Construction Algorithms for Higher Order Polynomial Lattice Rules

Transcription

1 Construction Algorithms for Higher Order Polynomial Lattice Rules Jan Baldeaux, Josef Dick, Julia Greslehner and Friedrich Pillichshammer Dedicated to Gerhard Larcher on the occasion of his 50th birthday Abstract Higher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. Recently it has been shown that higher order polynomial lattice point sets of good quality must exist. However, it was not shown how to construct such point sets avoiding an exhaustive search. This is the contribution of the present paper. We use a component-by-component approach to construct higher order polynomial lattice rules achieving optimal convergence rates for functions of arbitrarily high smoothness and at the same time under certain conditions on the weights strong polynomial tractability. In addition, we show how to combine a sieve-type algorithm with the component-by-component approach to construct higher order polynomial lattice rules adjusting themselves to the smoothness of the The support of the Australian Research Council under its Centre of Excellence Program is gratefully acknowledged. The support of the Australian Research Council under its Centre of Excellence Program is gratefully acknowledged. The author is supported by an Australian Research Council Queen Elizabeth II Research Fellowship. F.P. is supported by the Austrian Science Foundation FWF, Project S9609, that is part of the Austrian National Research Network Analytic Combinatorics and Probabilistic Number Theory.

2 2 integrand up to a certain given degree. Analogous results for higher order Korobov polynomial lattice point sets are presented as well. Introduction Quasi-Monte Carlo rules are equal weight integration formulas used to approximate integrals over the unit cube [0, ] s, where the dimension s is typically large. In particular, one approximates an integral [0,] s fx dx by Q N,s f = N N n=0 fx n where x 0,..., x N [0, s. Popular choices for the underlying integration nodes x 0,..., x N [0, s are either lattice point sets see [4, 5] or digital t, m, s-nets see [2, 4]; in this paper, we focus on digital nets. Recently, digital higher order nets were introduced by Dick [3] which include digital t, m, s-nets as special cases and have the appealing property that they can exploit the smoothness of the integrand under consideration. This is not possible with ordinary digital t, m, s-nets. To be more precise, if the integrand under consideration has square integrable mixed partial derivatives of order α in each variable, then digital higher order nets consisting of N points can produce integration errors converging at a rate of N α+ε with arbitrary small ε > 0. Having established the desirable properties of digital higher order nets, an important question is how to construct them. One possible answer to this question was given in [3, Section 4.4], where it was shown how to obtain digital higher order nets from classical digital nets. The aim of the present paper is to provide constructions of digital higher order nets independent of classical digital nets. As alternative construction we consider polynomial lattice rules which first have been introduced by Niederreiter [3, 4] as special cases of digital nets and later generalized in [8] as special cases of digital higher order nets. Quasi-Monte Carlo rules using such point sets as integration nodes are nowadays known as higher order polynomial lattice rules. We refer to [9, 4] for more information on polynomial lattice rules. In [8] the existence of higher order polynomial lattice rules achieving optimal convergence rates was established with an averaging argument. In addition, these rules can at the same time achieve strong polynomial tractability. However,

3 3 this approach is purely probabilistic and gives no hint of how to construct such point sets. A further nonconstructive existence result for good higher order polynomial lattice rules is presented in [6]. This approach uses the concept of a figure of merit. The results presented in these papers were very encouraging. In particular, for large values of α and s, the higher order polynomial lattice rules improved on the construction based on classical digital nets from [3, Section 4.4]. In this paper we find a construction procedure producing higher order polynomial lattice rules which perform well when applied to numerical integration. In particular, we consider two cases: Firstly, we use a componentby-component CBC approach an idea first used in [6] to produce higher order polynomial lattice rules achieving the optimal rate of convergence for functions having higher order mixed partial derivatives, see Algorithm and Theorem 3.. Furthermore, by combining the CBC approach with a sieve - type algorithm as used in [0] we can even construct higher order polynomial lattice rules which automatically adjust themselves to the smoothness of the integrand in terms of the convergence of the integration error within a certain arbitrarily high range; see Algorithm 2 and Theorem 4.2. We point out already here, that an analogous result for lattice rules is not known. In addition, we study a special case of higher order polynomial lattice rules, namely higher order Korobov polynomial lattice rules as introduced in [8]; see also [] for the classical definition. We find that analogous results as for the higher order polynomial lattice rules constructed using a CBC and CBC sieve approach can be established. The approaches to constructing digital higher order nets presented in this paper are direct, that is, we avoid employing classical digital nets, which earlier constructions see [3, Section 4.4] relied on. Furthermore, the higher order polynomial lattice rules are constructed to perform well when applied to numerical integration and without reference to the quality parameter t of the resulting digital higher order net. The structure of the paper is as follows: In Section 2 we recall higher order polynomial lattice rules, discuss the function space under consideration and present a result on numerical integration in this function space employing higher order polynomial lattice rules. In Section 3 we use a CBC approach to construct higher order polynomial lattice rules achieving optimal rates of convergence for functions of a given smoothness and in Section 4 we show how to construct higher order polynomial lattice rules achieving optimal convergence rates for a given range of smoothness parameters using a CBC

4 4 sieve algorithm. Finally, in Section 5, analogous results for higher order Korobov polynomial lattice rules are established. 2 Preliminaries In this section we introduce higher order polynomial lattice rules which can achieve arbitrarily high convergence rates, the function space under consideration, and a result on numerical integration in this function space when using higher order polynomial lattice rules. 2. Polynomial lattice rules for arbitrarily smooth functions For a prime b let Z b be the finite field with b elements and let Z b x be the field of formal Laurent series over Z b. Elements of Z b x are formal Laurent series, L = t l x l, l=w where w is an arbitrary integer and all t l Z b. Note that Z b x contains the field of rational functions over Z b as a subfield. Further let Z b [x] be the set of all polynomials over Z b. For an integer n let v n be the map from Z b x to the interval [0, defined by n v n t l x l = t l b l. l=w l=max,w The following definition of higher order polynomial lattice rules given in [8] is a slight generalization of the definition from [3], see also [4]. Definition 2. Let b be prime and let m n be integers. For a given dimension s, choose px Z b [x] with degpx = n and let q x,..., q s x Z b [x]. For 0 h < b m let h = h 0 + h b + + h m b m be the b-adic expansion of h. With each such h we associate the polynomial hx = m r=0 h r x r Z b [x].

5 5 Then S p,m,n q, where q = q x,..., q s x, is the point set consisting of the b m points hxq x hxqs x x h = v n,..., v n [0, s, px px for 0 h < b m. A quasi-monte Carlo rule using the point set S p,m,n q is called a polynomial lattice rule. Remark 2. Using similar arguments as for the classical case n = m, see [3, 4], it can be shown that the point set S p,m,n q is a digital net in the sense of [3] which can be seen as a generalisation of the classical definition of digital nets according to Niederreiter [2, 3, 4]. The generating matrices C,..., C s Z n m b of this digital net can be obtained in the following way: For j s consider the expansions q j x px = u j l x l Z b x, l=w j where w j Z. Then the elements c j l,r of the n m matrix C j over Z b are given by c j l,r = uj r+l Z b, for j s, l n, 0 r m. For the rest of the paper, we make use of the following notation: We write h for vectors over Zb and h for vectors over Z or R. Polynomials over Z b are denoted by hx and vectors of polynomials by hx. Furthermore, given an integer h with b-adic expansion h = r=0 h rb r, we denote the associated polynomial by hx, which is given by n hx = h r x r. r=0 For arbitrary kx = k x,..., k s x Z b [x] s and qx = q x,..., q s x Z b [x] s, we define the inner product kx qx = s k j xq j x Z b [x],

6 6 and we write qx 0 mod px if px divides qx in Z b [x]. We remark here that for our results only the degree of the polynomial px is important and not the specific choice of px itself we assume though that px is irreducible, but this assumption could be removed by a more complicated analysis. 2.2 Walsh functions and the function space W α,s,γ We now define the space of functions we are going to study. This function space is based on Walsh functions whose definition is recalled in the following. Let N 0 denote the set of nonnegative and N the set of positive integers. Each k N has a unique b-adic representation k = a i=0 κ ib i with digits κ i {0,..., b } for 0 i a, where κ a 0. For k = 0 we have a = 0 and κ 0 = 0. Similarly, each x [0, has a b-adic representation x = i= ξ ib i with digits ξ i {0,..., b } for i. This representation is unique in the sense that infinitely many of the ξ i must differ from b. We define the kth Walsh function in base b, wal k : [0, C by wal k x := exp2πiξ κ ξ a+ κ a /b. For dimension s 2 and vectors k = k,..., k s N s 0 and x = x,..., x s [0, s we define wal k : [0, s C by wal k x := wal kj x j. It follows from the definition above that Walsh functions are piecewise constant functions. For more information on Walsh functions, see, e.g., [, 8] or [9, Appendix A]. When studying integration errors resulting from the approximation of an integral based on a digital net or digital higher order net or a higher order polynomial lattice rule, it is convenient to consider the Walsh series of the integrand f. In particular, for f L 2 [0, ] s, the Walsh series of f is given by fx fkwal k x, k N s 0 where the Walsh coefficients fk are given by fk = fxwal k x dx. [0,] s

7 7 In general, the Walsh series given in equation need not converge to f, however, for the space of Walsh series W α,s,γ, which we define in the following, it does, see also [3]. For more details on the convergence of Walsh series, we refer to [3] or [9]. Throughout the paper we assume that b is a fixed prime, all polynomials are over Z b [x] and all Walsh functions are also considered in the same base b. The function space under consideration in this paper is the space W α,s,γ L 2 [0, ] s as introduced in [3]. Here γ = γ j is a sequence of positive, nonincreasing weights, which are introduced to model the importance of different variables for our approximation problem, see [7]. For s N let [s] := {,..., s} and for u [s] let γ u := j u γ j be the weight associated with the projection onto components whose index is contained in u. Given a positive integer k with base b expansion k = κ b a + κ 2 b a2 + +κ v b av, a v < < a, v, we define µ α k := a + +a minv,α. Furthermore we put µ α 0 := 0. For k N 0 and a weight γ > 0, we define a function { if k = 0, r α γ, k := γb µαk otherwise. If we consider a vector k N s 0 of the form k = k,..., k s, we set r α γ, k := r α γ j, k j. Definition 2.2 The space W α,s,γ L 2 [0, ] s consists of all Walsh series f = k N fkwal s k for which the norm 0 f Wα,s,γ := sup fk k N s r 0 α γ, k. 2 is finite. For α 2, the following property was shown in [3]: Let f : [0, ] s R be such that all mixed partial derivatives up to order α in each variable are square integrable, then f W α,s,γ. Furthermore, an inequality using a Sobolev type norm and the norm in equation 2 was shown in [3], see also [2, 4]. Consequently, the results we are going to establish in the following

8 8 for functions in W α,s,γ also apply automatically to smooth functions. The assumption α > is needed to ensure that the sum of the absolute values of the Walsh coefficients converges. For the case α =, which requires a different analysis, we refer to [7] or to [9]. 2.3 Numerical Integration in W α,s,γ We are interested in the worst-case error of multivariate integration in W α,s,γ using a quasi-monte Carlo rule Q b m,s, which is given by eq b m,s, W α,s,γ = The initial error is given by eq 0,s, W α,s,γ = sup I s f Q b m,sf. 3 f W α,s,γ f Wα,s,γ sup I s f = I s. f W α,s,γ f Wα,s,γ We denote the quasi-monte Carlo rule based on a polynomial lattice rule S p,m,n q by Q b m,sq and the associated worst-case integration error by e b m,αq, p. The next proposition gives information on this quantity. Proposition 2. Let b be a prime and α 2 an integer. Then the worstcase integration error for multivariate integration in W α,s,γ using the polynomial lattice rule S p,m,n q is given by where e b m,αq, p = k D pq r α γ, k, D p q := { k N s 0 \ {0} : kx qx ax mod px with degax < n m}. 4 Proof. Combine [3, equation 5.2] with the determination of the dual net D of a polynomial lattice from [8, Section 4].

9 3 Component-by-component construction of polynomial lattice rules We propose the following algorithm to construct a polynomial lattice rule that achieves higher order convergence. We remark that unlike the results presented in Section 4, we only deal with a fixed α in this section. For ease of notation, we proceed as follows: We use q = qx Z b [x], p = px Z b [x] and a = ax Z b [x]; also, if we consider the polynomial associated with an integer k, we use k = kx Z b [x]. We put G b,n := {q Z b [x] : degq < n}. We also make use of the following lemma, which appeared in a weaker and nonexplicit form as [8, Lemma 4.2]. Lemma 3. Let α 2 be an integer. Then for every /α < λ we have where rαγ, λ l γ λ C b,α,λ, l= α C b,α,λ := C b α b,α,λ + b λα b b λi, i= { α if λ =, C b,α,λ = b b α b λ α if λ <. b b λ b λ α Furthermore, the series l= rλ αγ, l diverges to as λ goes to /α from the right. Proof. Let l = λ b a + + λ v b av where v, 0 < a v < < a and λ i {,..., b }. We divide the sum over all l N into two parts, namely firstly where v α and secondly where v > α. For the first part we have α b v v= b λa + +a v 0<a v< <a 9

10 0 = α b v v= α b b λ v= =: Cb,α,λ. a b λa a =v a 2 =v v = a v b λa 2 b λav a v= { α if λ =, b b α b λ α if λ <, b b λ b λ α For the second part we have b α = = = = baα b λa + +a α 0<a α< <a b α b b α b b α b b α b λα b a b λa a =α a 2 =α a α= α i= α i= b aα b λaα b λi b λi. Hence, we have shown that a α =a α+ a α= a α b λa 2 a α= b λa α b aα b λaα b λα aα b aα b λaα b λa 2 a 2 =a 3 + b λa a =a 2 + As b α b λα b α λ b α γ b λα b i= γ λ C b,α,λ + α i= b λi rαγ, λ l b α b λα b l= α i= =: γ λ C b λi b,α,λ. whenever λ /α from the right we also obtain b λi the second assertion. Now we show that a component-by-component approach can be used to construct a polynomial lattice rule that achieves higher order convergence, where for d s, we set q d = q,..., q d. Note that we consider this vector instead of, q 2,..., q s, c.f. [5, Algorithm 4.3], as otherwise the

11 projection onto the first coordinate does not achieve a convergence rate of b αm, see also [8, Remark 2.3]. The component-by-component algorithm for a fixed α is summarised in Algorithm. Algorithm CBC algorithm for fixed α Require: b a prime, s, m N and weights γ = γ j j. : Choose an irreducible polynomial p Z b [x], with degp = n. 2: for d = to s do 3: find q d G b,n by minimising e b m,αq,..., q d, p as a function of q d. 4: end for 5: return q = q,..., q s. Theorem 3. Let b be prime, let s, n, m, α N, m n and let α 2. Let p Z b [x] be irreducible with degp = n. Suppose q,..., q s G s b,n is constructed using Algorithm. Then for all d =,..., s we have: e b m,αq,..., q d, p b minτm,n d + 3γ /τ j C b,α,/τ τ τ < α. Proof. We firstly show the result for d =. By Proposition 2., e b m,αq, p = r α γ, k. k D pq The algorithm chooses q as to minimise the worst-case error, so we have Hence for all /α < λ we have e b m,αq, p e b m,αq, p, q G b,n. e b m,αq, p λ e b n b m,αq, p λ. q G b,n Using an argument very similar to the one used in the proof of [8, Proposition 4.3], it can be shown that for all /α < λ e b m,αq, p λ e b n b m,αq, p λ γ λ C b,α,λ b m + b λn. q G b,n

12 2 Consequently, setting τ = /λ we obtain e b m,αq, p + 2γ λ C b,α,λ /λ b minm/λ,n + 3γ /τ C b,α,/τ τ b minmτ,n. We now assume that for some d < s we have q d Gd b,n such that d e b m,αq d, p b minτm,n + 3γ /τ j C b,α,/τ τ. We consider e b m,αq d, q d+, p = = k,k d+ D pq d,q d+ k D pq d r α γ, k + r α γ, kr α γ d+, k d+ k d+ = = e b m,αq d, p + θq d, q d+, where we set θq d, q d+ := k d+ = r α γ d+, k d+ r α γ d+, k d+ k N d 0 k,k d+ D pq d,q d+ k N d 0 k,k d+ D pq d,q d+ r α γ, k. r α γ, k We see from Algorithm that qd+ is chosen in such a way that the worst-case error e b m,αq d, q d+, p is minimised. Since the only dependence on q d+ is in θq d, q d+ we have θq d, q d+ θq d, q d+ for all q d+ G b,n. This implies that for all /α < λ we have θq d, qd+ λ θq b d, q n d+ λ q d+ G b,n = b n q d+ G b,n k d+ = r α γ d+, k d+ k N d 0 k,k d+ D pq d,q d+ r α γ, k λ

13 3 b n q d+ G b,n k d+ = k d+ = p k d+ + b n rαγ λ d+, k d+ k d+ = p k d+ rαγ λ d+, k d+ r λ αγ d+, k d+ k N d 0 k q d a mod p dega<n m q d+ G b,n k N d 0 k,k d+ D pq d,q d+ r α γ, k k N d 0 k q d +k d+q d+ a mod p dega<n m λ r α γ, k λ r λ αγ, k, where we used Jensen s inequality, which states that for a sequence a k of nonnegative reals we have a k λ a λ k for any 0 < λ. Now we have k d+ = p k d+ rαγ λ d+, k d+ γλ d+ C b,α,λ b λn where we used the following: rαγ, λ k = k= p k + k N d 0 k q d a mod p dega<n m k N d 0 \{0} k q d a mod p dega<n m γλ d+ C b,α,λ b λn + eb m,αq d, p λ, rαγ, λ b n l + l= l=0 b n k= p k r α γ, k λ r α γ, k λ r λ αγ, k + b n l.

14 4 For l > 0 we have r α γ, b n l b n r α γ, l. Further for k < b n the polynomial p never divides k since degp = n. Hence rαγ, λ k = k= p k rαγ, λ b n l b λn l= l= r λ αγ, l γλ C b,α,λ b λn. Next we consider the case where k d+ is not a multiple of p. Here we have b n q d+ G b,n b n Now we have Hence k d+ = p k d+ k d+ = p k d+ q d+ G b,n = k N d 0 rαγ λ d+, k d+ r λ αγ d+, k d+ q d+ G b,n k N d 0 \{0} k q d +k d+q d+ a mod p dega<n m k N d 0 r λ αγ, k r λ αγ, kb n m d = b n m + C b,α,λ γj λ. k,k d+ D pq d,q d+ r α γ, k k N d 0 k q d +k d+q d+ a mod p dega<n m r λ αγ, k a Z b [x] q d+ G b,n dega<n m k q d +k d+ q d+ a mod p λ r λ αγ, k. b n k d+ = p k d+ r λ αγ d+, k d+ q d+ G b,n k N d 0 \{0} k q d +k d+q d+ a mod p dega<n m r λ αγ, k

15 5 Consequently, b n k d+ = b m C b,α,λγ λ d+ r λ αγ d+, k d+ b n m d + C b,α,λ γj λ d + C b,α,λ γj λ. θq d, q d+ γ λ d+ C b,α,λ b λn + e b m,αq d, p λ + b m C b,α,λγ λ d+ γ d+ C /λ b,α,λ [ d + C b,α,λ γj λ /λ b λn + e b m,αq d, p λ + b m /λ d + C b,α,λ γj ] λ. We now set τ = /λ and use the induction hypothesis to obtain θq d, qd+ γ d+ Cb,α,/τ τ b + e n/τ b m,αq d, p /τ + d + C b m b,α,/τ γ /τ τ γ d+ Cb,α,/τ τ 3 d + 3γ /τ b minm,n/τ j C b,α,/τ = Finally, we have 3 τ b minτm,n γ d+c τ b,α,/τ d + 3γ /τ j C b,α,/τ τ. e b m,αq d+, p = e b m,αq d, p + θq d, qd+ d + 3γ /τ b minτm,n j C b,α,/τ τ = 3 τ + b γ d+c τ minτm,n b,α,/τ d + 3γ /τ C b,α,/τ τ b minτm,n + 3τ γ d+ C τ b,α,/τ j d + 3γ /τ C b,α,/τ τ j j τ

16 6 d+ + 3γ /τ b minτm,n j C b,α,/τ τ, where we again used Jensen s inequality. From Theorem 3. we obtain the following corollary. Corollary 3. Let b be prime, let s, n, m, α N, m n and α 2. Let p Z b [x] be irreducible with degp = n and suppose q G s b,n is constructed using Algorithm. We have where e b m,αq, p γ α δ j c s,α,γ,δ := c s,α,γ,δ b minα δm,n 0 < δ α, α δ + 3γ α δ j C b,α,. α δ Suppose <, then c s,α,γ,δ c,α,γ,δ < and we have e b m,αq, p c,α,γ,δ 0 < δ α. b minα δm,n Thus the worst-case error is bounded independently of the dimension. Under the assumption A := lim sup s s γ j/log s < we obtain c s,α,γ,α c η s 2C b,α,a+η and therefore e b m,αq, p c ηs 2C b,α,a+η η > 0, b m where c η depends only on η. Thus the worst-case error satisfies a bound which depends only polynomially on the dimension. Proof. The first part follows from Theorem 3. by setting τ = α δ. The second and the third part follow from the first part in exactly the same way as in the proof of [5, Corollary 4.5]. The above result shows that higher order polynomial lattice rules can achieve a worst-case error satisfying at the same time the almost optimal convergence rate and a bound which depends only polynomially or even does not depend on the dimension s the technical term for such a behavior is strong polynomial tractability. Until now it is not known whether this is possible for ordinary lattice rules.

17 4 Optimal convergence rates for a range of smoothness parameters In this section, we construct polynomial lattices which are optimal for a range of smoothness parameters; we use α and τ α to denote the smoothness, where 2 α β, τ α < α. We set A m,n,s,α,p λ := e λ b sn b,αq m s, p. q s G s b,n Proposition 4. For α 2 and /α < λ we have 2 A m,n,s,α,p λ + + γ λ b minm,λn j C b,α,λ. Proof. Using Proposition 2. and Jensen s inequality we obtain, A m,n,s,α,p λ b sn = q G s b,n k D pq rαγ, λ k b sn k N s 0 \{0} r λ αγ, k q G s b,n k q a mod p dega<n m 7. 5 In the case where all components of k are multiples of p every q satisfies the equation k q 0 mod p and hence we have b sn q G s b,n k q a mod p dega<n m = and the sum over all k which satisfy this condition is therefore bounded by k N s 0 \{0} k 0 mod p r λ αγ, k = + rαγ λ j, k. k=0 p k

18 8 Now we have rαγ λ j, k = k=0 p k rαγ λ j, b n l + l=0 l=0 b n k= p k r λ αγ j, k + b n l. For l > 0 we have r α γ j, b n l b n r α γ j, l and further for k < b n the polynomial p never divides k since degp = n. Hence rαγ λ j, k = + k=0 p k l= r λ αγ j, b n l + b λn rαγ λ j, l. l= Therefore, k N s 0 \{0} k 0 mod p r λ αγ, k + = =u [s] b λn + b λn γj λ C b,α,λ b u λn γ λ u C u b,α,λ + + γj λ C b,α,λ. In the case where there is at least one component of k which is not a multiple of p we have = b sn b m q G s b,n k q a mod p dega<n m and therefore this part of equation 5 is bounded by b m k N s 0 \{0} k 0 mod p r λ αγ, k b m k N s 0 \{0} r λ αγ, k + b m + γj λ C b,α,λ.

19 9 Altogether we now obtain that A m,n,s,α,p λ b γ λ m b λn j C b,α,λ γ λ b minm,λn j C b,α,λ as required. G s b,βm where Let α β and set n = βm. Let ν denote the equiprobable measure on. For c and τ < α β the following set is introduced: C b,α c, τ := { q G s b,βm : e b m,αq, p E b,α,γ,s,m c, τ }, 6 Furthermore, let C b,α c := E b,α,γ,s,m c, τ := 2τ c τ τ<α b τm C b,α c, τ + τ + γ /τ j C b,α,/τ. = { q G s b,βm : e b m,αq, p E b,α,γ,s,m c, τ τ < α }. 7 Note that the intersection τ<α C b,αc, τ can be understood as an intersection of finitely many sets since C b,α c, τ has only finitely many elements. Lemma 4. Let c and τ < α β, then we have νc b,α c, τ > c. Proof. We denote C b,α c, τ := G s b,βm \ C b,αc, τ. Then for all τ < α we have A m,βm,s,α,p /τ = e /τ b sβm b m,αq, p q G s b,βm > νc b,α c, τ 2c b m + + γ /τ j C b,α,/τ. Now using Proposition 4. we obtain νc b,α c, τ < c and the result follows.

20 20 Lemma 4.2 Let c, then we have Proof. Let τ < α be such that νc b,α c > c. E b,α,γ,s,m c, τ = inf E b,α,γ,s,mc, τ τ<α note that by Lemma 3. we have E b,α,γ,s,m c, τ whenever τ α and hence we can find τ with the demanded property. Then we have C b,α c, τ C b,α c, τ = C b,α c τ<α and hence the result follows from Lemma 4.. If we choose c = β in Lemma 4.2, then we obtain νc b,α β > β and consequently we have β β β ν C b,α β = ν C b,α β νc b,α β > 0. α=2 α=2 Hence we obtain the following theorem which establishes the existence of a q G s b,βm which achieves the optimal convergence rate for a range of α s. Theorem 4. Let β, m, s N, β 2 and let p Z b [x] with degp = βm. Then there exists a q G s b,βm such that τα e b m,αq, p 2τα β τα + + γ /τα j C b,α,/τα 8 b ταm for all 2 α β and for all τ α < α. The proof of Theorem 4. suggests that in principle we can find q which satisfies equation 8 for all 2 α β and all τ α < α by using a so-called sieve algorithm which will be explained in the following. Use a computer search to find β b βms + of the b βms vectors q in G s b,βm which satisfy α=2 e b m,2q, p E b,2,γ,s,m β, τ 2 τ 2 < 2,

21 2 and label this set T 2. By Lemma 4.2 we know that at least such a number of vectors exists. Then proceed by using a computer search to find 2β b βms + vectors q in T 2 which satisfy e b m,3q, p E b,3,γ,s,m β, τ 3 τ 3 < 3 and label this set T 3. Since 3 3 ν C b,α β = ν C b,α β α=2 α=2 3 νc b,α β > 2 β, we know that there are at least 2β b βms + values in T 2 to populate the set T 3. In the same way we proceed to construct the sets T 4,..., T β. Theorem 4. guarantees that T β is not empty and we may select q to be any vector from T β. This vector satisfies equation 8 for all 2 α β and all τ α < α. However, in practice such a search algorithm would not be applicable since it is much too time consuming. For this reason we show in the following how the sieve algorithm may be combined with the component-by-component CBC algorithm which makes the search algorithm applicable. Such an algorithm, we call it CBC sieve algorithm, is presented in Algorithm 2. For its statement we use the following notation: For 2 α β and p Z b [x] with degp = βm we define the following: for d = 0 and q G b,βm we set θ α 0, q := e b m,αq, p, and for d N, q d G d b,βm and q d+ G b,βm we set α=2 θ α q d, q d+ := e b m,αq d, q d+, p e b m,αq d, p. Furthermore, for short we use the notation M d,α,γ τ := b m d + 3βγ /τ j C b,α,/τ. Now we prove the following result.

22 22 Algorithm 2 CBC sieve algorithm for 2 α β Require: b a prime, s, m, β N, β 2, and p Z b [x] with degp = βm. : Set T,d := G b,βm for all d s and q 0 = 0. 2: for d = 0 to s do 3: for α = 2 to β do 4: perform a computer search to find α β b βm + elements q in T α,d+ to populate the set T α,d+, which is a subset of 5: if d = 0 then 6: { q T α,d+ : θ α 0, q } τα + 3γ /τα b ταm C b,α,/τα τα < α 7: else 8: { q T α,d+ : θ α q d, q 3βγ /τα d+ C τα } b,α,/τ α M d,α,γ τ α τα < α 9: end if 0: end for : Select q d+ T β,d+. 2: Set q d+ = q d, q d+. 3: end for 4: return q = q s. Theorem 4.2 Let s, m, β N, β 2, then Algorithm 2 constructs a vector q d Gd b,βm such that e b m,αq d, p b ταm d + 3βγ /τα C b,α,/τα τα for all τ α < α and for all 2 α β. To prove Theorem 4.2 we introduce the following set: for q d G d b,βm let F b,α c, q d be the set of all q d+ G b,βm such that θ α q d, q d+ j 3cγ /τα d+ C b,α,/τ α M d,α,γ τ α τα 9

23 23 for all τ α < α. Lemma 4.3 Let 2 α β and let c. q d G d b,βm such that Assume that there exists a for all τ α < α and for all 2 α β. Then e b m,αq d, p M d,α,γ τ α τα 0 ν F b,α c, q d > c. Proof. From the proof of Theorem 3. and using assumption 0 for all /α < λ we have b βm q d+ G b,βm θ α q d, q d+ λ γd+c λ b,α,λ b + e λαm b m,αq d, p λ + 3γ λ d+c b,α,λ M d,α,γ /λ. b m d + γj λ C b,α,λ From this the result follows in the same way as in the proof of Lemmas 4. and 4.2. Now we give the proof of Theorem 4.2. Proof. We proceed by induction on d and firstly show the result for d =. Having fixed d =, we proceed by induction on α. For q C b,α β we have see equation 7 e b m,αq, p + 3βγ/τα bταm C b,α,/τα τα τ α < α, for 2 α β. According to Lemma 4.2, νc b,α β > β, 2 α β, hence there are β b βm + elements to populate T 2,. Assume now that for 2 α < β, there are α β b βm + elements to populate T α,, hence νt α, > α β. We want to show that ν {q T α, : e b m,α+q, p M,α+,γ τ α+ τ α+ τ α+ < α + } > αβ,

24 24 which implies that there are αβ b βm + elements to populate T α+,. But {q T α, : e b m,α+q, p M,α+,γ τ α+ τ α+ τ α+ < α + } = T α, {q G b,βm : e b m,α+q, p M,α+,γ τ α+ τ α+ τ α+ < α + }, hence we get equation from the induction assumption and from Lemma 4.2. Thus we have proven the assertion for d =. We now assume that for d < s, the algorithm has found q d so that e b m,αq d, p M d,α,γ τ α τα 2 for all τ α < α and for all 2 α β and again we proceed by induction. According to Lemma 4.3, under the assumption 2, we have ν F b,α β, q d > β 2 α β, hence there are β b βm + elements to populate T 2,d+. We now assume that for 2 α < β, there are α β b βm + elements to populate T α,d+, hence νt α,d+ > α β. Since { q T α,d+ : θq d, q 3βγ /τ τα+ α+ d+ C b,α+,/τα+ M d+,α+,γ τ α+ } τ α+ < α + = T α,d+ F b,α+ β, q d we obtain from the inductive hypothesis and from Lemma 4.3 that { ν q T α,d+ : θq d, q 3βγ /τ α+ d+ C b,α+,/τα+ M d+,α+,γ τ α+ } τ α+ < α + > αβ, τα+ which implies that there are αβ b βm + elements to populate T α+,d+. Therefore Algorithm 2 finds a q d+ G b,βm such that θ α q d, q d+ 3βγ /τα d+ C τα b,α,/τ α M d,α,γ τ α

25 25 for all τ α < α and for all 2 α β. Using equation 2 we obtain, e b m,αq d, q d+, p = e b m,αq d, p + θ α q d, q d+ M d,α,γ τ α τα + 3βγ /τα d+ C b,α,/τ α τα M d+,α,γ τ α τα, for all τ α < α and for all 2 α β. 5 Optimal convergence rates for a range of smoothness parameters using Korobov polynomial lattice rules In this section we study a special case of polynomial lattice rules, namely Korobov polynomial lattice rules. We show the existence of higher order Korobov polynomial lattice rules which achieve optimal rates of convergence for a range of smoothness parameters and present an algorithm which shows how to construct such higher order Korobov polynomial lattice rules. This algorithm is the same as the sieve algorithm presented in Section 4, but due to the structure of Korobov polynomial lattice rules, the cost of such an algorithm is feasible. We now present the results which are used to establish the existence of a higher order Korobov polynomial lattice rule achieving optimal rates of convergence for a range of smoothness parameters and its construction. For the remainder of this section, we use φq := q, q 2,..., q s mod p, q G b,βm to denote the generating vector of the higher order Korobov polynomial lattice rule S p,m,βm φq and e b m,αφq, p to denote the corresponding worst-case error, 2 α β; we recall that α β and n = βm. As in Section 3 we point out that we use generating vectors φq := q, q 2,..., q s mod p instead of, q,..., q s see e.g. [5, Algorithm 4.6], as otherwise the projection onto the first coordinate does not achieve a convergence rate of b αm. We start with the following proposition, which is analogous to Proposition 4., where we set à m,n,s,α,p λ := e λ b n b,αφq, p. m q G b,n

26 26 Proposition 5. For α 2 and /α < λ we have [ ] Ã m,n,s,α,p λ s γ λ b minm,λn j C b,α,λ. Proof. Using Proposition 2. and Jensen s inequality we obtain, Ã m,n,s,α,p λ r b αγ, λ k n = q G b,βm k D pφq k N s 0 \{0} r λ αγ, k b n q G b,βm k φq a mod p dega<n m. 3 In the case where all components of k are multiples of p every q G b,βm satisfies the equation k φq 0 mod p and hence we have = b n k N s 0 \{0} k 0 mod p q G b,βm k φq a mod p dega<n m and the sum over all k which satisfy this condition is therefore bounded by rαγ, λ k + + γ λ b λn j C b,α,λ, see the proof of Proposition 4.. In the case where there is at least one component of k which is not a multiple of p we have b n q G b,βm k φq a mod p dega<n m sb m, because for any choice a there are at most s solutions q to k φq a mod p and there are b n m possible choices for a. Therefore this part of equation 3 is bounded by s b m + + γj λ C b,α,λ.

27 27 Altogether we now obtain that à m,n,s,α,p λ s b m + b λn s + b minm,λn γj λ C b,α,λ + γj λ C b,α,λ as required. Let ν denote the equiprobable measure on G b,βm. For c and τ < α β the following set is introduced: { } C b,α c, τ := q G b,βm : e b m,αφq, p Ẽb,α,γ,s,mc, τ, 4 where Ẽ b,α,γ,s,m c, τ := cτ s + τ b τm τ + + γ /τ j C b,α,/τ. Furthermore, let C b,α c := = τ<α C b,α c, τ 5 { } q G b,βm : e b m,αφq, p Ẽb,α,γ,s,mc, τ τ < α. Lemma 5. Let c and τ < α β. Then we have ν C b,α c, τ > c. Proof. The proof follows exactly along the lines of the proof of Lemma 4.. Lemma 5.2 Let c. Then we have ν C b,α c > c. Proof. The proof follows exactly along the lines of the proof of Lemma 4.2.

28 28 Algorithm 3 Korobov sieve algorithm Require: b a prime, s, m, β N, β 2, and p Z b [x] with degp = βm. : Set T := G b,βm. 2: for α = 2 to β do 3: perform a computer search to find α β b βm + elements q in T α to populate the set T α, which is a subset of { } q T α : e b m,αφq, p Ẽb,α,γ,s,mβ, τ α τ α < α 4: end for 5: Select q T β 6: return q. As in Section 4 we now introduce a sieve algorithm see Algorithm 3 which shows how to obtain a generating vector for a higher order Korobov polynomial lattice rule, which achieves optimal convergence rates for a range of smoothness parameters. The next theorem shows that Algorithm 3 does indeed produce such a vector. Theorem 5. Let s, m, β N, β 2. Then Algorithm 3 finds an element q G b,βm such that e b m,αφq, p s + τα τα β τα + + γ /τα j C b,α,/τα, b ταm for all τ α < α, 2 α β. Proof. We prove the result by induction on α. For α = 2, by Lemma 5.2, ν Cb,α β > β, 2 α β, so there are at least β b βm + vectors to populate the set T 2. We now assume that there are α β b βm + elements in the set T α, where 2 α < β, hence νt α > α β. We want to show that { } ν q T α : e b m,α+φq, p Ẽb,α+,γ,s,mβ, τ α+ τ α+ < α + > αβ, 6

29 29 which implies that there are αβ b βm + elements to populate the set T α+. Since { } q T α : e b m,α+φq, p Ẽb,α+,γ,s,mβ, τ α+ τ α+ < α + = T α C b,α+ β we obtain equation 6 from the induction assumption and from Lemma 5.2. References [] H.E. Chrestenson, A class of generalized Walsh functions, Pacific J. Math [2] J. Dick, Explicit constructions of quasi-monte Carlo rules for the numerical integration of high-dimensional periodic functions, SIAM J. Numer. Anal [3] J. Dick, Walsh spaces containing smooth functions and quasi-monte Carlo rules of arbitrary high order, SIAM J. Numer. Anal [4] J. Dick, The decay of the Walsh coefficients of smooth functions, Bull. Austral. Math. Soc [5] J. Dick, F. Kuo, F. Pillichshammer, I. Sloan, Construction algorithms for polynomial lattice rules for multivariate integration, Math. Comp [6] J. Dick, P. Kritzer, F. Pillichshammer, W. Ch. Schmid, On the existence of higher order polynomial lattices based on a generalized figure of merit, J. Complexity [7] J. Dick, F. Pillichshammer, Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces, J. Complexity [8] J. Dick, F. Pillichshammer, Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules, J. Complexity

30 30 [9] J. Dick, F. Pillichshammer, Digital Nets and Sequences, Cambridge University Press, Cambridge, 200 to appear. [0] J. Dick, F. Pillichshammer, B. Waterhouse, The construction of good extensible rank- lattices, Math. Comp [] N.M. Korobov, Properties and calculation of optimal coefficients, Dokl. Akad. Nauk SSSR In Russian. [2] H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math [3] H. Niederreiter, Low-discrepancy point sets obtained by digital constructions over finite fields, Czech. Math. J [4] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS NSF Series in Applied Mathematics 63, SIAM, Philadelphia, 992. [5] I. H. Sloan, S. Joe, Lattice methods for multiple integration, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 994. [6] I. Sloan, A. Reztsov, Component-by-component construction of good lattice rules, Math. Comp [7] I.H. Sloan, H. Woźniakowski, When are quasi-monte Carlo algorithms efficient for high dimensional integrals?, J. Complexity [8] J.L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math Author s Addresses: Jan Baldeaux and Josef Dick, School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia. janbaldeaux@gmail.com, josef.dick@unsw.edu.au Julia Greslehner and Friedrich Pillichshammer, Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria. julia.greslehner@gmx.at, friedrich.pillichshammer@jku.at