Construction algorithms for good extensible lattice rules

Transcription

1 Construction algorithms for good extensible lattice rules Harald Niederreiter and Friedrich Pillichshammer Dedicated to Ian H. Sloan on the occasion of his 70th birthday Abstract Extensible olynomial lattice rules have the roerty that the number N of oints in the node set may be increased while retaining the existing oints. It was shown by Hickernell and Niederreiter in a nonconstructive manner that there exist generating vectors for extensible integration lattices of excellent quality for N b,b 2,b 3,..., where b is a given integer greater than. Similar results were roved by Niederreiter for olynomial lattices. In this aer we rovide construction algorithms for good extensible lattice rules. We treat the classical as well as the olynomial case. AMS subject classification: K38, K45, 65C05, 65D30. Key words: Numerical integration, quasi-monte Carlo, extensible lattice rule, discreancy. Introduction We are interested in the aroximation of a multidimensional integral over the unit cube by a so-called quasi-monte Carlo QMC algorithm, which takes the average of function evaluations over well-chosen samle oints, i.e., [0,] s Fx dx N N k0 Fx k with x 0,...,x N [0, s. On first sight, this aroach looks quite simle, but the crux of this method is the choice of the samle oints x 0,...,x N to obtain good results for large function classes. In this aer we focus on two methods of constructing oint sets which work very well in many ractical alications. The first construction rincile is that of integration lattices, introduced indeendently by Hlawka [6] and Korobov [7] and studied extensively in recent years see for examle [, 26, 29, 32]. An integer vector a, the generating vector of the lattice rule, is used The research of H.N. was carried out while he was hosted by CNRS-FR229 FRUMAM at the Université de la Méditerranée in Marseille-Luminy. F.P. is suorted by the Austrian Science Foundation FWF, Project S9609, that is art of the Austrian National Research Network Analytic Combinatorics and Probabilistic Number Theory.

2 to generate N oints by {ka/n + } for k 0,...,N. The braces indicate that we take the fractional art of each comonent. The shift [0, s is either chosen 0 for eriodic functions or i.i.d. for noneriodic functions. QMC algorithms which use integration lattices as underlying oint sets are called lattice rules, or randomly shifted lattice rules if [0, s is chosen i.i.d.. In order to distinguish them from olynomial lattice rules introduced below, we will often refer to lattice rules as classical or usual lattice rules. The second construction method considered here is that of olynomial lattices, introduced by Niederreiter [25] see also [26, Section 4.4]. These are similar to integration lattices, but here we use olynomial arithmetic instead of integer arithmetic. Polynomial lattices are secial cases of so-called digital t, m, s-nets as introduced by Niederreiter in [24] see also [26, Chater 4] for an introduction to this toic. We give a detailed definition of olynomial lattices in Section 3. QMC algorithms which use olynomial lattices as underlying oint sets are called olynomial lattice rules. The quality of a olynomial lattice rule deends on the choice of its generating vector. Until now no exlicit constructions of good generating vectors are known excet for dimension s 2, and hence one has to resort to comuter search. Several construction algorithms for generating vectors have been introduced and analyzed, where most of them rely on a comonent-by-comonent aroach. See for examle [3, 2, 22, 30] in the classical case and [5, 6, 9] in the olynomial case. One major disadvantage of these construction methods is their deendence on the cardinality N of the resulting oint set. If one constructs a generating vector of a olynomial lattice rule of cardinality N with good quality, it does not mean that the same vector can be used to generate a olynomial lattice rule of good quality which uses N N oints. Although such a roerty would be very desirable for ractical alications [2], an extension in the number of oints has not been shown to be ossible with the algorithms known so far. That an extension of usual lattice rules in the number of oints is ossible at least in rincile was shown in [3]. Such lattice rules are nowadays called extensible olynomial lattice rules. A cornerstone in this area is the aer [5] by Hickernell and Niederreiter where for the first time the existence of good extensible usual lattice rules was roved. In detail, they showed for any integer b 2 the existence of a generating vector a which generates a lattice rule which is good for all cardinalities N b, b 2, b 3,.... Furthermore, Niederreiter [27] alied similar techniques to show the existence of good extensible olynomial lattice rules. We remark here that the olynomial lattice rules whose existence was shown in [5] res. [27] are also extensible in the dimension s. However, the roofs in [5] and [27] are nonconstructive. Although the existence of generating vectors yielding good extensible olynomial lattice rules is known, it remained an oen question how they can be exlicitly found in general. Several numerical investigations have been carried out in [2, 0, 4], but a roof that those algorithms yield good results has not been given. A first construction algorithm was rovided in [8], but the generating vectors constructed there yield good results only for a finite range of cardinalities b u, b u+,...,b u+v, where b 2 is an integer and u, v N have to be chosen in advance. One could say that these lattice rules are only finitely extensible, which does not meet our demand for extensible lattice rules in its full generality. It should be mentioned that in rincile also the choice v is ossible. But in this case the construction cost becomes so large that the algorithm is 2

3 useless for ractical alications. Only for finite v can the fast comonent-by-comonent aroach be incororated which makes the algorithm alicable. Furthermore we note that the algorithm in [8] requires some storage. A similar algorithm for finitely extensible olynomial lattice rules was given in [4]. In this aer we resent construction algorithms for extensible lattice rules where we treat the classical as well as the olynomial case. These algorithms in rincile work as follows: each comonent of a generating vector a is considered in its -adic exansion where is a rime. If we have already constructed the first n digits of each comonent such that the generating vector yields good results for N, 2,..., n, we search in each comonent for the n + st digit such that the generator vector yields good results also for N n+. In this way we find digit by digit a good generating vector for all N, 2, 3,.... In contrast to other algorithms, we do not have to sto at some a riori fixed v. Furthermore we mention already here that in our algorithm the search sace is the same in each ste and so it does not grow with n. The disadvantage of our results is that the bounds we obtain on the considered quality arameters are a bit weaker than those anticiated for examle in [5] and [27] or obtained in [8] and [4]. As far as we know, the only earlier aer where a similar idea was used but only for the classical case and for a different quality arameter is that of Korobov [8]. The rest of the aer is organized as follows. In Section 2 we deal with usual integration lattices. We recall the definition of lattice rules and some measures for the quality thereof. Then we introduce two algorithms for the construction of generating vectors of good quality with resect to these measures for all N, 2, 3,.... In Section 3 we treat the olynomial case with similar results. Finally, in Section 4 we resent some numerical results for the classical case. 2 Classical integration lattices In this section we consider classical integration lattices. As already mentioned in the revious section, these are defined as follows. Let s N the dimension and let N N. An integer vector a Z s, the generating vector of the lattice rule, is used to generate N oints by { } k N a + for k 0,...,N, where we use the notation introduced in Section. Our aim is to construct a generating vector a digit by digit such that the corresonding integration lattice is of good quality for N, 2, 3,.... We consider two quality measures for integration lattices. The first one is the worst-case error for QMC integration in a weighted Korobov sace res. the root mean-square worst-case error with resect to a shift in a weighted Sobolev sace and the second one is the discreancy. The worst-case error in a weighted Korobov sace. Let s N, α >, and γ γ j j. The ositive reals γ j are called weights, which are introduced to modify the imortance of different coordinate directions [3]. Furthermore, α is a smoothness arameter. As for examle in [32] we consider the weighted Korobov sace HK s,α,γ 3

4 with the reroducing kernel K s,α,γ x, y : h Z s r α h, γ e2πih x y, where denotes the standard inner roduct in R s and where for h h,..., h s Z s we ut r α h, γ : s j r αh j, γ j with { if h 0, r α h, γ : γ h α if h Z \ {0}. In [32] it was shown that the squared worst-case error for integration in HK s,α,γ, i.e., the worst erformance of a QMC algorithm over the unit ball of HK s,α,γ using a lattice oint a a,...,a s Z s, is given by e 2 N,s,α,γa + N + N N e2πika h/n r α h, γ s + γ j h α k0 h Z s N k0 j h Z h 0 e 2πikha j/n Remark It is easy to show or see [26, Chater 5] that we have e 2 N,s,α,γ a h Z s \{0} h a 0 mod N r α h, γ. Thus, in the unweighted case, i.e., if γ j for all j N, e 2 N,s,α,γ a is the same as the quantity P α a, N defined in [26, Definition 5.2]. Remark 2 If α 2 is an even integer, then the Bernoulli olynomial B α of degree α has the Fourier exansion B α x α+2/2 α! 2π α h Z h 0 e 2πihx h α for all x [0, ; see for examle [29, Aendix C]. Hence in this case we obtain e 2 N,s,α,γa + N N k0 j s α+2/2 2π α + γ j so that e 2 N,s,α,γ a can be calculated in ONs oerations. α!. { } kaj B α, N Remark 3 Consider a tensor roduct Sobolev sace H s,γ of absolutely continuous functions whose mixed artial derivatives of order in each variable are square integrable. The norm in the unanchored weighted Sobolev sace H s,γ see [9] is given by f Hs,γ u {,...,s} j u [0,]s u γj u fx dx {,...,s}\u [0,] x u u 4 2 dx u /2,

5 where u f/ x u denotes the mixed artial derivative with resect to all variables j u. As ointed out in [8], the root mean-square worst-case error ê N,s,γ for QMC integration in H s,γ using randomly shifted lattice rules, i.e., /2 ê N,s,γ a e 2 N,s,γ a, d, [0, s where e N,s,γ a, is the worst-case error for QMC integration in H s,γ using a randomly shifted integration lattice, is more or less the same as the worst-case error e N,s,2,γ in the weighted Korobov sace HK s,2,γ using the unshifted version of the lattice rules. In fact, we have ê N,s,2π 2 γa e N,s,2,γ a, where 2π 2 γ denotes the sequence of weights 2π 2 γ j j. For more information, see for examle [8, Section 2]. Thus, the results that will be shown in the following are valid for the root mean-square worst-case error for numerical integration in the Sobolev sace as well as for the worst-case error for numerical integration in the Korobov sace. Hence it suffices to state them only for e N,s,α,γ. Equation can be used to obtain results also for ê N,s,γ. The discreancy. For a oint set P {x 0,...,x N } consisting of N oints in [0, s, the discreancy is defined by N D N P su c J J x k λ s J N, k0 where the suremum is extended over all subintervals J of [0, s, c J denotes the characteristic function of J, and λ s J is the volume of J. If we take the suremum just over all subintervals J of [0, s with one vertex anchored at the origin, then we seak of the star discreancy and write DN P. Obviously, we always have D N P D NP. The star discreancy of a finite oint set is intimately related to the worst-case error of multivariate QMC integration of functions with bounded variation in the sense of Hardy and Krause. Hereby the basic error estimate is rovided by the so-called Koksma-Hlawka inequality. For more information, we refer to [20, Chater 2, Section 5] and [26, Chater 2]. For a generating vector a Z s, s 2, and an integer N 2, let P be the oint set consisting of the oints { } k x k N a for k 0,,..., N. Then for the discreancy D N P of P we have see [26, Theorem 5.6] D N P s N + Ra, N, 2 2 where Ra, N : h C s N h a 0 mod N rh 5

6 with and rh : s max {, h j } j for h h,...,h s Z s C s N : Z s N/2, N/2] s, C s N : C sn \ {0}. To get useful bounds on the discreancy D N P of P, it suffices to have good bounds on the quantity Ra, N. 2. Good extensible integration lattices with resect to the worstcase error We resent an algorithm which constructs digit by digit a generating vector which is good with resect to the worst-case error e N,s,α,γ for all N, 2, 3,.... Algorithm Let be a rime number and let Z : {0,,..., }.. Find a : a by minimizing e 2,s,α,γ a over all a Zs. For 2 one can choose a,..., Z s. 2. For n 2, 3,... find a n : a n + n z by minimizing e 2 n,s,α,γa n + n a over all a Z s, i.e., z argmin a Z s e2 n,s,α,γ a n + n a. Theorem Let n, s N, be a rime, and α >. Assume that a n Z s is constructed according to Algorithm. Then we have s { } e 2 α 2,s,α,γa n n + 2γ j ζα min n, α n, j where in the case 2 and 0 < γ j for all j N we can relace the fraction 2 by n. Here for α >, ζα denotes the Riemann zeta-function defined by ζα 2 n i i α. Before we rove this result, we state two remarks. Remark 4 The search for a in the first ste of Algorithm takes at least for even α Os s+ oerations. With a comonent-by-comonent construction this can be reduced to Os 2 2 oerations. In this case one gets a slightly weaker error bound in Theorem the term after the first roduct must be deleted. Alternatively, one can choose a,..., in all cases. But then the uer bound in Theorem has to be relaced by s e 2,s,α,γa n n + 2γ j ζα j { min n, α } α n excet in the case 2 and 0 < γ j for all j N. This follows immediately from the subsequent roof of Theorem. 6

7 Remark 5 We remark that it is not necessary to start Algorithm with item. If one has given an arbitrary generating vector a n0 for some n 0 N with squared worst-case error e 2 0,s,α,γa n n0, then this vector can be extended as well for n n 0 +, n 0 + 2,.... From the roof below it is easy to see that in this case we obtain e 2 n,s,α,γa n e 2 n 0,s,α,γa n0 min Now we give the roof of Theorem. Proof. First we show the result for n. We have e 2,s,α,γa e 2,s,α,γa s s a Z s h Z s \{0} h 0 mod 2 r α h, γ + { n n 0 +, h Z s \{0} h Z s \{0} h 0 mod α α r α h, γ r α h, γ } n n 0. a Z s h a 0 mod r α h, γ + r α h, γ h Z s \{0} h Z s \{0} s r α h, γ 2 + 2γ j ζα. h Z s \{0} For 2 and 0 < γ j for all j N and the choice a,...,, we can estimate e 2 2,s,α,γa directly. In this case we obtain e 2 2,s,α,γa + s h,...,h s j j r α h j, γ j h h h + +h s mod 2 r α h, γ s. Denote the innermost sum in the above exression by Σ. If h + +h s 0 mod 2, then we have Σ r α 2h, γ s αγ sζα. h If h + + h s mod 2, then we have Σ r α 2h +, γ s 2 r α 2h +, γ s h h0 2γ s h 2γ α 2h α s ζα 2 2 αγ sζα. h Altogether we obtain Σ 2 + γ sζα + h + +h s 2 γ sζα αγ sζα. h 7

8 Therefore we get Since we obtain e 2 2,s,α,γ a γ sζα r j α h, γ j h s s + 2 γ sζα αγ sζα + s + 2γ j ζα 2 j αγ sζα 2γ s ζα h e 2 2,s,α,γa + 2 j s h j h h r α h, γ j αγ jζα 2γ j ζα, s + 2γ j ζα + 2 j s j h r α h, γ j h. r α h, γ j 2 4 γ 2 α j ζα. 3 Now we claim that for any α > we have < 2 4 ζα < α This inequality is equivalent to which is in turn equivalent to 2 2 < ζα < 2, 2 α 2 α 2 j 2 < j0 2 α i i < α j0 j 2. 2 α This is now shown by comaring the three series above using a suitable grouing of terms. For instance, to show the uer bound on ζα, we comare the first term of the series for ζα with the first term of the last series, the 2nd + 3rd term of the series for ζα with the 2nd term of the last series, the 4th+5th+6th+7th term of the series for ζα with the 3rd term of the last series, and so on. In this way we find that the inequality 4 is indeed correct. Now, as 0 < γ j for all j N, 3 and 4 yield e 2 2,s,α,γa 2 s + 2γ j ζα s + 2 j s θ j α with < θ j α <, j 8

9 from which we obtain s e 2 2,s,α,γa + 2γ j ζα 2. j Hence the result follows for n and any rime. Now let n 2. With the reresentation of e 2 N,s,α,γ a from Remark we have e 2 n,s,α,γ a n e 2 s n,s,α,γ a n + n a a Z s s h Z s h 0 r α h, γ a Z s h a n + n a 0 mod n The inner sum is equal to the number of a Z s with n h a h a n mod n. For this we must have h a n 0 mod n, and then h a n h a n mod. Thus, e 2 n,s,α,γa n s h Z s, h 0 h a n 0 mod n r α h, γ. a Z s h a n h a n mod Consider the inner sum. If h 0 mod, then the inner sum is equal to s. If h 0 mod, then the inner sum is equal to 0 if h a n 0 mod n and equal to s if h a n 0 mod n. Thus, e 2 n,s,α,γ a n h Z s, h 0 mod h a n 0 mod n e2 n,s,α,γ a n r α h, γ + h Z s, h 0, h 0 mod h a n 0 mod n h Z s, h 0, h 0 mod h a n 0 mod n r α h, γ + r α h, γ and this holds for all n 2. If we insert this inequality for e 2 n,s,α,γ a n, then we obtain e 2 n,s,α,γa n 2e2 n 2,s,α,γ a n 2 2 h Z s, h 0, h 0 mod h a n 0 mod n h Z s, h 0, h 0 mod h a n 2 0 mod n 2 r α h, γ + r α h, γ + h Z s, h 0, h 0 mod h a n 0 mod n. h Z s, h 0, h 0 mod h a n 0 mod n r α h, γ. h Z s, h 0, h 0 mod h a n 2 0 mod n r α h, γ, r α h, γ Assume that h Z s, h 0 mod, i.e., h h, and h a n 2 0 mod n. Then we have h a n h a n 2 + n 2 a h a n 2 + n h a 0 mod n, 9

10 with some a Z s. Therefore we obtain e 2 n,s,α,γ a n 2e2 n 2,s,α,γ a n 2 2 Reeating this argument, we get e 2 n,s,α,γ a n n e2,s,α,γ a n h Z s, h 0, h 0 mod h a n 2 0 mod n 2 h Z s, h 0, h 0 mod h a 0 mod For the last sum in the above exression, we have h Z s, h 0, h 0 mod h a n 0 mod n r α h, γ With this uer bound, we obtain h Z s, h 0 h a n 0 mod n α h Z s, h 0 h a n 0 mod n r α h, γ + r α h, γ + r α h, γ h Z s, h 0, h 0 mod h a n 0 mod n h Z s, h 0, h 0 mod h a n 0 mod n h Z s, h 0 h a n 0 mod n r α h, γ. r α h, γ. r α h, γ r α h, γ αe2 n,s,α,γ a n. e 2 n,s,α,γ a n n e2,s,α,γ a + αe2 n,s,α,γ a n. With backward induction on n and invoking the uer bound for e 2,s,α,γ a, we get e 2 n,s,α,γ a n e 2,s,α,γ a + + n n 2+α 2αe2 n 2,s,α,γ a n 2. n 2 e 2,s,α,γa k0 n e 2,s,α,γa + n k+kα,s,α,γa n αe2 n k+kα k0 s + 2γ j ζα j { min n, } α 2 α n, and s e 2 2,s,α,γa n n + 2γ j ζα j if 0 < γ j for all j N. But this is the desired result. { min n, 2 α } 2 α 2 n 2.2 Good extensible integration lattices with resect to the discreancy We also resent an algorithm which uses the quantity R and hence the discreancy as the quality criterion. 0

11 Algorithm 2 Let be a rime and let Z : {0,..., }.. Find a : a by minimizing Ra, over all a Z s. 2. For n 2, 3,... find a n : a n + n z by minimizing Ra n + n a, n over all a Z s, i.e., z argmin a Z s Ra n + n a, n. Theorem 2 Let n and s 2 be integers and let 3 be a rime. Assume that a n Z s is constructed according to Algorithm 2. Then we have Ra n, n O n log + 2 sn with an imlied constant deending only on s. Before we give the roof of this result, we state some remarks and corollaries. Remark 6 Again we remark that it is not necessary to start Algorithm 2 with item. If one has given an arbitrary generating vector a n0 for some n 0 N, then this vector can be extended as well for n n 0 +, n 0 + 2,.... From the roof of Theorem 2 below it is easy to see that in this case we obtain Ra n, n log + 2sn n 0 Ra n n n0, n 0 + O n log + 2 sn n 0+, 0 where the imlied constant deends only on s. From 2 and Theorem 2 we immediately obtain a bound on the discreancy. This bound is good for large values of. Corollary Let n and s 2 be integers and let 3 be a rime. Assume that a n Z s is constructed according to Algorithm 2. Then for the discreancy D np n of the oint set P n consisting of the oints x k {k/ n a n }, k 0,,..., n, we have D np n O n log + 2 sn with an imlied constant deending only on s. From Theorem 2 we can also obtain a bound on the quantity P α, i.e., on the squared worst-case error for integration in HK s,α,γ in the unweighted case cf. Remark. Corollary 2 Let n and s 2 be integers and let 3 be a rime. Assume that a n Z s is constructed according to Algorithm 2 and that no comonent of a is zero. Then for α > we have P α a n, n O αn log + 2 sαn with an imlied constant deending only on s and α. Remark 7 The assumtion that no comonent of a is zero is not really a restriction. In Algorithm 2 the first ste could be relaced by: Choose an arbitrary vector a in Z s. It can be seen from the roof of Theorem 2 below that this would not disturb the asymtotic order of Ra n, n. Hence a can be chosen such that no comonent is equal to zero.

12 We give the roof of Corollary 2. Proof. As no comonent of a is zero, it follows that for all a n : a n,...,a s n Z s which are constructed according to Algorithm 2 we have gcda j n, n for all j s. Hence from [26, Theorem 5.5] we obtain the bound P α a n, n < Ra n, n α + + 2ζα αn s + n + 2ζα + 2α ζα αn s n + 2ζαs, where ζα denotes the Riemann zeta-function. The desired result follows by using Theorem 2 to estimate Ra n, n together with + 2ζα αn s O αn and n + 2ζα + 2α ζα αn s n + 2ζαs O αn, with imlied constants deending only on s and α. For the roof of Theorem 2 we need the following elementary result which is roved by standard arguments comare with [23]. Lemma For any odd integer k 3 we have k /2 b Now we give the roof of Theorem 2. b log k. Proof. For n a standard averaging argument see [23] and Lemma yield Ra, 2 log + s, 5 and so the desired result follows. Now let n 2. As in the roof of Theorem we have Ra n, n Ra s n + n a, n a Z s s h C s n s rh h C s n h a n 0 mod n h C s n, h 0 mod h a n 0 mod n h C s n h a n 0 mod n a Z s h a n + n a 0 mod n rh rh + rh + a Z s h a n h a n mod h C s n, h 0 mod h a n 0 mod n h C s n h a n 0 mod n rh rh. 2

13 Using rh rh for h Z s with h 0, we obtain with Ra n, n 2 Ra n, n + Σ 6 Σ : h C s n \C s n h a n 0 mod n rh. To bound Σ, we note that any h Cs n \ Cs n can be reresented uniquely in the form h h + n b with h C s n, b Cs. Therefore Σ h Cs n h a n 0 mod n b Cs r n b + b C s rh + n b h C s n h a n 0 mod n Next we observe that note that 3 is odd Σ s /2 r n b r n b b Cs b /2 s + 2 /2 2s /2 n b n b b In view of Lemma, this yields Now we consider Σ 2 Σ We write the inner sum in 8 as 2s log n h C s n h a n 0 mod n rh + n b b C s rh + n b : Σ + Σ 2. b Cs b + 2 n /2 b b s + 2 log s. 7 n b C s b C s Now for fixed h h,...,h s Cs n we have s rh + n b b C s j 3 rh + n b. 8 rh + n b rh. 9 b C. 0 rh j + n b.

14 If h j 0, then by Lemma, b C If h j < 2 n, then b C rh j + n b rh j + n b For the last sum we have and so /2 b b C b b 2 4 b C b C s rh j + r n b + 2 log n rh j + 2 log n. /2 b /2 rh j + b /2 rh j + b 2 + /2 b /2 b log , n b + h j + 2 n b 2n 2 b 2 h 2 j 2 n b 2n 2 b 2 2n 2 4 b 2 b + 2 n b h j + 3/2 b b /2 rh j + 2 /2 n b dx x + 2 rh j + n b rh j + 2 log n b b b 2. 4 The case 2 n < h j is symmetric to the above case, and so we obtain the same bound as above. Thus, in all cases we have rh j + n b rh j + 2 log n log 3 rh j In view of 9 and 0 this yields rh + n b log rh s, 3 b Cs and so Σ 2 log s Ra n, n. Using Σ Σ + Σ 2 and 7, we obtain Σ 2s log n + 2 log s + log n s Ra n, n. 4

15 Therefore by 6, Ra n, n log + 8 s Ra n, n 2s log + n log + 2s Ra n, n 2s log + n + 2 log + 2 log n n s. s By iterating this inequality and using 5, we get for all n 2, Ra n, n log + 2sn 2 log + s 2s log n + log + 2 sn j + 2 log n n j 2s log + 2 sn + n 2s log n + 2 log j2 s n 2 log + 2 sj. It is clear that the last exression is O n log +2 sn with an imlied constant deending only on s. 3 Polynomial lattices The construction of a olynomial lattice is quite similar to the construction of usual integration lattices, but now we use olynomial arithmetic over a finite field. Here we consider only olynomial lattices over the finite field Z of elements, where is a rime. For an introduction of olynomial lattices in their full generality, we refer to [25] and [26, Section 4.4]. Let be a rime and let Z x be the field of formal Laurent series over Z. Elements of Z x are formal Laurent series of the form L t l x l, lw where w is an arbitrary integer and all t l Z. Further let Z [x] be the set of all olynomials over Z. For an integer m, let υ m be the ma from Z x to the interval [0, defined by m υ m t l x l t l l, lw lmax,w where all t l Z {0,..., }. With the above notation we can now introduce olynomial lattices. For a given dimension s, we choose f Z [x] with degf and g g,...,g s Z [x] s. Then Pg, f is defined as the oint set consisting of the degf oints x h υ degf hg f,...,υ degf hgs f j0 [0, s, where h runs through all olynomials in Z [x] with degh < degf. The oint set Pg, f is called a olynomial lattice and g is called the generating vector of the olynomial lattice. QMC algorithms which use olynomial lattices as underlying oint sets 5 s

16 are called olynomial lattice rules. It is clear that g,...,g s are relevant only modulo f. Furthermore, it was shown by Niederreiter [25] that a olynomial lattice Pg, f is a digital t, m, s-net over Z, where m degf. For the determination of the quality arameter t and the corresonding generating matrices, we refer to [25] and [26, Section 4.4]. We could also consider randomly digitally shifted olynomial lattices. Let x x x 2 + and σ σ 2 + σ 2 + be the base reresentation of x res. σ in [0,. Then 2 the digitally shifted oint y x σ is given by y y + y 2 +, where y 2 i x i +σ i Z with addition modulo. For vectors x and σ in [0, s, we define the digitally shifted oint x σ comonentwise. Obviously, the shift deends on the base. If the shift σ [0, s is chosen i.i.d. and the same shift is alied to all oints of Pg, f, then we seak of a randomly digitally shifted olynomial lattice. QMC algorithms which use randomly digitally shifted olynomial lattices as underlying oint sets are called randomly digitally shifted olynomial lattice rules. Our aim is to construct a generating vector g coefficient by coefficient such that the corresonding olynomial lattice Pg, f is of good quality for f f, f 2, f 3,..., i.e., for cardinalities N degf, 2degf, 3degf,.... We remark here that the existence result of Niederreiter [27] is more general. Instead of the sequence f, f 2, f 3,..., Niederreiter investigated arbitrary divisibility chains of olynomials in Z [x]. We consider two quality measures for olynomial lattices. The first one is the worstcase error for QMC integration in a weighted Hilbert sace of functions which is based on Walsh functions res. the root mean-square worst-case error with resect to a digital shift σ in a weighted Sobolev sace and the second one is the star discreancy. The worst-case error in a weighted Hilbert sace. Let s N, α >, and γ γ j j with ositive real numbers γ j. Further let be a rime and denote by wal k, k N s 0, the -adic Walsh functions defined as follows. Let ω e 2πi/ C. For a nonnegative integer k with base reresentation k κ 0 + κ + + κ a a, the function wal k : R C, eriodic with eriod, is defined by wal k x ω κ 0x + +κ ax a+, where x [0, has base reresentation x x / + x 2 / 2 + unique in the sense that infinitely many of the x j must be different from. For dimensions s and for k k,...,k s N s 0, the s-dimensional -adic Walsh function wal k : R s C is defined as s wal k x : wal kj x j for all x x,...,x s R s. j For any integer s, the system {wal k : k N s 0 } is a comlete orthonormal system in L 2 [0, s. More information on Walsh functions can be found in [, 28, 33]. As in [5, 7, 9], we consider the weighted Hilbert function sace H wal,s,α,γ with reroducing kernel given by K wal,s,α,γ x, y ρ k N s α k, γ wal kxwal k y, 0 + 6

17 where for k k,...,k s N s 0 we ut ρ α k, γ s j ρ αk j, γ j with { if k 0, ρ α k, γ γ αψk if k N. Here, for k κ 0 + κ + + κ a a N with κ a 0, we ut ψ k a. Throughout this section we use the following notation. For arbitrary h h,...,h s and g g,..., g s in Z [x] s, we define the inner roduct h g : s h j g j, j and we write g 0 mod f if f divides g in Z [x]. Furthermore, we define for f Z [x] with degf, G f : {h Z [x] : degh < degf} and G s f : G f s. In [5] it was shown that the squared worst-case error of integration in H wal,s,α,γ using a olynomial lattice Pg, f, with f Z [x], degf, and g G s f, is given by e 2 degf,s,α,γg, f h Z[x] s \{0} h g 0 mod f ρ α h, γ. Here, for h h,...,h s Z [x] s, we ut ρ α h, γ s j ρ αh j, γ j with { if h 0, ρ α h, γ γ αdegh if h Z [x] \ {0}. Remark 8 It was shown in [7] see also [5] that e 2 g, f can be comuted in degf,s,α,γ Os degf oerations. Remark 9 Consider again the weighted Sobolev sace H s,γ from Remark 3 in Section 2. It was shown in [7] see also [5] that the mean-square worst-case error ê 2 for QMC degf,s,γ integration in H s,γ using a randomly digitally shifted olynomial lattice Pg, f is given by ê 2 degf,s,γg, f ρh, γ, h Z[x] s \{0} h g 0 mod f where for h h,...,h s Z [x] s we ut ρh, γ s j ρh j, γ j with { if h 0, ρh, γ 2γ 2r+ sin 2 κ rπ/ 3 if h κ 0 + κ x + + κ r x r with κ r 0. Note that because of the similarities between the worst-case error e degf,s,α,γg, f in the weighted Hilbert sace H wal,s,α,γ and the root mean-square worst-case error ê degf,s,γg, f in the Sobolev sace H s,γ, the following results will aly to both cases, though we will state them only for the Hilbert sace H wal,s,α,γ. 7

18 The star discreancy. The star discreancy res. discreancy has already been used in Section 2 as a quality criterion for usual integration lattices. For f Z [x] with degf and g G s f, we define the quantity R g, f h G s f\{0} h g 0 mod f ρh, where for h h,...h s Z [x] s we ut ρh s j ρh j with { if h 0, ρh πκ r+ sin 2 r if h κ 0 + κ x + + κ r x r with κ r 0. Then for the star discreancy D degf Pg, f of the olynomial lattice Pg, f we have see [6, Proosition 2.] D degf Pg, f s degf + R g, f. Hence good bounds on the quantity R g, f yield good bounds on the star discreancy of the olynomial lattice generated by g and f. We remark that the bound even holds without the square at the sine in the definition of the factors ρh. However, as shown in [6, Section 4], the slightly weaker version used here allows the comutation of R g, f in Os degf oerations. 3. Good extensible olynomial lattices with resect to the worst-case error We now resent an algorithm which constructs coefficient by coefficient a generating vector which is good with resect to the worst-case error e degf,s,α,γ for all f f, f 2, f 3,.... Algorithm 3 Let f Z [x] be irreducible.. Find g : g by minimizing e 2 degf,s,α,γ over all g Gs f. 2. For n 2, 3,... find g n : g n +f n a by minimizing e 2 n degf,s,α,γ g n +f n g, f n over all g G s f, i.e., a argmin g G s f e 2 n degf,s,α,γ g n + f n g, f n. Theorem 3 Let s, n N, be a rime, α >, and f Z [x] irreducible. Assume that g n G s fn is constructed according to Algorithm 3. Then we have s e 2 n degf,s,α,γ g n, f n + γ j µ α where µ α : α α. j Before we rove this result, we state two remarks. { min n, } α degf α degf 2 n degf, 8

19 Remark 0 The search for g in the first ste of Algorithm 3 takes Os degfs+ oerations. With a comonent-by-comonent construction see [5] this can be reduced to Os 2 2degf oerations. In this case we obtain a slightly weaker error bound in Theorem 3 the term after the first roduct must be deleted. Alternatively, in the first ste of Algorithm 3 one can choose an arbitrary vector g G s f. But then the error bound in Theorem 3 has to be relaced by s e 2 ndegf,s,α,γ g n, f n + γ j µ α j { min n, } α degf α degf This follows immediately from the subsequent roof of Theorem 3. n degf. Remark We remark that it is not necessary to start Algorithm 3 with item. If one has given an arbitrary generating vector g n0 for some n 0 N with squared worst-case error e 2 n 0 degf,s,α,γ g n 0, f n 0, then this vector can be extended as well for n n 0 +, n 0 +2,.... From the roof of Theorem 3 below it is easy to see that in this case we obtain e 2 ndegf,s,α,γ g n, f n e 2 n 0 degf,s,α,γ g n 0, f n 0 min Now we give the roof of Theorem 3. { n n 0 +, Proof. First we show the result for n. We have e 2 degf,s,α,γ g, f s degf s degf h Z[x] s \{0} h 0 mod f g G s f degf 2 degf h Z [x] s \{0} e 2 degf,s,α,γg, f ρ α h, γ ρ α h, γ + degf h Z [x] s \{0} which is the desired bound for n. Let n 2. Then we have e 2 n degf,s,α,γ g n, f n s degf s degf h Z [x] s \{0} ρ α h, γ } α degf α degf g G s f h g 0 mod f h Z[x] s \{0} h 0 mod f ρ α fh, γ + ρ α h, γ n n 0degf. degf h Z [x] s \{0} s + γ j µ α j e 2 n degf,s,α,γ g n + f n g, f n g G s f ρ α h, γ h Z [x] s \{0} 9 g G s f h g n +f n g 0 mod f n ρ α h, γ 2 degf,.

20 The inner sum is equal to the number of g G s f with f n h g h g n mod f n. For this we must have h g n 0 mod f n, and then h g h g f n n mod f. Thus, e 2 n degf,s,α,γ g n, f n sdegf h Z[x] s \{0} h g n 0 mod f n ρ α h, γ g G s f h g f n h g n mod f Consider the inner sum. If h 0 mod f, then the inner sum is equal to s degf. If h 0 mod f, then the inner sum is equal to 0 if h g n 0 mod f n and equal to sdegf if h g n 0 mod f n. Thus, e 2 ndegf,s,α,γ g n, f n degf h Z[x] s \{0}, h 0 mod f h g n 0 mod f n ρ α h, γ + degfe2 n degf,s,α,γ g n, f n degf + h Z[x] s \{0}, h 0 mod f h g n 0 mod f n ρ α h, γ, h Z[x] s \{0}, h 0 mod f h g n 0 mod f n. ρ α h, γ h Z[x] s \{0}, h 0 mod f h g n 0 mod f n and this holds for all n 2. If we insert this inequality for e 2 n degf,s,α,γ g n, f n, then we obtain e 2 n degf,s,α,γ g n, f n 2degfe2 n 2 degf,s,α,γ g n 2, f n 2 2degf + degf + h Z[x] s \{0}, h 0 mod f h g n 2 0 mod f n h Z[x] s \{0}, h 0 mod f h g n 0 mod f n ρ α h, γ. ρ α h, γ degf h Z[x] s \{0}, h 0 mod f h g n 2 0 mod f n 2 h Z[x] s \{0}, h 0 mod f h g n 0 mod f n ρ α h, γ ρ α h, γ Assume that h Z [x] s, h 0 mod f, i.e., h f h, and h g n 2 0 mod f n. Then we have h g n h g n 2 + f n 2 g h g n 2 + f n h g 0 mod f n, with some g G s f. Therefore we obtain e 2 ndegf,s,α,γ g n, f n 2degfe2 n 2 degf,s,α,γ g n 2, f n 2 2degf + h Z[x] s \{0}, h 0 mod f h g n 0 mod f n 20 ρ α h, γ. h Z[x] s \{0}, h 0 mod f h g n 2 0 mod f n 2 ρ α h, γ ρ α h, γ

21 Reeating this argument, we get e 2 ndegf,s,α,γ g n, f n n degfe2 degf,s,α,γ g, f n degf + h Z[x] s \{0}, h 0 mod f h g n 0 mod f n ρ α h, γ. For the last sum in the above exression, we have h Z[x] s \{0}, h 0 mod f h g n 0 mod f n ρ α h, γ With this uer bound, we obtain e 2 n degf,s,α,γ g n, f n h Z[x] s \{0} h g n 0 mod f n αdegf ρ α fh, γ h Z[x] s \{0} h g n 0 mod f n h Z[x] s \{0}, h 0 mod f h g 0 mod f ρ α h, γ αdegfe2 n degf,s,α,γ g n, f n. n degfe2 degf,s,α,γ g, f + αdegfe2 n degf,s,α,γ g n, f n. With backward induction on n and invoking the uer bound for e 2 g degf,s,α,γ, f, we get e 2 ndegf,s,α,γ g n, f n e 2 degf,s,α,γ g, f + n degf n 2+α degf + 2αdegfe2 n 2 degf,s,α,γ g n 2, f n 2. e 2 degf,s,α,γ g n 2, f k0 n e 2 degf,s,α,γ g, f ρ α h, γ + n k+kαdegf n α degfe2 degf,s,α,γ g, f n k+kαdegf k0 s + γ j µ α min j { n, } α degf α degf 2 ndegf, which is the desired bound. 3.2 Good extensible olynomial lattices with resect to the star discreancy We also resent an algorithm which uses the quantity R and hence the star discreancy as a quality criterion. 2

22 Algorithm 4 Let f Z [x] be monic and irreducible.. Find g : g by minimizing R g, f over all g G s f. 2. For n 2, 3,... find g n : g n + f n a by minimizing R g n + f n g, f n over all g G s f, i.e., a argmin g G s fr g n + f n g, f n. Theorem 4 Let s, n N, be a rime, f Z [x] monic and irreducible, and define ν,f : degf 2. Assume that g 3 n G s fn is constructed according to Algorithm 4. Then we have R g n, f n O degfn + ν,f s + n with an imlied constant deending only on s. Before we give the roof of this result, we state a remark and a corollary. Remark 2 Again we remark that it is not necessary to start Algorithm 4 with item. If one has given an arbitrary generating vector g n0 for some n 0 N, then this vector can be extended as well for n n 0 +, n 0 + 2,.... From the roof of Theorem 4 below it is easy to see that in this case we obtain Rg n, f n + ν,f s + n n 0 Rg degfn n 0 n0, f n 0 + O where the imlied constant deends only on s. degfn + ν,f s + n n 0+, From and Theorem 4 we immediately obtain a bound on the star discreancy. This bound is good only for small and olynomials f of large degree. Corollary 3 Let s, n N, be a rime, and f Z [x] monic and irreducible. Assume that g n G s fn is constructed according to Algorithm 4. Then we have D degfn Pg n, f n O degfn + ν,f s + n with an imlied constant deending only on s. For the roof of Theorem 4 we need the following result. Lemma 2 We have where ν,f : degf 2 3. Proof. We have and further h G f degf ρh + u0 h G s f ρh + ν,f s, ρh h G s f h G f u+u κ ρh sin 2 πκ s + degf 2 3, since κ sin 2 πκ 2 as shown, for examle, in [7, Aendix C]. 3 22

23 Now we give the roof of Theorem 4. Proof. First we show the result for n. We have R g, f R sdegf g, f sdegf degf g G s f h G s f\{0} h G s f\{0} ρh ρh degf + ν,f s, g G s f h g 0 mod f where we used Lemma 2 for the last equality. Hence the desired result follows for n. Now let n 2. As in the roof of Theorem 3 we have R g n, f n R sdegf g n + f n g, f n sdegf sdegf degf degf g G s f h G s f n \{0} h G s fn \{0} ρh h g n 0 mod f n h G s fn \{0}, h 0 mod f h g n 0 mod f n h G s fn \{0} h g n 0 mod f n g G s f h g n +f n g 0 mod f n ρh g G s f h g f n h g n mod f ρh + ρh + h G s fn \{0}, h 0 mod f h g n 0 mod f n h G s fn \{0} h g n 0 mod f n ρfh. ρh Since f is monic, we find that ρfh degf ρh for h Z [x] s with h 0. Hence we obtain R g n, f n 2 degfr g n, f n + Σ 2 degf with Σ : h G s fn \G s fn h g n 0 mod f n ρh. Any h G s fn \ G s fn can be reresented uniquely in the form Therefore Σ h h + f n b with h G s f n, b G s f \ {0}. h G s fn h g n 0 mod f n b G s f\{0} : Σ + Σ 2. ρh + f n b b G s f\{0} ρf n b + h G s fn \{0} b G s f\{0} h g n 0 mod f n 23 ρh + f n b

24 First we deal with Σ. By Lemma 2 we have Σ s + ρf n b b G f + ν s,f sν,f degfn degfn Now we turn to Σ 2. We have Σ 2 b G s f h G s fn \{0} b G s f\{0} h g n 0 mod f n Denoting the inner sum by Σ 3, we get Σ 3 ρh + f n b s ρh j degfn b G f\{0} + ν,f degfn b G f where h h,...,h s. If h j 0, then by Lemma 2 and since f is monic, b G f ρh j + f n b b G f ρf n b + If h j 0, then 0 degh j < degf n and b G f ρh j + f n b ρh j + b G f\{0} ρh j + ν,f, degfn ρh + f n b. s. ρb s ρh j + f n b ρh, ν,f degfn ρh j + ρh j + f n b ρh j + where we again used Lemma 2 and the assumtion that f is monic. Hence we obtain s Σ 3 ρh j + ν,f degfn ρh. j Since ρh j, it follows now that degfn and hence Σ 3 s j Altogether we find that Σ Σ + Σ 2 sν,f degfn ρh j + ν,f ρh j ρh ρh + ν,f s Σ 2 + ν,f s R g n, f n. ν,f degfn. b G f\{0} + ν s,f + + ν degfn,f s R g n, f n 24 ρf n b

25 and hence R g n, f n degf + ν,f s + R g n, f n + sν,f degfn Iterating this inequality, we get for all n 2, + ν s,f. degfn R g n, f n + ν,f s + n R degfn g, f n 2 + ν,f s j + +sν,f degf degfn j j0 + ν,f s + n degfn + sν,f + ν,f degfn degf s n ν,f s + j. j0 s ν,f degfn j Hence + R g n, f n ν,f s + n O with an imlied constant deending only on s. degfn 4 Numerical Results In this section we resent some numerical results for the classical case. We wrote a rogram for Algorithm with Mathematica. In each of the following examles we have chosen 2. In Figure we considered the unweighted case in dimension s 5, whereas in Figures 2 and 3 we investigated the weighted case in dimensions s 0 and s 5 with weights γ j j 2 in both cases. In all cases we tried different values of α. In contrast to our theoretical bound from Theorem, in all exeriments higher values of α show higher convergence rates for the worst-case errors. This reflects the widely held view that smoother functions are in general much easier to integrate than nonsmooth functions. However, the numerical results seem to suggest that the integration lattices constructed with Algorithm do not achieve the otimal rate for the worst-case error e N,s,α,γ in the first lace which would be of order ON α/2+ε for ε > 0. For examle, for α 2, the observed α for the convergence rate seems to be around.25 or for α 6, the observed α seems to be around 2.5. In each of the following figures we lotted on the x-axes the value log 2 2 n n and on the y-axes the base 2 logarithm of the corresonding squared worst-case error e 2 2 n,s,α,γ. References [] Chrestenson, H.E.: A class of generalized Walsh functions. Pacific J. Math. 5: 7 3,

26 Α 2 Α 4 Α 6 Α Figure : log 2 e 2 2 n,s,α,γ with s 5, α {2, 4, 6, 8}, γ j, and n 2,..., 5. 0 Α 2-5 Α 4-0 Α Figure 2: log 2 e 2 2 n,s,α,γ with s 0, α {2, 4, 6}, γ j 2 j, and n 2,..., Α 2 Α Figure 3: log 2 e 2 2 n,s,α,γ with s 5, α {2, 4}, γ j 2 j, and n 2,...,0. 26

27 [2] Cools, R., Kuo, F.Y. and Nuyens, D.: Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comut. 28: , [3] Dick, J.: On the convergence rate of the comonent-by-comonent construction of good lattice rules. J. Comlexity 20: , [4] Dick, J.: The construction of extensible olynomial lattice rules with small weighted star discreancy. Math. Com. 76: , [5] Dick, J., Kuo, F.Y., Pillichshammer, F. and Sloan, I.H.: Construction algorithms for olynomial lattice rules for multivariate integration. Math. Com. 74: , [6] Dick, J., Leobacher, G. and Pillichshammer, F.: Construction algorithms for digital nets with low weighted star discreancy. SIAM J. Numer. Anal. 43: 76 95, [7] Dick, J. and Pillichshammer, F.: Multivariate integration in weighted Hilbert saces based on Walsh functions and weighted Sobolev saces. J. Comlexity 2: 49 95, [8] Dick, J., Pillichshammer, F. and Waterhouse, B.J.: The construction of good extensible rank- lattices. Math. Com., to aear, [9] Dick, J., Sloan, I.H., Wang, X. and Woźniakowski, H.: Liberating the weights. J. Comlexity 20: , [0] Gill, H.S. and Lemieux, C.: Searching for extensible Korobov rules. J. Comlexity 23: , [] Hickernell, F.J.: Lattice rules: how well do they measure u? In: Hellekalek, P. and Larcher, G., eds., Random and Quasi-Random Point Sets, Lecture Notes in Statistics, vol. 38, , Sringer, New York, 998. [2] Hickernell, F.J.: My dream quadrature rule. J. Comlexity 9: , [3] Hickernell, F.J. and Hong, H.S.: Comuting multivariate normal robabilities using rank- lattice sequences. In: Scientific Comuting Hong Kong, 997, , Sringer, Singaore, 997. [4] Hickernell, F.J., Hong, H.S., L Ecuyer, P. and Lemieux, C.: Extensible lattice sequences for quasi-monte Carlo quadrature. SIAM J. Sci. Comut. 22: 7 38, [5] Hickernell, F.J. and Niederreiter, H.: The existence of good extensible rank- lattices. J. Comlexity 9: , [6] Hlawka, E.: Zur angenäherten Berechnung mehrfacher Integrale. Monatsh. Math. 66: 40 5, 962. [7] Korobov, N.M.: The aroximate comutation of multile integrals. Dokl. Akad. Nauk SSSR 24: , 959. In Russian 27

28 [8] Korobov, N.M.: On the calculation of otimal coefficients. Soviet Math. Dokl. 26: , 982. [9] Kritzer, P. and Pillichshammer, F.: Constructions of general olynomial lattices for multivariate integration. Bull. Austral. Math. Soc. 76: 93 0, [20] Kuiers, L. and Niederreiter, H.: Uniform Distribution of Sequences. John Wiley, New York, 974; rerint, Dover Publications, Mineola, NY, [2] Kuo, F.Y.: Comonent-by-comonent constructions achieve the otimal rate of convergence for multivariate integration in weighted Korobov and Sobolev saces. J. Comlexity 9: , [22] Kuo, F.Y. and Joe, S.: Comonent-by-comonent construction of good lattice rules with a comosite number of oints. J. Comlexity 8: , [23] Niederreiter, H.: Existence of good lattice oints in the sense of Hlawka. Monatsh. Math. 86: , 978. [24] Niederreiter, H.: Point sets and sequences with small discreancy. Monatsh. Math. 04: , 987. [25] Niederreiter, H.: Low-discreancy oint sets obtained by digital constructions over finite fields. Czechoslovak Math. J. 42: 43 66, 992. [26] Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. No. 63 in CBMS-NSF Series in Alied Mathematics. SIAM, Philadelhia, 992. [27] Niederreiter, H.: The existence of good extensible olynomial lattice rules. Monatsh. Math. 39: , [28] Rivlin, T.J. and Saff, E.B.: Joseh L. Walsh Selected Paers. Sringer, New York, [29] Sloan, I.H. and Joe, S.: Lattice Methods for Multile Integration. Clarendon Press, Oxford, 994. [30] Sloan I.H., Kuo, F.Y. and Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev saces. SIAM J. Numer. Anal. 40: , [3] Sloan, I.H. and Woźniakowski, H.: When are quasi-monte Carlo algorithms efficient for high-dimensional integrals? J. Comlexity 4: 33, 998. [32] Sloan, I.H. and Woźniakowski, H.: Tractability of multivariate integration for weighted Korobov classes. J. Comlexity 7: , 200. [33] Walsh, J.L.: A closed set of normal orthogonal functions. Amer. J. Math. 55: 5 24,

29 Authors Addresses: Harald Niederreiter, Deartment of Mathematics, National University of Singaore, 2 Science Drive 2, Singaore nied@math.nus.edu.sg Friedrich Pillichshammer, Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria. friedrich.illichshammer@jku.at 29