Explicit constructions of quasi-monte Carlo rules for the numerical integration of high dimensional periodic functions

Transcription

1 Explicit constructions of quasi-monte Carlo rules for the numerical integration of high dimensional periodic functions By J. Dick at Sydney Astract In this paper we give first explicit constructions of point sets in the s dimensional unit cue yielding quasi-monte Carlo algorithms which achieve the optimal rate of convergence of the worst-case error for numerically integrating high dimensional periodic functions. In the classical measure P α of the worst-case error introduced y Koroov in 1959 the convergence is of O(N min(α,d) (log N) sα ) for every even integer α 1, where d is a parameter of the construction which can e chosen aritrarily large and N is the numer of quadrature points. This convergence rate is known to e est possile up to some log N factors. We prove the result for the deterministic and also a randomized setting. The construction is ased on a suitale extension of digital (t, m, s)-nets over the finite field Z. Keywords: Numerical integration, quasi-monte Carlo method, digital net, digital sequence, lattice rule. MSC 1991: primary: 11K38, 11K45, 65C5; secondary: 65D3, 65D3; 1 Introduction In 1959 Koroov [13] and independently in 196 Hlawka [11] introduced a quadrature formula suited for numerically integrating high dimensional periodic functions. More precisely, we want to approximate the high dimensional integral [,1] s f(x) dx (where f is assumed to e periodic with period 1 in each coordinate) y a quasi-monte Carlo rule, i.e., an equal weight quadrature rule Q N,s (f) N 1 N 1 n f(x n), where x,..., x N 1 [, 1] s are the quadrature points. Specifically, Koroov and Hlawka suggested the quadrature rule Q N,g,s (f) N 1 N 1 n f({ng/n}), where for a vector of real numers x (x 1,..., x s ) we define {x} as the fractional part of each component of x, i.e., {x j } x j x j x j (mod 1) and where g Z s is an integer vector. The quadrature rule Q N,g,s is called lattice rule and g is called the generating vector (of the lattice rule). The monographs [1, 14, 18, 6] deal partly or entirely with the approximation of such integrals. (Note that the assumption that the integrand f is periodic is not really a restriction since there are transformations which transform non-periodic functions into periodic ones such that the smoothness of the integrand is preserved, see for example [6].) The author is supported y the Australian Research Council under its Center of Excellence Program. 1

2 Dick, Explicit constructions of quasi-monte Carlo rules To analyze the properties of a quadrature rule one considers then the worst-case error sup f BH f(x) dx Q [,1] s N,s (f), where B H denotes some class of functions. In the classical theory the class ε s α of periodic functions has een considered where one demands that the asolute value of the Fourier coefficients of the function decay sufficiently fast (see [1, 14, 6, 18]. This leads us to the classical measure of the quality of lattice rules P α sup f ε s α f(x) dx Q [,1] s N,s (f), which then for a lattice rule with generating vector g (g 1,..., g s ) can also e written as P α P α (g, N) h α, h Z s \{} h g ( mod N) where h (h 1,..., h s ), h g h 1 g h s g s and h s j1 max(1, h j ). (Later on in this paper we prefer to use the more contemporary notation of reproducing kernel Hilert spaces, in our case so-called Koroov spaces, (see Section.3), ut as is well understood (and as is also shown in Section.3) the results also apply to the classical prolem.) By averaging over all generating vectors g several existence results for good lattice rules which achieve P α O(N α (log N) αs ) have een shown, see [1, 13, 14, 19, 18, 6]. By a lower ound of Sharygin [5] this convergence is also known to e essentially est possile, as he showed that the worst-case error is at least of order N α (log N) s 1. But, except for dimension s, no explicit generating vectors g which yield a small worstcase error are known. For s 3 one relies on computer search to find good generating vectors g and many such search algorithms have een introduced and analysed, especially recently, see [13, 7, 8, 33]. On the other hand one can of course also use some other quadrature rule Q N,s (f) N 1 n ω nf(x n ) to numerically integrate functions in the class ε s α. In this case the worstcase error in the class ε s α for a quadrature rule with weights ω,..., ω N 1 and points {x,..., x N 1 } [, 1) s is given y (1) P α ({x,..., x N 1 }) N 1 n,m ω n ω m h Z s \{} e πih (xn xm) h α. Such a construction of quadrature rules is due to Smolyak [3] and is nowadays called sparse grid, see [8]. Those quadrature rules are sums over certain products of differences of one-dimensional quadrature rules. In principle any one-dimensional quadrature rule can e chosen as asis, leading to different quadrature rules. In many cases the weights ω n of such quadrature rules are not known explicitly ut can e precomputed. But even if the underlying one-dimensional quadrature rule has only positive weights, it is possile that some weights in Smolyak s quadrature rules are negative, which can have a negative impact on the staility of the quadrature formula. In general, quadrature formulae for which all weights are equal and N 1 n ω n 1, that is, ω n N 1 for all n,..., N 1, are to e preferred. As mentioned aove, such quadrature rules are called quasi-monte Carlo rules, to which we now switch for the remainder of the paper. As the weights for quasi-monte Carlo rules are given y N 1 the focus lies on the choice of the quadrature points. Constructions of quadrature points have een introduced with the aim to distriute the points as evenly as possile over the unit cue. An explicit construction of well distriuted point sets in the unit cue has een introduced

3 Dick, Explicit constructions of quasi-monte Carlo rules 3 y Sool [31] in A similar construction was estalished y Faure [7] in 198 efore Niederreiter [17] (see also [18]) introduced the general concept of (t, m, s)-nets and (t, s)- sequences and the construction scheme of digital (t, m, s)-nets and digital (t, s)-sequences. For such point sets it has een shown that the star discrepancy (which is a measure of the distriution properties of a point set) is O(N 1 (log N) s 1 ), see [18]. From this result it follows that those point sets yield quasi-monte Carlo algorithms which achieve a convergence of O(N (log N) s ) for functions in the class ε s α for all α. This result holds in the deterministic and randomized setting. For smoother functions though, i.e., larger values of α in the class ε s α, one can expect higher order convergence. For example, if the partial derivatives up to order two are square integrale then one would expect an integration error of O(N 4 (log N) c(s) ) for some c(s) > depending only on s in the function class ε s α, and in general, if the mixed partial derivatives up to order α/ exist and are square integrale then one would expect an integration error in ε s α of O(N α (log N) c(s,α) ) for some c(s, α) > depending only on s and α. But until now (t, m, s)-nets and (t, s)-sequences have only een shown to yield a convergence of at est O(N (log N) s ) (or O(N 3+δ ) for any δ > if one uses a randomization method called scramling, see [3]) in ε s α, even if the integrands satisfy stronger smoothness assumptions. In this paper we show that a modification of digital (t, m, s)-nets and digital (t, s)- sequences introduced y Niederreiter [17, 18] yields point sets which achieve the optimal rate of convergence of the worst-case error P α O(N min(α,d) (log N) sα ) for any integer α 1 and where d N is a parameter of the construction which can e chosen aritrarily large. We too use the digital construction scheme introduced y Niederreiter [17, 18] for the construction of (t, m, s)-nets and (t, s)-sequences, ut our analysis of the worst-case error shows that the t-value does not provide enough information aout the point set. Hence we generalize the definition of digital (t, m, s)-nets and digital (t, s)- sequences to suit our needs. This leads us to the definition of digital (t, α, β, m, s)-nets and digital (t, α, β, s)-sequences. For α β 1 those definitions reduce to the case introduced y Niederreiter, ut are different for α > 1. Susequently we prove that quasi- Monte Carlo rules ased on digital (t, α, β, m, s)-nets and digital (t, α, β, s)-sequences achieve the optimal rate of convergence. Further we give explicit constructions of digital (t, α, min(α, d), m, s)-nets and digital (t, α, min(α, d), s)-sequences, where d N is a parameter of the construction which can e chosen aritrarily large. Digital (t,, m, s)-nets and digital (t,, s)-sequences over Z can also e used for nonperiodic function spaces where one uses randomly shifted and then folded point sets using the aker s transformation (see [3]). Our analysis and error ounds for α here also apply for the case considered in [3] (with different constants though), hence yielding useful constructions also for non-periodic function spaces where one uses the aker s transformation. Using a digital (t, α, m, s)-net with a scramling algorithm (see [3]) on the other hand does not improve the performance in non-periodic spaces compared to (t, m, s)-nets. Further, the constructions here can also e used for algorithms to solve (periodized) integral equations, see for example [1], Chapter 1 for more ideas. In the following we summarize some properties of the quadrature rules: The quadrature rules introduced in this paper are equal weight quadrature rules which achieve the optimal rate of convergence up to some log N factors and we

4 Dick, Explicit constructions of quasi-monte Carlo rules 4 show the result for deterministic and randomly digitally shifted quadrature rules. The upper ound for the randomized quadrature rules even improves upon the est known upper ound (more precisely, the power of the log N factor) for lattice rules for the worst-case error in ε s α for all dimensions s and even integers α (compare Corollary 3 to Theorem in [19]). The construction of the underlying point set is explicit. They automatically adjust themselves to the optimal rate of convergence in the class ε s α as long as α is an integer such that α d, where d is a parameter of the construction which can e chosen aritrarily large. The underlying point set is extensile in the dimension as well as in the numer of points, i.e., one can always add some coordinates or points to an existing point set such that the quality of the point set is preserved. Tractaility and strong tractaility results (see [9]) can e otained for weighted Koroov spaces. The outline of the paper is as follows. In the next section we introduce the necessary tools, namely Walsh functions, the digital construction scheme upon which the construction of the point set is ased on and Koroov spaces. Further we also introduce the worst-case error in those Koroov spaces and we give a representation of this worst-case error for digital nets in terms of the Walsh coefficients of the reproducing kernel. In Section 3 we give the definition of digital (t, α, β, m, s)-nets and digital (t, α, β, s)-sequences. Further we prove some propagation rules for those digital nets and sequences. In Section 4 we give explicit constructions of digital (t, α, β, m, s)-nets and digital (t, α, β, s)-sequences and we prove some upper ounds on the t-value. We then show, Section 5, that quasi- Monte Carlo rules ased on those digital nets and sequences achieve the optimal rate of convergence of the worst-case error in the Koroov spaces. The results are ased on entirely deterministic point sets. Section 6 finally deals with randomly digitally shifted digital (t, α, β, m, s)-nets and (t, α, β, s)-sequences and we show similar results for the mean square worst-case error in the Koroov space for this setting. The Appendix is devoted to the analysis of the Walsh coefficients of the Walsh series representation of B α ( x y ), where B α is the Bernoulli polynomial of degree α. Preliminaries In this section we introduce the necessary tools for the analysis of the worst-case error and the construction of the point sets. In the following let N denote the set of natural numers and let N denote the set of non-negative integers..1 Walsh functions In the following we define Walsh functions in ase, which are the main tool of analyzing the worst-case error. First we give the definition for the one-dimensional case.

5 Dick, Explicit constructions of quasi-monte Carlo rules 5 Definition 1 Let e an integer and represent k N in ase, k κ a 1 a κ with κ i {,..., 1}. Further let ω e πi/. Then the k-th Walsh function wal k : [, 1) {1, ω,..., ω 1 } in ase is given y wal k (x) ω x 1κ + +x aκ a 1, for x [, 1) with ase representation x x x + (unique in the sense that infinitely many of the x i are different from 1). Definition For dimension s, x (x 1,..., x s ) [, 1) s and k (k 1,..., k s ) N s we define wal k : [, 1) s {1, ω,..., ω 1 } y s wal k (x) wal kj (x j ). j1 As we will always use Walsh functions in ase we will in the following often write wal instead of wal. We introduce some notation. By we denote the digit-wise addition modulo, i.e., for x iw x i i and y iw y i i we define x y z i i, iw where z i {,..., 1} is given y z i x i + y i (mod ) and let denote the digit-wise sutraction modulo. In the same manner we also define a digit-wise addition and digitwise sutraction for non-negative integers ased on the -adic expansion. For vectors in [, 1) s or N s the operations and are carried out component-wise. Throughout the paper we always use ase for the operations and. Further we call x [, 1) a -adic rational if it can e written in a finite ase expansion. In the following proposition we summarize some asic properties of Walsh functions. Proposition 1 1. For all k, l N and all x, y [, 1), with the restriction that if x, y are not -adic rationals then x y is not allowed to e a -adic rational, we have wal k (x) wal l (x) wal k l (x), wal k (x) wal k (y) wal k (x y).. We have 1 wal (x) dx 1 and 1 wal k (x) if k >. 3. For all k, l N s we have the following orthogonality properties: { 1, if k l, wal k (x)wal l (x) dx [,1), otherwise. s 4. For any f L ([, 1) s ) and any σ [, 1) s we have f(x σ) dx f(x) dx. [,1) s [,1) s 5. For any integer s 1 the system {wal k : k (k 1,..., k s ), k 1,..., k s } is a complete orthonormal system in L ([, 1) s ). The proofs of are straightforward and for a proof of the remaining items see [] or [3] for more information.

6 Dick, Explicit constructions of quasi-monte Carlo rules 6. The digital construction scheme The construction of the point set used here is ased on the digital construction scheme introduced y Niederreiter, see [18]. Definition 3 Let integers m, s 1 and e given. Let R e a commutative ring with identity such that R and let Z {,..., 1}. Let C 1,..., C s R m m with C j (c j,k,l ) 1 k,l m. Further, let ψ l : Z R for l,..., m 1 and µ j,k : R Z for j 1,..., s and k 1,..., m e ijections. For n,..., m 1 let n m 1 l a l(n) l, with all a l (n) Z, e the ase digit expansion of n. Let n (ψ (a (n)),..., ψ m 1 (a m 1 (n))) T and let y j (y j,1,..., y j,m ) T C j n for j 1,..., s. Then we define x j,n µ j,1 (y j,1 ) µ j,m (y j,m ) m for j 1,..., s and n,..., m 1 and the n-th point x n is then given y x n (x 1,1,..., x s,n ). The point set {x,..., x m 1} is called a digital net (over R ) (with generating matrices C 1,..., C s ). For m we otain a sequence {x, x 1,...}, which is called a digital sequence (over R ) (with generating matrices C 1,..., C s ). Niederreiter s concept of a digital (t, m, s)-net and a digital (t, s)-sequence will appear as a special case in Section 3. Apart from Section 3 and Section 4, where we state the results using Definition 3 in the general form, we use only a special case of Definiton 3, where we assume that is a prime numer, we choose R the finite field Z and the ijections ψ l and µ j,k from Z to Z are all chosen to e the identity map. We remark that throughout the paper when Walsh functions wal, digit-wise addition, digit-wise sutraction or digital nets are used in conjunction with each other we always use the same ase for each of those operations..3 Koroov space Historically the function class ε s α has een used. In this paper we use a more contemporary notation y replacing the function class ε s α with a reproducing kernel Hilert space H α called Koroov space. The worst-case error expression (1) will almost e the same for oth function classes and hence the results apply for oth cases. A reproducing kernel Hilert space H over [, 1) s is a Hilert space with inner product, which allows a function K : [, 1) s R such that K(, y) H, K(x, y) K(y, x) and f, K(, y) f(y) for all x, y [, 1) s and all f H. For more information on reproducing kernel Hilert spaces see [1], for more information on reproducing kernel Hilert spaces in the context of numerical integration see for example [5, 9]. The Koroov space H α is a reproducing kernel Hilert space of periodic functions. Its reproducing kernel is given y K α (x, y) h Z s e πih (x y) h α, where α > 1/ and h s j1 max(1, h j ). The inner product in the space H α is given y () f, g α h Z s h α ˆf(h)ĝ(h),

7 Dick, Explicit constructions of quasi-monte Carlo rules 7 where ˆf(h) f(x)e πih x dx [,1) s are the Fourier coefficients of f. The norm is given y f α f, f 1/ α. Note that for α a natural numer and any x (, 1) we have B α (x) ( 1)α+1 (α)! (π) α h e πihx h α, where B α is the Bernoulli polynomial of degree α. Hence, for α a natural numer we can write ( s K α (x, y) 1 + ) e πih(x j y j ) s ) (1 ( 1) α (π)α h α (α)! B α( x j y j ). j1 h j1 Let now (3) K α (x, y) 1 + h e πih(x y) h α 1 ( 1) α (π)α (α)! B α( x y ). Then we have K α (x, y) s K α (x j, y j ), j1 where x (x 1,..., x s ) and y (y 1,..., y s ). Hence the Koroov space is a tensor product of one-dimensional reproducing kernel Hilert spaces. Though α > 1/ can in general e any real numer we restrict ourselves to integers α 1 for most of this paper. The ounds on the integration error for H α with α 1 a real numer still apply when one replaces α with α, as in this case the unit all of H α given y {f H α : f α 1} is contained in the unit all {f H α : f α 1} of H α as f α f α. Hence it follows that integration in the space H α is easier than integration in the space H α. In general, the worst-case error e(p, H) for multivariate integration in a normed space H over [, 1] s with norm using a point set P is given y e(p, H) sup f(x) dx Q P (f), f H, f 1 [,1] s where Q P (f) N 1 x P f(x) and N P is the numer of points in P. If H is a reproducing kernel Hilert space with reproducing kernel K we will write e(p, K) instead of e(p, H). It is known that (see for example [9]) (4) e (P, K) K(x, y) dx dy [,1) N s N 1 n [,1) s K(x n, y) dy + 1 N where P {x,..., x N 1 }. Hence for the Koroov space H α we otain (5) e (P, K α ) N N 1 n,h K α (x n, x h ). N 1 n,l K(x n, x l ),

8 Dick, Explicit constructions of quasi-monte Carlo rules 8 Therefore it follows that e (P, K α ) P α and hence our results aslo apply to the classical setting introduced y Koroov [13]. It follows from Proposition 1 that K α can e represented y a Walsh series, i.e., let (6) K α (x, y) r,α (k, l)wal k (x)wal l (y), where k,l N s r,α (k, l) K α (x, y)wal k (x)wal l (y) dx dy. [,1) s As the kernel K α is a product of one-dimensional kernels it follows that r,α (k, l) s j1 r,α(k j, l j ), where k (k 1,..., k s ) and l (l 1,..., l s ) and r,α (k, l) 1 1 K α (x, y)wal k (x)wal l (y) dx dy. For a digital net with generating matrices C 1,..., C s let D D(C 1,..., C s ) e the dual net given y D {k N s \ {} : C T 1 k C T s k s }, where for k (k 1,..., k s ) with k j κ j, + κ j,1 + we set k j (κ j,,..., κ j,m 1 ) T. Further, for u {1,..., s} let D u D((C j ) j u ). We have the following theorem. Theorem 1 Let C 1,..., C s Z m m e the generating matrices of a digital net P m and let D denote the dual net. Then for any α > 1/ the square worst-case error in H α is given y e (P m, K α ) r,α (k, l). Proof. From (5) and (6) it follows that In [5] it was shown that Hence we have e (P m, K α ) m x P m k,l N s wal k (x) e (P m, K α ) 1 + k,l D r,α (k, l) 1 m x,y P m { 1 if k D {}, otherwise. k,l D {} r,α (k, l). wal k (x)wal l (y). In the following we will show that r,α (, ) 1 and r,α (, k) r,α (k, ) if k from which the result then follows. Note that it is enough to show those identities for the

9 Dick, Explicit constructions of quasi-monte Carlo rules 9 one dimensional case. We have wal (x) 1 for all x [, 1) and hence r,α (, k) (1 + h Z\{} 1 wal k (y) dy + h α e πih(x y) )wal k (y) dx dy h Z\{} 1 wal k (y) dy. 1 h α e πihx dxe πihy wal k (y) dy It now follows from Proposition 1 that r,α (, ) 1 and r,α (, k) for k >. The result for r,α (k, ) can e otained in the same manner. Hence the result follows. In the following lemma we otain a formula for the Walsh coefficients r,α. Lemma 1 Let e an integer and let α > 1/ e a real numer. coefficients r,α (k, l) for k, l N are given y The Walsh where β h,k 1 e πihx wal k (x) dx. Proof. We have r,α (k, l) 1 1 h Z\{} r,α (k, l) h Z\{} h Z\{} β h,k β h,l h α, h α e πih(x y) wal k (x)wal l (y) dx dy 1 h α e πihx wal k (x) dx 1 e πihy wal l (y) dy. The result follows. It is difficult to calculate the exact value of r,α (k, l) in general, ut for our purposes it is enough to otain an upper ound. Note that r,α (k, k) is a non-negative real numer. Lemma Let e an integer and let α > 1/ e a real numer. coefficients r,α (k, l) for k, l N are ounded y The Walsh r,α (k, l) r,α (k, k)r,α (l, l). Proof. Using Lemma 1 we otain r,α (k, l) The result follows. h Z\{} β h,k β h,l h α h Z\{} β h,k h α h Z\{} β h,l h α r,α(k, k)r,α (l, l).

10 Dick, Explicit constructions of quasi-monte Carlo rules 1 In the following we will write r,α (k) instead of r,α (k, k) and also r,α (k) instead of r,α (k, k). Lemma 3 Let C 1,..., C s Z m m e the generating matrices of a digital net P m and let D denote the dual net. Then for any natural numer α the worst-case error in H α is ounded y e(p m, K α ) r,α (k). k D Proof. From Theorem 1 and Lemma it follows that e (P m, K α ) k,l D r,α (k, l) ( k D ) r,α (k, k) and hence the result follows. For α 1 a natural numer we can write the reproducing kernel in terms of Bernoulli polynomials of degree α. Then for k 1 we have α+1 (π)α r,α (k) ( 1) (α)! 1 1 B α ( x y )wal k (x)wal k (y) dx dy. Note that the Bernoulli polynomials of even degree α are of the form B α (x) c α x α + c α 1 x (α 1) + + c + cx α 1, for some rational numers c α,..., c, c with c α, c. Let (7) I j (k) 1 1 x y j wal k (x)wal k (y) dx dy. As mentioned aove, r,α (k) is a real numer such that r,α (k) for all k 1 and α > 1/, hence it follows that for any natural numer α we have r,α (k) (π)α (α)! ( cα I α (k) + c α 1 I (α 1) (k) + + c I (k) + ci α 1 ). Using Lemma 9 and Lemma 1 from the Appendix we otain the following lemma. Lemma 4 Let, α N with. For k N with k κ 1 a κ ν aν 1 where ν 1, κ 1,..., κ ν {1,..., 1} and 1 a ν < < a 1 let q,α (k) a 1 a min(ν,α). Then for any natural numer α and any natural numer there exists a constant C,α > which depends only on and α such that r,α (k) C,α q,α(k) for all k 1. Let now q,α () 1. For k (k 1,..., k s ) N s we define q,α (k) s j1 q,α(k j ). We have the following lemma.

11 Dick, Explicit constructions of quasi-monte Carlo rules 11 Lemma 5 Let m 1, and α e natural numers and let D m,u {1,..., m 1} u. Then we have r,α (k) k D u {1,...,s} D u (1 + αm C,α (α + )) s u C u,α (1 + α + ) u Q,m,u,α(C 1,..., C s ) +(1 + αm C,α (α + )) s 1, where C,α is the constant from Lemma 4 and where Q,m,u,α(C 1,..., C s ) q,α (k). k D m,u Proof. Every k N s can e uniquely written in the form k h + m l with h {,..., m 1} s and l N s. Thus we have r,α (k) r,α ( m l) + r,α (h + m l). k D For the first sum we have l N s \{} l N s \{} h D m l N s r,α ( m l) 1 + r,α ( m l) 1 + l N s ( By using Lemma 15 from the Appendix and Lemma 4 we otain that l r,α ( m l)) s. r,α ( m l) 1 + r αm,α (l) 1 + αm C,α q,α (l). l l1 We need to show that l1 q,α(l) α +. Let l l 1 c l ν cν 1 for some ν 1 with 1 c 1 < < c ν and l 1,..., l ν {1,..., 1}. First we consider the sum over all those l for which 1 ν α. This part of the sum is ounded y l1 α ( 1) ν ν1 If ν > α we have c 1 1 c 1 ν c ν 1 c ν 1 1 c ν1 c 1... c ν α ( 1) ν ( c ) ν α. ν1 c1 ( 1) α c 1 1 c 1 α+1 c α 1 ( 1) α c α 1 1 c α c 1 1 c 1 α+1 c α 3 ( 1) α ( c ) α 1. c1 c 1 c α cα 1 c α c α 1 3 c c c1 (c α 1 ) c 1 c α 1

12 Dick, Explicit constructions of quasi-monte Carlo rules 1 Thus we otain l1 q,α(l) α +. Further we have h D m l N s r,α (h + m l) s h D m j1 l r,α (h j + m l), where h (h 1,..., h s ). By using Lemma 15 from the Appendix and Lemma 4 we otain r,α ( m l) 1 + αm C,α q,α (l) 1 + αm C,α (α + ). l Let now < h j < m. From Lemma 4 we otain r,α (h j + m l) C,α q,α (h j + m l) C,α q,α (h j )q,α (l). From aove we have l q,α(l) 1 + α + and hence l1 r,α (h j + m l) q,α (h j )C,α q,α (l) C,α (1 + α + )q,α (h j ). l Thus we otain r,α (h + m l) h D m l N s u {1,...,s} h u D m j u,u l u {1,...,s} l r,α (h j + m l) j u l r,α ( m l) (1 + αm C,α (α + )) s u C u,α (1 + α + ) u h u D m j u,u q,α (h j ), where h u (h j ) j u. The result follows. In [5] it was shown that the square worst-case error for numerical integration in the Koroov space can at est e of O(N α (log N) s 1 ), where N is the numer of quadrature points. Hence Lemma 5 shows that it is enough to consider only Q,m,u,α (C 1,..., C s ) in order to investigate the convergence rate of digitally shifted digital nets. 3 (t, α, β, m, s)-nets and (t, α, β, s)-sequences The t value of a (t, m, s)-net is a quality parameter for the distriution properties of the net. A low t value yields well distriuted point sets and it has een shown, see for example [6, 18], that a small t value also guarantees a small worst-case error for integration in Soolev spaces for which the partial first derivatives are square integrale. In the following we will show how the definition of the t value needs to e modified in order to otain faster convergence rates for periodic Soolev spaces for which the partial derivatives up to order α are square integrale. It is the aim of this definition to translate the prolem of minimizing the worst-case error into an algeraical prolem

13 Dick, Explicit constructions of quasi-monte Carlo rules 13 concerning the generating matrices. (This definition can therefore also e used in an computer search algorithm, where one could for example search for the polynomial lattice with the smallest t(α) value which in turn yields a small worst-case error for integration of periodic functions.) For natural numers α 1, Lemma 4 suggests to define the following metric µ,α (k, l) µ,α (k l) on N s which is an extension of the metric introduced in [16], see also [4] (for α 1 we asically otain the metric in [16, 4]). Here µ,α () and for k N with k κ ν aν κ 1 a1 1 where 1 a ν < < a 1 and κ i {1,..., 1} let µ,α (k) a 1 + +a min(α,ν). For a k N s with k (k 1,..., k s ) let µ,α (k) µ,α (k 1 )+ +µ,α (k s ). Then we have q,α (k) µ,α(k). Hence in order to otain a small worst-case error in the Koroov space H α, we need digital nets for which min{µ,α (k) : k D} is large. We can translate this property into a linear independence property of the row vectors of the generating matrices C 1,..., C s. We have the following definition. Definition 4 Let m, α 1 e natural numers, let < β α e a real numer and let t βm e a natural numer. Let R e a ring with elements and let C 1,..., C s R m m with C j (c j,1,..., c j,m ) T. If for all 1 i j,νj < < i j,1 m, where ν j m for all j 1,..., s, with the vectors i 1,1 + + i 1,min(ν1,α) + + i s,1 + + i s,min(νs,α) βm t c 1,i1,ν1,..., c 1,i1,1,..., c s,is,νs,..., c s,is,1 are linearly independent over R then the digital net with generating matrices C 1,..., C s is called a digital (t, α, β, m, s)-net over R. Further we call a digital (t, α, α, m, s)-net over R a digital (t, α, m, s)-net over R. If t is the smallest non-negative integer such that the digital net generated y C 1,..., C s is a digital (t, α, β, m, s)-net, then we call the digital net a strict digital (t, α, β, m, s)-net or a strict digital (t, α, m, s)-net if α β. Remark 1 Using duality theory (see []) it follows that for a digital (t, α, β, m, s)-net we have min k D µ,α (k) > βm t and for a strict digital (t, α, β, m, s)-net we have min k D µ,α (k) βm t + 1. Hence digital (t, α, β, m, s)-nets with high quality have a large value of βm t. Definition 5 Let α 1 and t e integers and let < β α e a real numer. Let R e a ring with elements and let C 1,..., C s R with C j (c j,1, c j,,...) T. Further let C j,m denote the left upper m m sumatrix of C j. If for all m > t/β the matrices C 1,m,..., C s,m generate a digital (t, α, β, m, s)-net then the digital sequence with generating matrices C 1,..., C s is called a digital (t, α, β, s)-sequence over R. Further we call a digital (t, α, α, s)-sequence over R a digital (t, α, s)-sequence over R. If t is the smallest non-negative integer such that the digital sequence generated y C 1,..., C s is a digital (t, α, β, s)-sequence, then we call the digital sequence a strict digital (t, α, β, s)-sequence or a strict digital (t, α, s)-sequence if α β. Remark Note that the definition of a digital (t, 1, m, s)-net coincides with the definition of a digital (t, m, s)-net and the definition of a digital (t, 1, s)-sequence coincides with the definition of a digital (t, s)-sequence as defined y Niederreiter [18]. Further note that the t-value depends on α and β, i.e., t t(α, β) or t t(α) if α β.

14 Dick, Explicit constructions of quasi-monte Carlo rules 14 In the following theorem we estalish some propagation rules. Theorem Let P e a digital (t, α, β, m, s)-net over a ring R and let S e a digital (t, α, β, s)-sequence over a ring R. Then we have: (i) P is a digital (t, α, β, m, s)-net for all 1 β β and all t t β m and S is a digital (t, α, β, s)-sequence for all 1 β β and all t t. (ii) P is a digital (t, α, β, m, s)-net for all 1 α m where β β min(α, α )/α and t t min(α, α )/α and S is a digital (t, α, β, s)-sequence for all 1 α m where β β min(α, α )/α and t t min(α, α )/α. (iii) Any digital (t, α, m, s)-net is a digital ( tα /α, α, m, s)-net for all 1 α α and every digital (t, α, s)-sequence is a digital ( tα /α, α, s)-sequence for all 1 α α. Proof. Note that it follows from Definition 5 that we need to prove the result only for digital nets. The first part follows trivially. To prove the second part choose an α such that 1 α α. Then choose aritrary 1 i j,νj < < i j,1 m with ν j m such that i 1,1 + + i 1,min(ν1,α ) + + i s,1 + + i s,min(νs,α ) mβ min(α, α ) α We need to show that the vectors t min(α, α ). α c 1,i1,ν1,..., c 1,i1,1,..., c s,is,νs,..., c s,is,1 are linearly independent over R. This is certainly the case as long as i 1,1 + + i 1,min(ν1,α) + + i s,1 + + i s,min(νs,α) βm t. Indeed we have i 1,1 + + i 1,min(ν1,α) + + i s,1 + + i s,min(νs,α) α min(α, α ) (i 1,1 + + i 1,min(ν1,α ) + + i s,1 + + i s,min(νs,α )) α mβ t min(α, α ) min(α, α ) α mβ t, and hence the second part follows. The third part is just a special case of the second part. Remark 3 Note y choosing α 1 in part (iii) of Theorem it follows that digital (t, α, m, s)-nets and digital (t, α, s)-sequences are also well distriuted point sets if the value of t is small, see [18].

15 Dick, Explicit constructions of quasi-monte Carlo rules 15 4 Constructions of digital (t, α, β, m, s)-nets and digital (t, α, β, s)-sequences In this section we show how digital (t, α, β, m, s)-nets and digital (t, α, β, s)-sequences can e constructed. Let d 1 and let C 1,..., C sd e the generating matrices of a digital (t, m, sd)-net. Note that many explicit examples of such generating matrices are known, see for example [7, 18, 1, 31] and the references therein. For the construction of a (t, α, β, m, s)-net any of the aove mentioned explicit constructions can e used, ut as will e shown elow the quality of the (t, α, β, m, s)-net otained depends on the quality of the underlying digital (t, m, sd)-net on which our construction is ased on. Let C j (c j,1,..., c j,m ) T for j 1,..., sd, i.e., c j,l are the row vectors of C j. Now let the matrix C (d) j e made of the first rows of the matrices C (j 1)d+1,..., C jd, then the second rows of C (j 1)d+1,..., C jd and so on till C (d) j is an m m matrix, i.e., C (d) j (c (d) j,1,..., c(d) j,m )T where c (d) j,l c u,v with l (v j)d + u, 1 v m and (j 1)d < u jd for l 1,..., m and j 1,..., s. In the following we will show that the matrices C (d) 1,..., C s (d) generating matrices of a digital (t, α, min(α, d), m, s)-net. are the Theorem 3 Let d 1 e a natural numer and let C 1,..., C sd e the generating matrices of a digital (t, m, sd)-net over some ring R with elements. Let C (d) 1,..., C s (d) e defined as aove. Then for any α 1 the matrices C (d) 1,..., C s (d) are generating matrices of a digital (t, α, min(α, d), m, s)-net over R with s(d 1) min(α, d) t min(α, d) t +. Proof. Let C (d) j (c (d) j,1,..., c(d) j,m )T for j 1,..., s. Further let i 1,1,..., i 1,ν1,..., i s,1,..., i s,νs e such that 1 i j,νj < < i j,1 m and i 1,1 + + i 1,min(ν1,α) + + i s,1 + + i s,min(νs,α) min(α, d)m t. We need to show that the vectors c (d) 1,i 1,1,..., c (d) 1,i 1,ν1,..., c (d) s,i s,1,..., c (d) s,i s,νs are linearly independent over R. For j 1,..., s let U j {c (d) j,i j,νj,..., c (d) j,i j,1 }. The vectors in the set U j stem from the matrices C (j 1)d+1,..., C jd. For j 1,..., s and d j (j 1)d+1,..., jd let e dj denote the largest index such that (e dj j)d+d j {i j,νj,..., i j,1 } and if for some d j there is no such e dj we set e dj (asically this means e dj is the largest integer such that c dj,e dj U j ). Let d α, then we have d((e (j 1)d+1 1) (e jd 1) + ) + L j l1 l i j,1 + + i j,min(νj,d) where (x) + max(x, ) and L j {(j 1)d + 1 d j jd : e dj > }. Hence we have d((e (j 1)d+1 1) (e jd 1) + ) + l L j l1 d(e (j 1)d e jd ) L j d + L j (L j + 1)/ d(d 1) d(e (j 1)d e jd ).

16 Dick, Explicit constructions of quasi-monte Carlo rules 16 Thus it follows that d(e e sd ) s d(d 1) (i j,1 + + i j,min(νj,α)) + s j1 d(d 1) dm t + s and therefore e e sd m t d + sd 1 m t. Thus it follows from the (t, m, sd)-net property of the digital net generated y C 1,..., C sd that the vectors c (d) 1,i 1,1,..., c (d) 1,i 1,ν1,..., c (d) s,i s,1,..., c (d) s,i s,νs are linearly independent. Let now d > α. Then we have d((e (j 1)d+1 1) (e jd 1) + )) + L j l1 l i j,1 + +i j,min(νj,α)+(d α)i j,min(νj,α), where again L j {(j 1)d+1 d j jd : e dj > }. Hence we have d((e (j 1)d+1 1) (e jd 1) + ) + l L j l1 d(e (j 1)d e jd ) L j d + L j (L j + 1)/ d(d 1) d(e (j 1)d e jd ). Note that i 1,min(ν1,α) + + i s,min(νs,α) m t/α and hence we have s (i j,1 + + i j,min(νj,α) + (d α)i j,min(νj,α)) αm t + (d α)(m t/α) dm dt/α. j1 Thus it follows that d(e e sd ) s d(d 1) (i j,1 + + i j,min(νj,α) + (d α)i j,min(νj,α)) + s j1 dm dt α + sd(d 1) and therefore e e sd m t α + sd 1 m t. Thus it follows from the (t, m, sd)-net property of the digital net generated y C 1,..., C sd that the vectors c (d) 1,i 1,1,..., c (d) 1,i 1,ν1,..., c (d) s,i s,1,..., c (d) s,i s,νs are linearly independent and hence the result follows. Note that the construction and Theorem 3 can easily e extended to (t, α, β, s)- sequences. Indeed, let d 1 and let C 1,..., C sd e the generating matrices of a digital (t, sd)-sequence. Again many explicit generating matrices are known, see for example [7, 18, 1, 31]. Let C j (c j,1, c j,,...) T for j 1,..., sd, i.e., c j,l are the row vectors of C j. Now let the matrix C (d) j e made of the first rows of the matrices C (j 1)d+1,..., C jd, then the second rows of C (j 1)d+1,..., C jd and so on, i.e., C (d) j (c (j 1)d+1,1,..., c jd,1, c (j 1)d+1,,..., c jd,,...) T. The following theorem states that the matrices C (d) 1,..., C (d) s are the generating matrices of a digital (t, α, min(α, d), s)-sequence.

17 Dick, Explicit constructions of quasi-monte Carlo rules 17 Theorem 4 Let d 1 e a natural numer and let C 1,..., C sd e the generating matrices of a digital (t, sd)-sequence over some ring R with elements. Let C (d) 1,..., C s (d) e defined as aove. Then for any α 1 the matrices C (d) 1,..., C s (d) are generating matrices of a digital (t, α, min(α, d), s)-sequence over R with s(d 1) min(α, d) t min(α, d) t +. The last result shows that (t, α, β, m, s)-nets indeed exist for any < β α and for m aritrarily large. We have even shown that digital (t, α, β, m, s)-nets exist which are extensile in m and s. This can e achieved y using an underlying (t, sd)-sequence which is itself extensile in m and s. If the t value of the original (t, m, s)-net or (t, s)- sequence is known explicitly then we also know the t value of the digital (t, α, β, m, s)-net or (t, α, β, s)-sequence. Furthermore it has also een shown how such digital nets can e constructed in practise. In the following we investigate for which values of t, α, s, digital (t, α, s)-sequences over Z exist. We need some further notation (see also [], Definition 8..15). Definition 6 For given integers s, α 1 and prime numer let d (s, α) e the smallest value of t such that a (t, α, s)-sequence over Z exists. We have the following ound on d (s, α). Corollary 1 Let s, α 1 e integers and e a prime numer. Then we have ( ) s α 1 log ( 1)s d (s, α) α(s 1) ) s 1 α( + α + s 1 α(α 1). Proof. The lower ound follows from part (iii) of Theorem y choosing α 1 and using a lower ound on the t-value for (t, s)-sequences (see [1]). The upper ound follows from Theorem 4 y choosing d α and using Theorem of []. 5 A ound on the worst-case error in H α for digital (t, α, β, m, s)-nets and digital (t, α, β, s)-sequences In this section we prove an upper ound on the worst-case error for integration in the Koroov space H α using digital (t, α, β, m, s)-nets and (t, α, β, s)-sequences. Lemma 6 Let α e a natural numer, let e prime and let C 1,..., C s Z m m e the generating matrices of a digital (t, α, β, m, s)-net over Z with m > t/β. Then we have Q,m,u,α(C 1,..., C s ) u α βm+t (βm + ) u α 1, where Q,m,u,α is defined in Lemma 5.

18 Dick, Explicit constructions of quasi-monte Carlo rules 18 Proof. We otain a ound on Q,m,α,{1,...,s}, for all other susets u the ound can e otained using the same arguments. We first partition the set D m,{1,...,s} into parts where the highest digits of k j are prescried and we count the numer of solutions of C T 1 k 1 + +C T s k s. For j 1,..., s let now i j,α < < i j,1 m with i j,1 1. Note that we now allow i j,l < 1, in which case the contriutions of those i j,l are to e ignored. This notation is adopted in order to avoid considering many special cases. Now we define D m,{1,...,s}(i 1,1,..., i 1,α,..., i s,1,..., i s,α ) {k D m,{1,...,s} : k j κ j,1 i j, κ j,α i j,α 1 + l j with l j < i j,α 1 and 1 κ j,l < for j 1,..., s}, where just means that the contriutions of i j,l < 1 are to e ignored. Then we have (8) Q,m,{1,...,s},α(C 1,..., C s ) i m 1,α 1 1 i 1,1 1 i 1,α 1 m i s,1 1 i s,α 1 1 i s,α1 D m,{1,...,s} (i 1,1,..., i 1,α,..., i s,1,..., i s,α ) i 1,1+ +i 1,α + +i s,1 + +i s,α. Some of the sums aove can e empty in which case we just set the corresponding summation index i j,l. Note that y the (t, α, β, m, s)-net property we have D m,{1,...,s}(i 1,1,..., i 1,α,..., i s,1,..., i s,α ) as long as i 1,1 + +i 1,α + +i s,1 + +i s,α βm t. Hence let now i 1,1,..., i s,α m e given such that i 1,1,..., i s,1 1, i j,α < < i j,1 m for j 1,..., s and where if i j,l < 1 we set i j,l and i 1,1 + + i 1,α + + i s,1 + + i s,α > βm t. We now need to estimate D m,{1,...,s} (i 1,1,..., i 1,α,..., i s,1,..., i s,α ), that is we need to count the numer of k D m,{1,...,s} with k j κ j,1 i j, κ j,α i j,α 1 +l j such that C T 1 k 1 + +C T s k s. There are at most ( 1) αs choices for κ 1,1,..., κ s,α (we write at most ecause if i j,l < 1 then the corresponding κ j,l does not have any effect and therefore need not to e included). Let now 1 κ 1,1,..., κ s,α < e given and define g κ 1,1 c T 1,i 1,1 + + κ 1,α c T 1,i 1,α + + κ s,1 c T s,i s,1 + + κ s,α c T s,i s,α, where we set c T j,l if l < 1. Further let B (c T 1,1,..., c T 1,i 1,α 1,..., c T s,1,..., c T s,i s,α 1). Now the task is to count the numer of solutions l of B l g. As long as the columns of B are linearly independent the numer of solutions can at most e 1. By the (t, α, β, m, s)- net property this is certainly the case if (we write (x) + max(x, )) (i 1,α 1) (i 1,α α) (i s,α 1) (i s,α α) + α(i 1,α + + i s,α ) βm t,

19 Dick, Explicit constructions of quasi-monte Carlo rules 19 that is, as long as i 1,α + + i s,α βm t. α Let now i 1,α + + i s,α > βm t. Then y considering the rank of the matrix B and α the dimension of the space of solutions of B l it follows the numer of solutions of B l g is smaller or equal to i 1,α+ +i s,α (βm t)/α. Thus we have D,{1,...,s}(i m 1,1,..., i 1,α,..., i s,1,..., i s,α ) if s α j1 l1 i j,l βm t, ( 1) αs if s α j1 l1 i j,l > βm t and s j1 i j,α βm t ( 1) αs i 1,α+ +i s,α (βm t)/α if s α j1 l1 i j,l > βm t and s j1 i j,α > βm t We estimate the sum (8) now. Let S 1 e the sum in (8) where i 1,1 + + i s,α > βm t and i 1,α + + i s,α βm t. For an l > βm t let A α 1(l) denote the numer of admissile choices of i 1,1,..., i s,α such that l i 1,1 + + i s,α. Then we have S 1 ( 1) αs αsm lβm t+1 A 1 (l) l. We have A 1 (l) ( ) l+sα 1 sα 1 and hence we otain ( ) l + sα 1 1 S 1 ( 1) sα sα βm+t 1 sα 1 l lβm t+1 ( βm t + sα sα 1 where the last inequality follows from a result y Matou sek [15], see also [6], Lemma 6. Let S e the part of (8) for which i 1,1 + +i s,α > βm t and i 1,α + +i s,α > βm t, α i.e., we have (9) S ( 1) sα m i 1,1 1 ms ( 1) sα (βm t)/α i 1,α 1 1 m i 1,1 1 i 1,α 1 m i s,1 1 i 1,α 1 i 1,α 1 1 i s,α 1 1 m i s,1 1 i s,α1 i s,α 1 (βm t)/α ), i 1,1+ +i 1,α 1 + +i s,1 + +i s,α 1 i 1,1+ +i 1,α 1 + +i s,1 + +i s,α 1 i s,α 1 1 where in the first line aove we have the additional conditions i 1,1 + +i s,α > βm t and i 1,α + +i s,α > βm t. From the last inequality and i α 1,α l + +i s,α l > i 1,α + +i s,α for l 1,..., α 1 it follows that i 1,1 + +i 1,α 1 + +i s,1 + +i s,α 1 (βm t)(1 α 1 ) + 1. Let A (l) denote the numer of admissile choices of i 1,1,..., i 1,α 1,..., i s,1,..., i s,α 1 such that ) l i 1,1 + + i 1,α i s,1 + + i s,α 1. Note that we have A (l). Then we have ( l+s(α 1) 1 s(α 1) 1 S ms ( 1) sα (βm t)/α l (βm t)(1 α 1 ) +1 ms ( 1) sα (βm t)/α (1 1 ) s(α 1) βm+t 1 (βm t)/α 1, α. α ( ) l + s(α 1) 1 1 s(α 1) 1 l ( (βm t)(1 α 1 ) + s(α 1) s(α 1) 1 ),

20 Dick, Explicit constructions of quasi-monte Carlo rules where the last inequality follows again from a result y Matou sek [15], see also [6], Lemma 6. Hence we have ( ) (βm t)(1 α S m s sα βm+t 1 ) + s(α 1). s(α 1) 1 Note that we have Q,m,α,{1,...,s} (C 1,..., C s ) S 1 +S. Let a 1 and e integers then we have ( ) a + ( 1 + a ) (1 + a). i i1 Therefore we otain S 1 sα βm+t 1 (βm t + ) sα 1 and S sα βm+t m s (βm t + ) s(α 1) 1. Thus we have Q,m,α,{1,...,s}(C 1,..., C s ) sα βm+t (βm + ) sα 1, from which the result follows. The following theorem is an immediate consequence of Lemma 5 and Lemma 6. Theorem 5 Let e prime, α e a natural numer and let C 1,..., C s Z m m e the generating matrices of a digital (t, α, β, m, s)-net over Z with m > t/β. Then the worst-case error in the Koroov space H α is ounded y e,m,α (C 1,..., C s ) (1 + αm C,α (α + ) + C,α (1 + α + )(βm + ) α ) s βm t (βm + ) +(1 + αm C,α (α + )) s 1, where C,α > is the constant in Lemma 4. Remark 4 By the lower ound of Sharygin [5] we have that the worst-case error in the Koroov space H α is at most O(N α (log N) s 1 ). Hence it follows from Theorem 5 that for a digital (t, α, β, m, s)-net with β > α we must have t O((β α)m). Thus in order to avoid having a t-value which grows with m we added the restriction β α in Definition 4. Further, this also implies that a digital (t, α, β, s)-sequence with t < cannot exist if β > α, hence β α is in this case a consequence of the definition rather than a restriction. Remark 5 Lemma 3 also holds for digital nets which are digitally shifted y an aritrary digital shift σ [, 1) s and hence it follows that Theorem 5 also holds in a more general form, namely for all digital (t, α, β, m, s)-net which are digitally shifted. Theorem 5 shows that we can otain the optimal convergence rate for natural numers α y using a digital (t, α, m, s)-net. The constructions previously proposed (for example y Sool, Faure, Niederreiter or Niederreiter-Xing) have only een shown to e (t, 1, m, s)-nets and it has een proven that they achieve a convergence of the worst-case error of O(N 1 (log N) s 1 ). If constructions y Sool, Faure, Niederreiter or Niederreiter- Xing are also (t, α, m, s)-nets with α > 1 is not known and is open for future investigation. We can use Theorem 5 to otain the following corollary.

21 Dick, Explicit constructions of quasi-monte Carlo rules 1 Corollary Let e prime and let C (d) 1,..., C s (d) Z e the generating matrices of a digital (t(a), a, min(a, d), s)-sequence S over Z for any integer a 1. Then for any real α 1 there is a constant C,s,α >, depending only on, s and α, such that the worst-case error in the Koroov space H α using the first N m points of S is ounded y e,m,α (C (d) 1,..., C (d) ) C,s,α s t( α ) (log N)s α 1 N min( α,d). Remark 6 The aove corollary shows that the digital (t, α, min(α, d), s)-sequences constructed in Section 4 achieve the optimal convergence (apart from maye some log N factor) of P α of O(N α (log N) sα ) as long as α is an integer such that 1 α d. If α > d we otain a convergence of O(N d (log N) sα ). 6 A ound on the mean square worst-case error in H α for digital (t, α, β, m, s)-nets and digital (t, α, β, s)- sequences To comine the advantages of random quadrature points with those of deterministic quadrature points one sometimes uses a comination of those two methods, see for example [6, 1, 15, 3]. The idea is to use a random element which preserves the essential properties of a deterministic point set. We call the expectation value of the square worstcase error of such randomized point sets mean square worst-case error. In the following we introduce the randomization considered here. 6.1 Randomization In the following we introduce a randomization scheme called digital shift of depth m (see [6, 15]). Let P N {x,..., x N 1 } [, 1) s with x n (x 1,n,..., x s,n ) and x j,n x j,n,1 1 + x j,n, + for n,..., N 1 and j 1,..., s. Let σ j,1,..., σ j,m {, 1} and δ j,n [, m ) e i.i.d. for j 1,..., s and n,..., N 1. Then the randomly digitally shifted point set P N,σm {z,..., z N 1 }, z n (z 1,n,..., z s,n ) using a digital shift of depth m, is then given y z j,n (x j,n,1 σ j,1 ) (x j,n,m σ j,m ) m + δ j,n for j 1,..., s and n,..., N 1, where x j,n,k σ j,n x j,n,k + σ j,n (mod ) (note that all additions of the digits are carried out in the finite field Z ). Susequently let P N {x,..., x N 1 } and let P N,σm e the digitally shifted point set P N using the randomization just descried. 6. The mean square worst-case error in the Koroov space In this section we will analyze the expectation value of e (P N,σm, K α ), which we denote y ẽ (P N, K α ) E[e (P N,σm, K α )], with respect to the random digital shift descried aove. We call ẽ (P N, K α ) the mean square worst-case error.

22 Dick, Explicit constructions of quasi-monte Carlo rules From (5) and the linearity of the expectation value we have ẽ (P N, K α ) E[e (P N,σm, K α )] N N 1 n,l j1 s E[K α (z j,n, z j,l )]. In order to compute E[K α (z j,n, z j,l )] we need the following lemma, which was already shown in [6], Lemma 3. Hence we omit a proof. Lemma 7 Let x 1, x [, 1) and let z 1, z [, 1) e the points otained after applying an i.i.d. random digital shift of depth m to x 1 and x. Then we have { walk (x E[wal k (z 1 )wal l (z )] 1 )wal k (x ) if k l < m, otherwise. where Recall that r,α (k, l) K α (x 1, x ) 1 1 r,α (k, l)wal k (x 1 )wal l (x ), k,l K α (x 1, x )wal k (x 1 )wal l (x ) dx 1 dx. Let z 1, z e otained y applying an i.i.d. random digital shift of depth m to x 1, x. Using Lemma 7 and the linearity of expectation we otain E[K α (z 1, z )] m 1 k r,α (k)wal k (x 1 )wal k (x ), where r,α (k) r,α (k, k) and r,α () 1. On the other hand if x 1 x the same shift is applied to oth variales of K α, in this case we have E[K α (z, z)] 1 + h e πih(x x) h α 1 + ζ(α), where ζ(λ) h1 h λ. Therefore we otain E[e (P N,σm, K α )] 1 + (1 + ζ(α))s N + 1 N N 1 n,l n l s j1 m 1 k r,α (k)wal k (x j,n )wal k (x j,l ) and y adding and sutracting ( m 1 k r,α (k)) s N 1 we otain E[e (P N,σm, K α )] 1 + (1 + ζ(α))s ( m 1 k r,α (k)) s + 1 N N 1 s m 1 n,l j1 k N r,α (k)wal k (x j,n )wal k (x j,l ).

23 Dick, Explicit constructions of quasi-monte Carlo rules 3 Using the inclusion-exclusion principle, Lemma 13 and Lemma 15 in the Appendix we otain ( ) m 1 s (1 + ζ(α)) s r,α (k) k k N s \{,...,m 1} s r,α (k) u {1,...,s}( 1) u +1 j u u {1,...,s} r,α (k j ) k j m j u k j r,α (k j ) ( 1) u +1 ( αm ζ(α)) u (1 + ζ(α)) s u (1 + ζ(α)) s (1 + ζ(α) αm ζ(α)) s. Further we have s m 1 r,α (k)wal k (x j,n )wal k (x j,l ) 1 + j1 k k {,..., m 1} s \{} r,α (k)wal k (x n x l ), where we write r,α (k) s j1 r,α(k j ) for k (k 1,..., k s ). We have shown the following theorem. Theorem 6 Let e a natural numer and let α > 1/ e a real numer. Then the mean square worst-case error for integration in the Koroov space H α using the point set P N randomized y a digital shift of depth m is given y E[e (P N,σm, K α )] (1 + ζ(α))s (1 + ζ(α) αm ζ(α)) s N + r,α (k) 1 N 1 wal N k (x n x l ). k {,..., m 1} s \{} By choosing m in the theorem aove we otain the average over all point sets consisting of N points. In this case the mean square worst-case error is given y E[e (P N,σ, K α )] (1 + ζ(α))s 1. N By choosing m we otain the usual digital shift. The theorem aove shows that the mean square worst-case error is in this case given y E[e (P N,σ, K α )] r,α (k) 1 N k N s \{} N 1 n,l n,l wal k (x n x l ). In the following we closer investigate the mean square worst-case error for digital nets randomized with a digital shift of depth m. For a digital net let D m D {,..., m 1}, where D is the dual net. Susequently we will often write ẽ,m,p,α (C 1,..., C s ) to denote the mean square worst-case error E[e(P m,σ p, K α )], where P m is a digital net with generating matrices C 1,..., C s and m points and P m,σ p is the digital net P m randomized with a digital shift of depth p. If p is chosen to e equal to m we will write ẽ,m,α (C 1,..., C s ).