2 K. ENTACHER 2 Generalized Haar function systems In the following we x an arbitrary integer base b 2. For the notations and denitions of generalized

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1 BIT 38 :2 (998), 283{292. QUASI-MONTE CARLO METHODS FOR NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES II KARL ENTACHER y Deartment of Mathematics, University of Salzburg, Hellbrunnerstr. 34 A-52 Salzburg, Austria. Karl.Entacher@sbg.ac.at Abstract. The resent aer contains a comarison of dierent classes of multivariate Haar series that have been studied with resect to numerical integration, new roerties of E s -classes and numerical results. AMS subject classication: Primary 65D3, 42C. Key words: Multivariate Haar series, numerical integration, generalized Haar functions, low-discreancy oint sets, quasi-monte Carlo methods. Introduction ~ In [3] we studied classes of multivariate Haar series f to the base b 2, called b Es (C)-classes, and their numerical integration using (t; m; s)-nets P = (x n ) N? n= to the base b as node sets for the integration rule Z N? (.) f(x) dx f(x n ): [;[ s N n= Sobol' [] already studied secial classes of Haar series to the base 2 with resect to numerical integration. Searching an otimal node set to integrate his classes, he dened the (t; m; s)-nets to base 2 (originally called P -nets). Niederreiter generalized the (t; m; s)-nets to arbitrary bases and gave ecient construction methods and an exhaustive theory (e.g. see [8]). The resent aer comletes the studies in [3] in the sense that we comare the dierent classes of multivariate Haar series that have been studied with resect to numerical integration. In Section 3 we will consider Sobol's S - and H -classes (the S -classes will be dened for arbitrary integer bases b 2) and comare them to our b E ~ s (C)-classes. The S - and b E ~ s (C)-classes are dened using resolution deendence only. Hence the basic ideas in Section 3 robably may be extended to other wavelet systems on comact intervals. Section 4 contains a comarison of the integration error estimates for these classes and numerical results. Finally, in Section 5 we will give sulementing roerties of b E ~ s (C)-classes. Received March 997. Revised Juli 997. Communicated by Tom Lyche. y Research suorted by the Austrian Science Foundation (FWF), roject no. P43-MAT.

2 2 K. ENTACHER 2 Generalized Haar function systems In the following we x an arbitrary integer base b 2. For the notations and denitions of generalized Haar functions relative to base b, see [3, 4]. In this section we recall the basic notations: For P P an integer k and an arbitrary number x 2 [; [, let k = j= k jb j and x = j= x jb?j?, k j ; x j 2 f; ; : : : ; b? P g, be the b-adic exansions P of k and g? x in base b. For g 2 N we dene k(g) := j= k jb j g? and x(g) := j= x jb?j?. Further let k() := and x() := : The suort of a given Haar function h k, k, is equal to an elementary b-adic interval. We now dene sets of integers k, for which such intervals have the same length (resolution). Definition 2.. () Let g be a nonnegative integer. Then (g) := fk 2 N : b g k < b g+ g. Further, let (?) := fg, and the sets N := N [ fg, and N = N [ f?g: (2) If g = (g ; : : : ; g s ), s 2 and g i 2 N, then (g) := Q s i= (g i). Definition 2.2. Let e b : Z b! K, where Z b = f; : : : ; b? g is the least residue system modulo b, and K := fz 2 C : jzj = g, denote the function e b (a) := ex(2i a b ); (a 2 Z b): The k-th Haar function h k ; k, to the base b is dened as follows: If k =, then h (x) := 8x 2 [; [. If k 2 (g), g, then b? h k (x) := b g 2 e b (a k g ) Dk (a)(x); a= with elementary b-adic intervals D k (a) := [(b k(g)+a)=b g+ ; (b k(g)+a+)=b g+ [. The k-th normalized Haar function H k on [; [ is dened as H := h and, if k 2 (g), g, then H k := b? g 2 h k : Hence, the suort D k of the k-th Haar function h k is given as the following elementary S b-adic interval : If k =, then D := [; [: If k 2 (g), g, then b? D k := a= D k(a) = [k(g)=b g ; (k(g) + )=b g [. Definition 2.3. Let H b := fh k : k := (k ; : : : ; k s ) 2 N s g denote the Haar function system to the base b on the s-dimensional torus [; [ s ; s. The k-th Haar function h k is dened as h k (x) := Q s i= h k i (x i ); x = (x ; : : : ; x s ) 2 [; [ s : The normalized version H k is dened in the same way and the suorts of h k and H k are dened as D k := Q s i= D k i : 3 Classes of multivariate Haar series Consider f 2 L ([; [ s ; s ), where s denotes the Lebesgue measure on the s-dimensional unit cube [; [ s, s 2. For k = (k ; : : : ; k s ) 2 N s let r be the number of comonents k i with k i ( k j = for j 6= i). Therefore a Haar series s f of f has the form (3.) s f (x) := ^f() + s r= i <:::<ir s k ij = ^f r (k) h ki (x i ) h kir (x ir )

3 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 3 with Haar coecients (3.2) ^f r (k) := Z [;[ s f(x) h ki (x i ) h kir (x ir ) dx Let f s f and s f absolutely convergent. Using a oint set P = (x n ) N? n=, x n = (x (n) ; : : : ; x(n) s ) 2 [; [ s, in rule (.) to aroximate the integral we obtain the following form of the integration error (3.3) with (3.4) R N (f; P) = s r= s r= i <:::<ir s i <:::<irs g ij = k ij = ^f r (k) N b 2 (gi +:::+gir ) N? n= h ki (x(n) i ) h kir (x (n) i r ) j ^f r (k)j js (r) N (H k; P)j; S (r) N (H k; P) := N? H ki N (x(n) i ) H kir (x (n) i r ): n= It follows that the oint set P enters the integration error only with the order of magnitude of the \Weyl" sums S (r) N (H k; P), hence deends only on the Haar function system. The so called (t; m; s)-nets to the base b are oint sets P, consisting of N = b m oints, which satisfy certain local conditions suitable to the local denition of the Haar functions. It follows that the Weyl sums vanish for a large set of resolutions g near the origin, more recisely S N (h k ; P) = for all k 2 (g) and g with s i= (g i + ) m? t: Further, for arbitrary (t; m; s)-nets to the base b for the remaining resolutions it follows that js N (H k ; P)j b?(gi +:::+gir ) for all k 2 (g), and g with m?t < P s i= (g i +) < m?t+r, and if P s i= (g i +) m?t+r, then there are at most (b? ) r b m vectors k 2 (g) with js N (H k ; P)j 6=. In the latter case we have js N (H k ; P)j b t?m (see [3, Lemma 4.2]). For secial construction methods these estimates may be imroved. Examles are given in [2]. For the general denition of (t; m; s)-nets and ecient construction methods see [, 6, 8]. The main goal in numerical integration is to nd large classes of functions which guarantee best ossible integration errors. If we, for examle, consider classes of multivariate Haar series with in a certain sense bounded Haar coef- cients we are able to estimate the integration error (3.3). In [3] we observed Haar series where the Haar coecients above are bounded in the following way (3.5) j ^f r (k)j C b?(gi +:::+gir ) =: C b (k): Haar series with this roerty are called b s (C)-classes. For the exact denition and roerties see [3, Sect. 2.2]. This aroach is due to Korobov [5] who studied

4 4 K. ENTACHER multivariate Fourier series. We changed Korobov's condition slightly in order to get a resolution deendency only. Sobol' [] studied more general classes called S,, classes. We will dene these classes for arbitrary bases b 2. Let < and q such that = + =q =. Alying Holder's inequality in (3.3), 2 j ^f r (k)j js (r) N (H k; P)j j ^f r (k)j q js (r) N (H k; P)j q 5 ; yields a more general estimate of R N (f; P). This estimate suggests the following denition, see also [,. 33]. Definition 3.. For < let 2 A (i;:::;ir) (s f ) := b 2 (gi +:::+gir ) 4 3 (3.6) j ^f r (k)j 5 : g ij = A Haar series s f belongs to the class S (b) (A i;:::;ir ) if A (i;:::;ir) (s f ) A i;:::;ir, for uer bounds A i;:::;ir > (which only deend on the resolutions g i ; : : : ; g ir ) and for all i < : : : < i r s, r s. If s f 2 S (b) (A i;:::;ir ) then s f converges absolutely and uniformly (see [,. 34]). Further, for < <, we have S (b) (A i ;:::;i r ) S (b) (A i;:::;ir ) S (b) S (b) (A i ;:::;i r ). The next roosition shows the relation between b E ~ s (C)- and -classes. Proosition 3.. If :=? 2? >, then b E ~ s (C) S (b) (A i;:::;ir ) for A i;:::;ir = C (b? ) r b b? The roof is veried easily if we estimate A (i;:::;ir) (s f ) using (3.5). The inclusion in Proosition 3. is strong. For examle let s = and A >. The Haar series s f with ( A 6 ^f(k) := b g 2 2 g : 2 k = b g ; g : otherwise belongs to S (b) (A) and not to any class b E ~ s (C) for > =2 + =. Sobol' also studied classes H (L i;:::;ir ). Let s =. A class H (L), <, L > consists of all functions f : [; [?! R, where 8 x; y 2 [; [ : jf(x)?f(y)j Ljx?yj. The multidimensional case H (L i;:::;ir ), which diers from general Holder - classes, is dened in [,. 36]. From [,. 4: (4.25)] we obtain j ^f r (k)j L i;:::;i r ry j= 2?(gi j +)(+ 2 )? 2 max i ;:::;i r L i;:::;ir 2 (+) 2 (+ 2 )(gi +:::+gir ) ; r

5 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 5 and therefore the following roosition. Proosition 3.2. Let < and s 2, then H (L i;:::;i r ) s (D) with D := max i ;:::;i r L i;:::;i r 2 (+) : The inclusion again is strong. An examle of our 2 s (C)-classes which does not belong to H (L i;:::;i r ) is the ste function () in [3, Sect. 2.3]. 4 Estimates of the integration error and numerical results In this section we comare integration error estimates for the dierent classes of Sect. 3 if we use (t; m; s)-nets to calculate the integral aroximation (.). Theorem 4.. Let P = fx ; : : : ; x N? g be a (t; m; s)-net to the base b 2. (a) If f 2 b s (C) and A := P i= is? b ( 2?)i, then (b) If f 2 S (b) R N (f; P) C A ( + 2s? )b (+ 2 )s (log b) s? b (+ 2 )t (A i;:::;i r ) and q with = + =q =, then R N (f; P) (b? ) s q (b t + b t+s ) A N ; A := s r= (log N)s? N (? 2 ) : i <:::<irs A i;:::;i r : Remark 4.. Note that our estimates for S (b) (A i;:::;i r )-classes are valid for arbitrary (t; m; s)-nets in base b and they are more recise than those of Sobol' (see [,. 228,239]). Using P -nets to integrate H (L i;:::;i r )-classes, Sobol' [,. 239] obtained an estimate of the integration error in the same order of magnitude as in (a). The estimate in (a) is best ossible for b E ~ s (C)-classes [2]. Proof. The roof of Part (a) is given in [3, Part (c), Sect. 4.]. To rove Part (b) we aly Holder's inequality in (3.3). Then, from the estimates of the Weyl sums given above, we obtain R N (f; P) 2 s m?t? (b? ) r q b ( 2? )(gi +:::+gir ) 4 3 j ^f r (k)j 5 + r=2 s r= i <:::<irs g i ;:::;g ir = m?t?r<g i +:::+g ir <m?t i <:::<irs g i ;:::;g ir = g i +:::+g ir m?t For f 2 S (b) (A i;:::;i r ) it follows that R N (f; P) b?m= s and this yields the result. (b?) r q b (t? m ) b 2 (gi +:::+gir ) 2 4 r=2 + b?m= s r= (b? ) r=q b (t+r)= (b? ) r=q b t i <:::<irs i <:::<irs A i;:::;i r ; A i;:::;i r j ^f r (k)j 3 5

6 6 K. ENTACHER As a numerical examle we consider the function F : [; [ s?! R, with (4.) F (x ; : : : ; x s ) := (x + : : : + x s ) n ; n 2 N: This function belongs to 2 3 2s (C)-classes. Using digital (t; m; s)-nets P from the Salzburg Tables [6] to calculate the aroximation (.), we obtained the results in Figure 4. for the integration error R N (F; P), N = The olynomials roviding the nets and the corresonding quality arameters t are given in [6, Table 2]. The grahics demonstrates the behavior of the error (logarithmic scale in base ) for increasing n and dimensions 3 s 5. Log of integration error n = 2 n = 3 n = 4 n = Dimension Figure 4.: Integration errors with (t; m; s)-nets. 5 Further results for b s (C)-classes In this section we resent two results for b s (C)-classes which sulement the studies given in [3]. From a basic roerty of digital (t; m; s)-nets we immediately obtain the following result (comare also to [7, Thm. 2 (b)]). Proosition 5.. For all > =2, C > and all b and s 2 we have: For all t and m for which there exists a digital (t; m; s)-net in base b, there is such a net and a function f 2 b s (C) with b (? 2 ) R N (f; P) = (b? ) b (? 2 )? b(? 2 )t N (? 2 ) : Proof. From the roof of [9, Lemma 3c] we get the following roerty of digital (t; m; s)-nets: For all t and m for which there exists a digital (t; m; s)- net in base b, there is such a net P t where the rst m? t rows of C () equal the rst m? t rows of the identity matrix and the remaining rows are zero vectors. Therefore the rst coordinates x (n), n < bm, of P t cover the set f; =b m?t ; : : : ; (b m?t? )=b m?t g.

7 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 7 Consider the Haar series P k2n s ^f(k) h k (x); x 2 [; [ s where ^f(k) := ( C b g : k 2 (g); g = (g;?; : : : ;?); g : otherwise : For this function we get R N (f; P t ) = g= k2(g) From [3, Lemma 4.] we obtain the result. C b (? 2 )g N N? n= H k (x (n) ) : In the following we will observe [3, Thm. 3.2] in reverse direction. The result is an analogue to [7, Lemma 8] where generalized Walsh functions are used instead of Haar functions. Note that in our case there are no further limitations on the arameters > =2, due to the local structures of Haar functions. Theorem 5.2. If f 2 B s (C) with B = b L ; L 2, and > =2, then f 2 b E ~ s (C C(; b) s ) where C(; b) := B(+ 2 ) B b (+ 2 ) B? a= sin a : B Proof. Let B = b L, L 2, for a given base b 2. The result is roven by induction in a similar way as [3, Thm. 3.2]. To avoid too many indices, we shall denote the Haar coecients of a given function f with resect to the system H B by ^f(n), hence with argument n. Further we will use the argument k to signify that ^f(k) denotes the Haar coecient of f with resect to the Haar system H b. We start with dimension s =. For B g n < B g+, g 2 N, the Haar function h n (B) 2 H B has a nite Haar series with resect to the system H b, more recisely (5.) h (B) n (x) = B g+? k=b g d h (B) n (k) h k (x): Therefore, if f 2 B (C), > =2, we obtain the following form of ^f(k), (5.2) ^f(k) = B g+? n=b g ^f(n) d h (B) n (k) for B g k < B g+ : Consider integers n; k with B g n < B g+, g and k 2 (g) with g = Lg + j; j < L. Further let us dene E a := D k (a) = [ (a) ; (a+) [ with (a) := k(g) b g + a ; a < b: bg+

8 8 K. ENTACHER Therefore we have (5.3) dh (B) n (k) = b g b? 2 a= e b (a k g ) d Ea (n): The latter result and (5.2) imly (5.4) j ^f(k)j b g 2 C B g B g+? n=b g b? jd Ea (n)j: a= From Hellekalek [4, Lemma 3.2], we observe that we may have d Ea (n) 6= only if n(g) 2 fb g (a) (g); B g (a+) (g)g, and in this case it follows that jd Ea (n)j B (? g 2?) = sin ng B. Hence we have to analyze for which Bg n < B g+ and a b the equation n(g) = B g (a) (g) holds. Comaring the B-adic exansions of n(g) and B g (a) (g) imlies that for arbitrary a 2 f; : : : ; bg there are B? such numbers n (n(g) is xed and n g varies in f; : : : ; B? g). Using (5.4) we observe that (5.5) (5.6) j ^f(k)j C B?(? 2 )g b ( g 2 +) B C C(; b) b g ; B? j= sin j B and this yields the result for dimension s =. As an induction hyothesis, we assume that f 2 B s? (C) imlies f 2 b s? (C C(; b) s? ). Therefore f(x) = k2n s? ^f b (k) h k (x) = n2n s? ^f B (n) h (B) n (x); x 2 [; [ s? ; with j ^f b (k)j C C(; b) s? b (k) and j ^f B (n)j C B (n). Let f 2 B s (C). This yields j ^f B (n ; : : : ; n s )j C B (n ) B (n), n = (n 2 ; : : : ; n s ), and therefore f(x ; : : : ; x s ) = n ^f B (n) h (B) n (x) A n2n s? {z } 2 B s? (C B (n )) h (B) n (x ); x = (x 2 ; : : : ; x s ): By induction, we get f(x ; : : : ; x s ) = n k2n s? ^f (n) b (k) h k (x) A h (B) n (x );

9 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 9 with j (n) ^f b (k)j C B (n ) C(; b) s? b (k). On the other hand f(x ; : : : ; x s ) = k2n s? h k (x) ^f (n) b (k) h (B) n (x ) n = {z } 2 B (C b (k) C(; b) s? ) Alying the result for dimension s = nishes the roof. Acknowledgments The author wishes to thank the Lab-grou, Salzburg, for suorting his work, and esecially the head of the grou Peter Hellekalek for many helful suggestions. Many thanks to I.M. Sobol' for interesting conversations and for a genuine Russian version of his book []. Secial thanks also to Prof. Peter Zinterhof who heled me to translate imortant arts of this book. REFERENCES. P. Bratley, B.L. Fox, and H. Niederreiter. Imlementation and tests of lowdiscreancy sequences. ACM Transactions on Modeling and Comuter Simulation, 2:95{23, K. Entacher. Generalized Haar function systems, digital nets and quasi-monte Carlo integration. In H.H. Szu, editor, Wavelet Alications III, Proc. SPIE 2762, 996. Available on the internet at htt://random.mat.sbg.ac.at. 3. K. Entacher. Quasi-Monte Carlo methods for numerical integration of multivariate Haar series. BIT, 37(4):846{86, P. Hellekalek. General discreancy estimates III: the Erdos-Turan-Koksma inequality for the Haar function system. Monatsh. Math., 2:25{45, N.M. Korobov. Number-Theoretic Methods in Aroximate Analysis. Fizmatgiz, Moscow, 963. (in Russian). 6. G. Larcher, A. Lau, H. Niederreiter, and W.Ch. Schmid. Otimal olynomials for (t; m; s)-nets and numerical integration of Walsh series. SIAM J. Numer. Analysis, 33:2239{2253, G. Larcher, W.Ch. Schmid, and R. Wolf. Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series. Mathl. Comut. Modelling, 23(8/9):55{67, H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelhia, W.Ch. Schmid and R. Wolf. Bounds for digital nets and sequences. Acta Arith., 78:377{399, I.M. Sobol'. Multidimensional Quadrature Formulas and Haar Functions. Izdat. \Nauka", Moscow, 969. (in Russian). :

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