; Qsc;bm ) ew (Hwav ; n) n? [log n](s?1)=2 ; bm n
0; min er (Hwav ; Qsc;bm ) er (Hwav ; n) n? ?1=2 ; min ea(Hwav ; Qsc;bm ) ea (Hwav ; n) n? ? [log n](s?1)=2 ; 0; > 1=2: bm n bm n
Remark. Usually the gap between worst-case and random-case n-th minimal errors is not more than of order n1=2 . This is the rst time that the authors have encountered a situation where this gap is larger, namely, ew (Hwav ; n)=er (Hwav ; n) is of the order n1=2 [log n](s?1)=2 .
18
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
References [Hei93] [HH99] [Hic96] [HW01] [HY01] [Lar98] [Loh01] [Nie92] [Nov88] [NP01] [Owe95] [Owe97a] [Owe97b] [Owe98] [Owe99] [Rit00] [YH01a] [YH01b] [YM99] [Yue99]
S. Heinrich, Random approximation in numerical analysis, Functional analysis. Proceedings of the Essen Conference, held in Essen, Germany, November 24 - 30, 1991. (K. D. Bierstedt, A. Pietsch, W. M. Ruess, and D. Vogt, eds.), Lecture Notes in Pure and Applied Mathematics, vol. 150, Marcel Dekker, New York, 1993, pp. 123{171. F. J. Hickernell and H. S. Hong, The asymptotic eciency of randomized nets for quadrature, Math. Comp. 68 (1999), 767{791. F. J. Hickernell, The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul. 6 (1996), 274{296. F. J. Hickernell and H. Wozniakowski, The price of pessimism for multidimensional quadrature, J. Complexity 17 (2001), to appear. F. J. Hickernell and R. X. Yue, The mean square discrepancy of scrambled (t; s)sequences, SIAM J. Numer. Anal. 38 (2001), 1089{1112. G. Larcher, Digital point sets: Analysis and applications, Random and Quasi-Random Point Sets (P. Hellekalek and G. Larcher, eds.), Lecture Notes in Statistics, vol. 138, Springer-Verlag, New York, 1998, pp. 167{222. W.-L. Loh, On the asymptotic distribution of scrambled net quadrature, 2001, submitted for publication. H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMSNSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992. E. Novak, Deterministic and stochastic error bounds in numerical analysis, Lectures Notes in Math., no. 1349, Springer-Verlag, Berlin, 1988. H. Niederreiter and G. Pirsic, The microstructure of (t; m; s)-nets, J. Complexity 17 (2001), to appear. A. B. Owen, Randomly permuted (t; m; s)-nets and (t; s)-sequences, Monte Carlo and Quasi-Monte Carlo Methods in Scienti c Computing (H. Niederreiter and P. J.-S. Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995, pp. 299{ 317. A. B. Owen, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997), 1884{1910. A. B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Stat. 25 (1997), 1541{1562. A. B. Owen, Scrambling Sobol' and Niederreiter-Xing points, J. Complexity 14 (1998), 466{489. A. B. Owen, Monte Carlo, quasi-Monte carlo, and randomized quasi-Monte Carlo, Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.), Springer-Verlag, Berlin, 1999. K. Ritter, Average-case analysis of numerical problems, Lecture Notes in Mathematics, vol. 1733, Springer-Verlag, Berlin, 2000. R. X. Yue and F. J. Hickernell, The discrepancy of digital nets, 2001, submitted to J. Complexity. R. X. Yue and F. J. Hickernell, Integration and approximation based on scramble sampling in arbitrary dimensions, J. Complexity 17 (2001), to appear. R. X. Yue and S. S. Mao, On the variance of quadrature over scrambled nets and sequences, Statist. Probab. Lett. 44 (1999), 267{280. R. X. Yue, Variance of quadrature over scrambled unions of nets, Statist. Sinica 9 (1999), 451{473.
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
19
FB Informatik, Universitat Kaiserslautern, PF 3049, D-67653 Kaiserslautern, Germany
E-mail address :
[email protected]
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
E-mail address :
[email protected]
College of Mathematical Science, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China
E-mail address :
[email protected]
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE Abstract. This article considers the error of the scrambled equidistribution
quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, Hwav . The asymptotic orders of the errors are derived for the case of the scrambled (; t; m; s)-nets and (t; s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands Hwav .
1. Introduction Base b scrambling quadrature, proposed by Owen [Owe95], is a hybrid method of Monte Carlo and quasi-Monte Carlo methods. One starts with a low discrepancy set or sequence and randomly permutes the digits of the points. When applied to a (t; m; s)-net or a (t; s)-sequence [Nie92, Lar98] the result is a new net or sequence with the same t-value. Since each scrambled point is uniformly distributed over the unit cube, the resulting scrambled quadrature rule is unbiased. Moreover, the quadrature error may be estimated by performing a number of replications of the scrambled rule. Furthermore, there is a central limit theorem for (0; m; s)-nets as m tends to in nity [Loh01]. The purpose of this article is to show that scrambled net quadrature rules are optimal for spaces of functions that are multidimensional Haar wavelet series. This means that the asymptotic convergence rate for scrambled net quadrature is the highest possible for any quadrature rule. This result holds for worst-case, randomcase and average-case error analyses, and the asymptotic convergence rates are obtained explicitly. The integration problem studied here is integration over the s-dimensional unit cube:
I (f ) =
Z
[0;1)s
f (x)dx:
Received by the editor October 8, 2001. 1991 Mathematics Subject Classi cation. 65C05, 65D30. Key words and phrases. Quasi-Monte Carlo methods, Monte Carlo methods, high dimensional integration, lower bounds. This work was partially supported by a Hong Kong Research Grants Council grant HKBU/2030/99P, by Hong Kong Baptist University grant FRG/97-98/II-99, and by Shanghai NSF Grant 00JC14057. 1
c 1997 American Mathematical Society
2
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
Quadrature rules to approximate this integral take the form:
Q(f ; P; fwi g) =
(1)
n X i=1
wi f (xi )
for some set of nodes P = fx1 ; : : : ; xn g [0; 1)s and some set of weights fwi g = fw1 ; : : : ; wn g. Quasi-Monte Carlo quadrature methods choose P to be a set of points evenly distributed over the integration domain and wi = n?1 for all i. The quality of a quadrature rule can be assessed by a worst-case, random-case, or average-case error analysis [HW01]. Let H denote some separable Hilbert space of measurable functions, and let B be the unit ball in H, i.e., B = ff 2 H : kf kH 1g. The quadrature error for a speci c integrand in H and a speci c quadrature rule Q is given by Err(f ; Q) = I (f ) ? Q(f ; P; fwi g). Suppose that Q is random, i.e., the nodes, weights, and number of function evaluations are all chosen randomly. Speci cally, let Q be chosen from some sample space, Qn , equipped with some probability distribution, , where the average number of function evaluations is n. (Deterministic quadrature rules are the case where Qn has a single element.) In the average case we assume that H is equipped with a probability measure such that E kf k2H = 1. The worst-case, random-case, and average-case error criteria for a given Hilbert space of integrands and type of rules are: worst-case ew (H; Qn ) := Qrms (2a) sup jErr(f ; Q)j ; 2Q (2b) (2c)
n f 2B
er (H; Qn ) := sup Qrms jErr(f ; Q)j ; 2Q
random-case:
ea (H; Q
average-case:
f 2B
n
rms jErr(f ; Q)j : n ) := Qrms 2Qn f 2H
The operator rms means root mean square (or the respective expression with the upper integral if the function to which rms is applied is not measurable). The optimal error criteria for a Hilbert space are de ned as the in ma of the above with respect to all possible quadrature rules: ex (H; n) := inf (3) ex(H; Qn ); x 2 fw; r; ag: Q n
A sequence of random quadrature rules (Qnm )m=0;1;2;::: is said to be optimal if it has the same asymptotic order as the best possible quadrature rules. Speci cally, one has worst-case, random-case, and average-case optimality, respectively, if there exists some nonzero constant C independent of n such that for all n = 1; 2; : : : min ex (H; Qnm ) Cex (H; n); n n m
where x = w; r; a, respectively. It is possible for a sequence of quadrature rules to be optimal for one of the above criteria and not for the others. This article is concerned with Haar wavelet spaces, Hwav , de ned in Section 2, and quadrature rules based on scrambled (; t; m; s)-nets and (t; s)-sequences in base b and weights all chosen to be wi = 1=n. It is shown in Corollary 11 that scrambled net rules, Qsc;n , are optimal for integrating functions in Hwav in the worst, random, and average cases. This is done by showing that for some constants Cup and Clo and some function hx (n), (4) min ex (Hwav ; Qsc;nm ) Cup hx (n); n = 1; 2; : : : ; n n m
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
3
in Theorem 8, and
Clo hx (n) ex(Hwav ; n); n = 1; 2; : : : ; in Theorems 9 and 10. Here x 2 fw; r; ag again. The formula for hx (n) is found
(5)
explicitly, so the asymptotic convergence rate is known. Equation (4) can be derived by applying the results of [Hic96, HH99, HY01] for worst-case and average-case error and the results of [Owe97a, Owe97b, Owe98, Yue99, YM99] for random-case error. The problem of tractability (unbounded dimension) is not considered here but has been studied by [YH01b]. Obtaining lower bounds relies on constructing test functions (or systems of such, with a measure on them, in the random-case) that cannot be integrated too well by any quadrature rule. This approach is explained in [Nov88] and [Hei93]. There are a couple of reasons for considering the Hilbert space Hwav . First, the technical details of computing the upper bounds in (4) for scrambled nets are relatively easy for this space compared to other spaces. In other words, Hwav seems to be the natural space for studying scrambled net quadrature. Second, it is known that for many other spaces, H, computing ew (H; Qsc;n ) is equivalent to computing ew (Hwav ; Qsc;n ) [HY01]. A similar relationship holds for average-case error analysis, but unfortunately the situation is not so simple for random-case error analysis. 2. Function Spaces Spanned by Haar Wavelets This section de nes the multidimensional spaces of Haar wavelets that are used in [Owe97a] and elsewhere. Let b be an integer greater than one that will denote the base of the Haar wavelets and the (t; s)-sequences. De ne the univariate basic wavelet functions
(x) = b1=2 1bbxc= ? b?1=2 1bxc=0 ;
= 0; 1; : : : ; b ? 1; where 1fg denotes the characteristic function, and bxc denotes the oor function of x or greatest integer less than or equal to x. For integers 0 and 0 < b the
dilated and translated versions of the above functions are the following univariate wavelets: (x) = bk=2 (b x ? ) = b(k?1)=2 (b1bbk+1 xc=b + ? 1bbk xc= ):
For each subset u of the coordinate axes f1; : : : ; sg, let juj denote the cardinality of u, and let u denote the complement, f1; : : : ; sg ? u. For each r 2 u let r , r and r be integers with r 0, 0 r < br and 0 r < b. De ne the vectors = (r )r2u , = ( r )r2u , and = ( r )r2u . Let u be a product over r 2 u of the dilated and translated wavelets, i.e., u (x) :=
Y
r2u
P
r r r (xr )
= b(jj?juj)=2
Y
r2u
(b1bbr +1 xr c=b r + r ? 1bbr xr c= r );
where jj = r2u r . For u = ; we take by convention u (x) = ; (x) = 1. The wavelets de ned above are not orthogonal nor linearly independent, but they
4
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
are nearly so. As observed in [Owe97a], b?1 X
(6) (7)
r =0
Z
u (x) = 0;
8r 2 u; 8u 6= ;; ; ; u?frg Y
u (x) u0 0 0 0 (x) dx = uu0 0 0 ( r r0 ? b?1 ); s [0;1) r2u
where is the Kronecker delta function. From [Owe97a] any function f 2 L2 ([0; 1)s ) can be represented as X f= (8) f^u u ; u;; ;
where the coecients f^u are given by Z (9) f (x) u (x) dx: f^u := s [0;1)
The coecients satisfy
X
u;; ;
f^u2 < 1
and the series (8) converges in L2 ([0; 1)s ). Note that f^; is simply I (f ), the integral of the function. Because the wavelets are not linearly independent, they do not form a basis, and the series expression in (8) for f is not unique. By (6) and (9) it follows that (10)
b?1 X
r =0
f^u = 0; 8r 2 u; 8u 6= ;; ; ; u?frg :
This condition ensures uniqueness of the representation in (8) in the sense that whenever X f= cu u ; u;; ;
where the cu satisfy and it follows that
X
u;; ; b?1 X
r =0
cu = 0
c2u < 1
8r 2 u; 8u 6= ;; ; ; u?frg ;
cu = f^u
8u; ; ; :
The space of integrands, Hwav , considered in this article consists of all wavelet series whose coecients converge to zero quickly enough. Let !u denote some positive scalars de ned for all u and , satisfying (11) sup !u < 1: De ne (12)
u
(
Hwav = f
2 L2 ([0; 1)s ) :
X
u;; ;
!?1 f^2
u u
)
0 one can only obtain upper bounds on the gain coecients [Owe97a, HY01]. In the case of digital nets and sequences, precise formulae for the gain coecients can be obtained [NP01, YH01a], but these o er no improvement in the asymptotic decay rates over the upper bounds on the gain coecients for general nets and sequences. The results mentioned in this paragraph are summarized in the following two lemmas. Lemma 6. Let P be a (; t; m; s)-net in base b. Then the gain coecients of P have the following upper bounds: 8 > = 0; jj < m ? t ? juj; > > ijuj < h b +1 ?u (P ) > bt h b?1 ; i m ? t ? juj < jj m ? t; > > :
bt
bjuj +(b?2)juj 2(b?1)juj
; jj > m ? t:
For the case of t = 0, the gain coecients can be computed exactly as
?u
j+juj?1 juj (?1) hbjuj? ? bm?jj i 1 ?m+jX (b ? 1)juj =1 ( = 0; jj m ? juj; 1; jj m + 1:
(P ) = ?0?net (; m) = u
Let P contain the rst n terms of a (t; s)-sequence in base b. Let n be expressed as n = 0 + 1 b + + q bq = 0q q?1 1 0 (base b); where q = blogb (n)c and where m 2 f0; 1; : : : ; b ? 1g. Then the gain coecients of P have the following upper bounds:
juj+1 juj+1 b + 1 b + 1 1 t +1 t min( t + j j + j u j ;q +1) b : ?u (P ) n b b b?1 b?1 For t = 0, these gain coecients can be computed precisely as
q+1 X ?u (P ) = nb(b1? 1) bm Cm ?0u?net (1; m);
where C0 = b0 (b ? 0 ),
m=0
Cm = m?1 (m?1 ? 1) + bm (b ? m ) ? 2m?1 m + 2(m?1 ? m )b?m+1 and
Cq+1 = q (q ? 1) + 2q b?q
mX ?2 h=0
q?1 X h=0
h bh; m = 1; : : : ; q;
h bh:
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
11
Lemma 7. Let a > 0 be a scalar, and let u be a subset of f1; : : : ; sg. If P is a (; t; m; s)-net in base b, then
n?1
where n then
= bm . If
X
b?ajj ?u (P ) = O(n?1?a [log n]s?1 );
P consists of the rst n points of a (t; s)-sequence in base b, 8 >
O(n?2 [log n]s ); a = 1; : ? 2 s ? 1 O(n [log n] ); a > 1;
with constants in the O-notation not depending on n and P . According to Theorem 1 the asymptotic convergence rates of the three error criteria depend, not only on the decay rates of the gain coecients, which are given in the lemma above, but also on the decay rates of the the !u (and u for the average case). From here on it is assumed that ?s ( b ? 1) j u j ? 2 j j (24) !u = 0 1 b juj b?2 jj ; ; u = 1 + 1 ? b1?2 where 0 ; 1 ; are positive numbers, and is non-negative. The summability condition in (14) for worst-case analysis is equivalent to > 1=2, while condition (11) is equivalent to 0. Condition (17) for the average-case analysis is equivalent to > 1=2. The constants and determine the asymptotic rates of decay of the three error criteria. Let Qsc;n denote Owen's scrambled quasi-Monte Carlo quadrature using a (; t; m; s)-net with n = bm points as the node set, where = 0; 1; : : : ; b ? 1 and m = 0; 1; : : : . Theorem 8. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Then for scrambled (; t; m; s)-nets in base b the three error criteria have the following asymptotic orders: (25a) ew (Hwav ; Qsc;n ) = O(n? [log n](s?1)=2 ); > 1=2; (25b) er (Hwav ; Qsc;n ) = O(n? ?1=2 ); 0; (25c) ea (Hwav ; Qsc;n ) = O(n? ? [log n](s?1)=2 ); 0; > 1=2: Proof. This theorem is obtained by applying the formulas in Lemma 7 to those of Theorem 1 under the assumption of (24). The key step is to note that the value of a in the term b?ajj determines the asymptotic rate of decay. The formulae given in Theorem 4 have been evaluated numerically for scrambled (; 0; m; s)-nets and (0; s)-sequences in base b, where s = 1; : : : ; 9, and the base b equal to all prime powers between s and 9 inclusively. For the (; 0; m; s)-net, the sample size is n = bm with m 0, 1 < b, while for the (0; s)-sequence, n can be any integer. The !u and u are assumed to be as follows: (26) !u = (6b)?juj b?2jj ; u = (bs ? 1)?1 (1 ? b?1)juj b?2jj : Conditions (14) and (17) are automatically satis ed. Some of these calculations are shown in Figures 1 and 2. The three error criteria for simple Monte Carlo quadrature as given in Theorem are also shown for comparison. The error criteria
12
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
for scrambled net or sequence quadrature are never larger than the corresponding errors of the simple Monte Carlo quadrature rules. Moreover, as the dimension s increases, ew (Hwav ; Qsc;n ) increases, er (Hwav ; Qsc;n ) stays about the same, and ea (Hwav ; Qsc;n ) decreases. 5. Lower Bounds for the Error Criteria In the previous section the asymptotic convergence rates of worst-case, randomcase and average-case quadrature errors were determined for scrambled net quadrature applied to integrands in Hwav . In this section it is shown that these convergence rates are optimal, i.e., they are the best possible for any random quadrature rule. Theorem 9. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Then the worst-case error and average-case criteria for this space as de ned in (3) have the lower bounds (27) ew (Hwav ; n) Cw n? [log n](s?1)=2 ; > 1=2; a ? ? ( s ? 1) = 2 (28) e (Hwav ; n) Ca n [log n] ; 0; > 1=2; for some constants Cw and Ca that are independent of n. Proof. This proof proceeds by constructing a function in Hwav that cannot be integrated very well by any quadrature rule. Let n be any given positive integer. Choose an integer m such that (29) bm?1 < 2n bm : For each s-vector = (1 ; : : : ; s ) of non-negative integers and each s-vector ` = (`1 ; : : : ; `s ) of integers 0 `r < bj , de ne the elementary interval in base b as s ` ` + 1 Y r; r (30) : B = `
r=1
br br
It is clear that the set of all B` is a partition of [0; 1)s into bjj elementary intervals of volume b?jj . Let P = fx1 ; : : : ; xn g [0; 1)s be any given set of points. We de ne the function f for each with j j = m by (
f (x) = 1; for all x 2 B` with B` \ P = ;; 0; otherwise:
Then f 2 Hwav and
I (f ) =
Z
[0;1)
1 ?m m f (x) dx b (b ? n) 2 s P
since 2n bm from (29). Now we de ne f0 = j j=m f . Then f0 2 Hwav , f0(xi ) = 0, for i = 1; : : : ; n, and X I (f ) 12 m s+?s 1? 1 c1 ms?1 ; (31) Err(f0 ; Q) = I (f0 ) = jj=m where c1 is a positive constant.
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
13
1
10
0
10
−1
10
−2
10
−3
10
eworst sc−wav
−4
10
−5
10
erand sc−wav
−6
10
eavg sc−wav
−7
10
s=4, b=11 −8
10
−9
10
0
1
10
10
2
3
10
10
4
5
10
10
n=λ bm 1
10
0
10
−1
10
−2
10
−3
worst
10
esc−wav
−4
10
−5
10
erand sc−wav
−6
10
−7
10
eavg sc−wav
s=7, b=11 −8
10
−9
10
0
10
1
10
2
3
10
10
4
10
5
10
n=λ bm
Figure 1. Performance of scrambled (; 0; m; s)-net quadrature for the space Hwav with the !u and u speci ed in (26). The three reference lines are for ew (Hwav ; Qmc;n ) (solid), er (Hwav ; Qmc;n ) (dashed) and ea (Hwav ; Qmc;n ) (dot-dashed), re-
spectively.
14
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE 1
10
0
10
−1
10
eworst sc−wav
−2
10
−3
10
erand sc−wav eavg sc−wav
−4
10
−5
s=4, b=4
10
−6
10
0
1
10
2
10
3
10
10
n 1
10
0
10
−1
10
eworst sc−wav −2
10
erand
−3
sc−wav
10
−4
10
−5
eavg sc−wav
s=7, b=7
10
−6
10
0
10
1
2
10
10
3
10
n
Figure 2. Performance of scrambled (0; s)-sequence quadrature for the space Hwav with the !u and u speci ed in (26). The three reference lines are for ew (Hwav ; Qmc;n ) (solid), er (Hwav ; Qmc;n ) (dashed) and ea (Hwav ; Qmc;n ) (dot-dashed), re-
spectively.
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
15
To complete the proof the norm of f0 must be estimated. For any juj-vector and any s-vector , the notation means that r r for all r 2 u. The wavelet series expansion for the function f de ned above is a nite sum:
f (x) =
X
;
f^;u u (x); where f^ ;u =
Z
[0;1)s
f (x) u (x) dx:
Note that the set of support for u is B(;0) by de nition (30), where (; 0) is the s-vector obtained by padding with zeros for all r 2= u. Since kf k1 = 1 and k u k1 = b(jj?juj)=2 (b ? 1)juj , it follows that ^ ;u
f
=
Z ( B(;0)
f x) u (x) dx
b?jj b(jj?juj)=2 (b ? 1)juj = (b ? 1)juj b?(jj+juj)=2 :
Based on the above upper bound, the square norm of f0 can now be written as follows:
kf0 k2Hwav = hf0; f0 iHwav =
X
X
j j=j0 j=m
X
X
hf ; f0 iHwav
!u?1 f^;u f^ 0 ;u
j j=j0 j=m uf1;::: ;sg0 min( ; ) X X 0?1 1?juj b2 jj(b ? 1)2juj uf1;::: ;sg ; 0 0 0jjm j j=j j=m
0?1 max(1; 1?s )(b ? 1)2s ?
m X p=0
3
2
X
6
b2 p 64
X
uf1;::: ;sg ; 0 jj=p j?(;0)j=j0 ?(;0)j=m?p
7
175 :
Given u, and , there are m?sp?+1s?1 (m ? p + s ? 1)s?1 ways to choose a non?s negative integer s-vector ? (; 0) with cardinality m ? p . Moreover, there are q ? subsets u with juj = q, and for each of them p+q?q?1 1 ways to construct a with jj = p. Since s X q=0
s q
p + q ? 1 2s (p + s ? 1)s?1 2s (m + s ? 1)s?1 ; q?1
16
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
the upper bound on kf0 k2Hwav may be written as
kf0 k2Hwav 0?1 max(1; 1?s )(b ? 1)2s 2s (m + s ? 1)s?1
(32)
m X p=0
(m ? p + s ? 1)2(s?1) b2 p
0 max(1; 1?s )(b ? 1)2s 2s (m + s ? 1)s?1 b2 m
m X
(q + s ? 1)2(s?1) b?2 q
q=0 s ? 1 2 c2 m b m :
for some c2 depending on b, s, and , but not on m. Letting f = f0 = kf0kHwav , it follows that f lies in the unit ball, Bwav . Moreover, from (29), (31) and (32) it follows that s?1
jErr(f ; Q)j pc cm1 ms?1 b2 m = pcc1 m(s?1)=2 b? m Cw n? [log n](s?1)=2 2
2
for some constant Cw . This implies (27). Formula (28) then follows from Lemma (2).
Theorem 10. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Then the random-case error criterion for this space as de ned in (3) has the lower bound
(33) er (Hwav ; n) Cr n? ?1=2 ; 0; for some constant Cr that is independent of n. Proof. It is sucient to consider the case with s = 1, since integration in the space Hwav with s = 1 is no harder than integration in the respective Hwav with s > 1 (the one dimensional Hwav can be identi ed with the subspace of the multidimensional Hwav consisting of functions depending only on the rst variable, or equivalently, of functions f with f^u = 0 for all u except u = ; and u = f1g). By convention the weights !u in (24) for the one dimensional case are written as !; = 0 and ! = 0 b?2 , for = 0; 1; 2; : : : . The square norm of the function in Hwav with s = 1 is ?1 b?1 1 bX X X ? 1 2 ! ?1 ; 2 2 f^ kf k = f^; !; + =0 =0 =0
where
f^; =
Z
1
0
f (x) dx and f^ =
Z
0
1
f (x) (x) dx:
Let n; m be taken as in (29) in the proof of the previous theorem. Again de ne basic intervals B = ` ; ` + 1 ; ` = 0; 1; : : : ; bm ? 1; m`
bm bm
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
17
and let g` = 1Bm` (x) be the indicator function of Bm` . It follows that I (g` ) = b?m. De ne
g=
m ?1 bX
`=0
"`g` ; where "` 2 f1; ?1g:
The norm of this function can be bounded in a similar manner as was done in the proof of the previous theorem. First the series coecients of g are bounded as follows: jg^;j 1 and
jg^ j =
Z ( +1)b? b?
g(x) (x) dx b(?1)=2(b ? 1)b? = b?(+1)=2(b ? 1):
This gives an upper bound on the norm of g: ?1 b?1 m bX X
?1 X Hwav = !; +
kgk2
=0 =0 =0
2 ! ?1 ?1 + ?1 (b ? 1)2 g^ 0 0
m X =0
b2 c23 b2 m ;
for some constant c3 that is independent of m. Now consider the following functions f` = g` =(c3 b m ). These f` have disjoint supports and satisfy
I (f` ) c3 b?( +1)m ;
m ?1
bX
`=0
`
"` f 1 for any "` 2 f1; ?1g:
From [Nov88, Section 2.2.4, Proposition 1] it follows that er(Hwav ; n) c4 b?( +1)m(bm )1=2 = c4 b?( +1=2)m Cr n? ?1=2 : This completes the proof of (33). Theorems 8, 9, and 10 may be combined to summarize the main results of this article, namely that scrambled net quadrature is optimal for Haar wavelet series in all three settings. We use the standard notation an bn for non-negative sequences an and bn meaning that there are constants clo ; cup > 0 such that clo bn an cupbn . Corollary 11. Let the space Hwav be de ned as in Section 2 with !u and u given as in (24). Let Qsc;n denote Owen's scrambled quasi-Monte Carlo quadrature using a (; t; m; s)-net with bm points as the node set, where = 0; 1; : : : ; b ? 1 and m = 0; 1; : : : . Then, > 1=2; min ew (Hwav ; Qsc;bm ) ew (Hwav ; n) n? [log n](s?1)=2 ; bm n
0; min er (Hwav ; Qsc;bm ) er (Hwav ; n) n? ?1=2 ; min ea(Hwav ; Qsc;bm ) ea (Hwav ; n) n? ? [log n](s?1)=2 ; 0; > 1=2: bm n bm n
Remark. Usually the gap between worst-case and random-case n-th minimal errors is not more than of order n1=2 . This is the rst time that the authors have encountered a situation where this gap is larger, namely, ew (Hwav ; n)=er (Hwav ; n) is of the order n1=2 [log n](s?1)=2 .
18
STEFAN HEINRICH, FRED J. HICKERNELL, AND RONG-XIAN YUE
References [Hei93] [HH99] [Hic96] [HW01] [HY01] [Lar98] [Loh01] [Nie92] [Nov88] [NP01] [Owe95] [Owe97a] [Owe97b] [Owe98] [Owe99] [Rit00] [YH01a] [YH01b] [YM99] [Yue99]
S. Heinrich, Random approximation in numerical analysis, Functional analysis. Proceedings of the Essen Conference, held in Essen, Germany, November 24 - 30, 1991. (K. D. Bierstedt, A. Pietsch, W. M. Ruess, and D. Vogt, eds.), Lecture Notes in Pure and Applied Mathematics, vol. 150, Marcel Dekker, New York, 1993, pp. 123{171. F. J. Hickernell and H. S. Hong, The asymptotic eciency of randomized nets for quadrature, Math. Comp. 68 (1999), 767{791. F. J. Hickernell, The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul. 6 (1996), 274{296. F. J. Hickernell and H. Wozniakowski, The price of pessimism for multidimensional quadrature, J. Complexity 17 (2001), to appear. F. J. Hickernell and R. X. Yue, The mean square discrepancy of scrambled (t; s)sequences, SIAM J. Numer. Anal. 38 (2001), 1089{1112. G. Larcher, Digital point sets: Analysis and applications, Random and Quasi-Random Point Sets (P. Hellekalek and G. Larcher, eds.), Lecture Notes in Statistics, vol. 138, Springer-Verlag, New York, 1998, pp. 167{222. W.-L. Loh, On the asymptotic distribution of scrambled net quadrature, 2001, submitted for publication. H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMSNSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992. E. Novak, Deterministic and stochastic error bounds in numerical analysis, Lectures Notes in Math., no. 1349, Springer-Verlag, Berlin, 1988. H. Niederreiter and G. Pirsic, The microstructure of (t; m; s)-nets, J. Complexity 17 (2001), to appear. A. B. Owen, Randomly permuted (t; m; s)-nets and (t; s)-sequences, Monte Carlo and Quasi-Monte Carlo Methods in Scienti c Computing (H. Niederreiter and P. J.-S. Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995, pp. 299{ 317. A. B. Owen, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997), 1884{1910. A. B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Stat. 25 (1997), 1541{1562. A. B. Owen, Scrambling Sobol' and Niederreiter-Xing points, J. Complexity 14 (1998), 466{489. A. B. Owen, Monte Carlo, quasi-Monte carlo, and randomized quasi-Monte Carlo, Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.), Springer-Verlag, Berlin, 1999. K. Ritter, Average-case analysis of numerical problems, Lecture Notes in Mathematics, vol. 1733, Springer-Verlag, Berlin, 2000. R. X. Yue and F. J. Hickernell, The discrepancy of digital nets, 2001, submitted to J. Complexity. R. X. Yue and F. J. Hickernell, Integration and approximation based on scramble sampling in arbitrary dimensions, J. Complexity 17 (2001), to appear. R. X. Yue and S. S. Mao, On the variance of quadrature over scrambled nets and sequences, Statist. Probab. Lett. 44 (1999), 267{280. R. X. Yue, Variance of quadrature over scrambled unions of nets, Statist. Sinica 9 (1999), 451{473.
OPTIMAL QUADRATURE FOR HAAR WAVELET SPACES
19
FB Informatik, Universitat Kaiserslautern, PF 3049, D-67653 Kaiserslautern, Germany
E-mail address :
[email protected]
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
E-mail address :
[email protected]
College of Mathematical Science, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China
E-mail address :
[email protected]
