APPLIED MATHEMATICS REPORT AMR04/2 DIAPHONY, DISCREPANCY, SPECTRAL TEST AND WORST-CASE ERROR. J. Dick and F. Pillichshammer

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1 APPLIED MATHEMATICS REPORT AMR4/2 DIAPHONY, DISCREPANCY, SPECTRAL TEST AND WORST-CASE ERROR J. Dick and F. Pillichshammer January, 24

2 Diaphony, discrepancy, spectral test and worst-case error Josef Dick and Friedrich Pillichshammer February 3, 24 Abstract In this paper various measures for the uniformity of distribution of a point set in the unit cube are studied. We show how the diaphony and spectral test based on Walsh functions appear naturally as the worst-case error of integration in certain Hilbert spaces which are based on Walsh functions. Furthermore, it has been shown that this worst-case error equals to the root mean square discrepancy of an Owen scrambled point set. Further we prove that the diaphony in base 2 coincides, for a special choice of weights, with the root mean square worst-case error for integration in certain weighted Sobolev spaces. This connection has also a geometrical interpretation, which leads to a geometrical interpretation of the diaphony in base 2. Furthermore we also establish a connection between the diaphony and the root mean square weighted L 2 discrepancy of randomly digitally shifted points. 1 Introduction In applications, like numerical integration using Monte Carlo or quasi-monte Carlo algorithms, where random number generators or low discrepancy sequences are used, the success of the algorithm often depends on the distribution properties of the underlying point set. Hence there is a need to measure the quality of such point sets. (In many cases point sets in the s dimensional unit cube [, 1) s are considered.) Subsequently various equidistribution measures have been introduced and analyzed. Some of them stem from numerical integration where the worst-case integration error has been analyzed. For a function space H with norm the worst-case error e(h, P N ) using a quasi-monte Carlo rule Q(P N, f) = 1 N n= f(x n), where P N = {x,..., x } [, 1) s, is given by e(h, P N ) := sup f H, f 1 f(x) dx 1 [,1) N s n= f(x n ). (1) For a given function space and norm, this worst-case error then only depends on the point set used. In some cases this worst-case error can be related to the discrepancy of The first author is supported by the Australian Research Council under its Center of Excellence Program. The second author is supported by the Austrian Research Foundation (FWF), Project S

3 the point set, see [9, 18]. The discrepancy can be defined via the discrepancy function. For a point set P N = {x,..., x } [, 1) s and α 1,..., α s 1, the discrepancy function is defined by (α 1,..., α s ) := A N([, α 1 ) [, α s )) N α 1 α s, where A N ([, α 1 ) [, α s )) denotes the number of indices h with x h [, α 1 ) [, α s ). Hence, for fixed α 1,..., α s 1, (α 1,..., α s ) measures the difference between the proportion of points in the box [, α 1 ) [, α s ) and the volume of this box. By taking a norm of (α 1,..., α s ) we obtain a measurement of the irregularity of distribution. For example, the classical L 2 discrepancy of a point set P N is given by (see for example [1]) ( ) 1/2 L 2 (P N ) := 2 (α) dα. (2) [,1) s Worst-case errors for various function spaces and numerous notions of discrepancy have been studied to a great extent, especially in connection with low discrepancy sequences. Further measures of the irregularity of distribution comprise the diaphony and the spectral test. Those two notions are based on wavelets. We confine ourselves here to Walsh functions, which are introduced in Section 2. The diaphony and spectral test have mainly been studied in connection with random number generators, see for example [6]. For a point set P N we define the diaphony F N (P N ) by F N (P N ) := c r 2 (k) S k (P N ) 2 k N s k where S k (P N ) is a certain function based on Walsh functions, r(k) is a function of k and c is such that F N (P N ) 1 (for more details see Section 3). The diaphony corresponds to the 2-norm, whereas the spectral test σ N (P N ) corresponds to the supremums norm, that is, σ N (P N ) := sup r(k) S k (P N ). k N s k In this paper we study various connections of the distribution measures introduced above. This allows us for example to give geometrical interpretations of the diaphony, thereby answering a question raised by Hellekalek [6]. Further detials are given in the outline of the paper below. In Section 2 we introduce Walsh functions and state some of their basic properties. Those functions are the wavelets used in the definition of the diaphony. In Section 3 we will show how the diaphony and the spectral test naturally arise from numerical integration of functions in certain function spaces based on Walsh functions. Such function spaces have previously been studied in [2, 3]. The diaphony and spectral test can be expressed by the worst-case error for integration in those function spaces for a particular choice of parameters determining those function spaces. We remark that the notion of worst-case error is far more general, hence the diaphony and spectral test can be generalized naturally using the general form of the worst-case error. Further, this 1/2, 2

4 worst-case error is related to the root mean square discrepancy of Owen scrambled point sets, see [3]. Section 4 relates the diaphony in base 2 to the mean square worst-case error of integration in weighted Sobolev spaces using randomly digitally shifted point sets. The weights γ := (γ j ) j 1, where γ j are real numbers, modify the importance of the irregularity of the projections. This worst-case error has also a geometrical interpretation, which yields a geometrical interpretation of the diaphony in base 2. Furthermore, the diaphony in base 2 is also related to the root mean square weighted L 2,γ discrepancy of randomly digitally shifted point sets. The weighted L 2,γ discrepancy is a generalized version of the classical L 2 discrepancy (2). We show that for a certain choice of weights γ j the diaphony and root mean square weighted L 2,γ discrepancy coincide apart from a certain factor. It has been shown that the weighted L 2,γ discrepancy itself coincides with the worst-case error of integration in certain weighted Sobolev spaces. Hence we can build another bridge between diaphony and numerical integration of functions from those weighted Sobolev spaces. 2 Walsh functions In this section we define Walsh functions in base b and state some of their basic properties. For more information on Walsh functions see for example [1, 14, 15, 19]. In the following let N denote the set of non-negative integers. Definition 1 Let b 2 be an integer. For a non-negative integer k with base b representation k = κ a 1 b a κ 1 b + κ, with κ i {,..., b 1}, we define the Walsh function b wal k : [, 1) C by bwal k (x) := e 2πi(x 1κ +...+x aκ a 1 )/b, for x [, 1) with base b representation x = x 1 b + x (unique in the sense that b 2 infinitely many of the x i must be different from b 1). If it is clear which base b is chosen we will simply write wal k. Note that Walsh functions are piecewise constant. We can define Walsh functions also for higher dimensions, which is done in the following definition. Definition 2 For dimension s 2, x 1,..., x s [, 1) and k 1,..., k s N we define bwal k1,...,k s : [, 1) s C by bwal k1,...,k s (x 1,..., x s ) := s bwal kj (x j ). For vectors k = (k 1,..., k s ) N s and x = (x 1,..., x s ) [, 1) s we write j=1 bwal k (x) := b wal k1,...,k s (x 1,..., x s ). Again, if it is clear which base we mean we simply write wal k (x). 3

5 We introduce some notation. By we denote the digit-wise addition modulo b, i.e., for x = i=w and y = i=w we have x i b i x y := y i b i i=w z i b i, where z i := x i + y i (mod b), and by we denote the digit-wise subtraction modulo b, i.e., x y := i=w z i b i, where z i := x i y i (mod b). In the following proposition we summarize some basic properties of Walsh functions, see also [1, 11, 14]. Proposition 1 Let b 2 be an integer. 1. For all k, l N and all x, y [, 1) we have wal k (x) wal l (x) = wal k l (x), wal k (x) wal k (y) = wal k (x y) and wal k (x) wal l (x) = wal k l (x), wal k (x) wal k (y) = wal k (x y). 2. We have 1 wal (x) dx = 1 3. For all k, l N s and 1 wal k (x) dx = if k >. 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 2 ([, 1) s ) and any σ [, 1) s we have f(x) dx = f(x σ) dx. [,1) s [,1) s 5. For any integer s 1 the system { b wal k1,...,k s : k 1,..., k s } is a complete orthonormal system in L 2 ([, 1) s ). 3 Diaphony, spectral test and worst-case error in H wal In this section we give the precise definition of the b-adic diaphony (see [5] or [8]). We show how the connection to numerical integration of functions from the Hilbert space H wal naturally arises from this definition. By using a different norm we also show how the spectral test directly relates to the worst-case error of integration in H wal. 4

6 Let b 2 be an integer. The b-adic diaphony of a point set P N = {x,..., x } is defined as 1/2 1 2 F b,n (P N ) := r 2 1 (1 + b) s 1 b(k) bwal k (x h ), N where r b (k) := s j=1 r b(k j ) and k N s k h= r b (k) := { 1 if k =, b a if b a k < b a+1 where a N. (3) Note that the b-adic diaphony is scaled such that F b,n (P N ) 1 for all N N, especially we have F b,1 (P 1 ) = 1. If it is clear which integer b we mean we simply speak of diaphony instead of b-adic diaphony. If b = 2 we also speak of dyadic diaphony. Note that Hence we have 1 N h= ((1+b) s 1)F 2 b,n (P N) = k N s k where We define an inner product by where wal k (x h ) r 2 b (k) 1 2 = 1 h,i= K wal (x, y) := k N s f, g wal := k N s h,i= wal k (x h )wal k (x i ). wal k (x h )wal k (x i ) = 1+ 1 r 2 b(k)wal k (x)wal k (y). r 2 b (k) ˆf wal (k)ĝ wal (k), ˆf wal (k) := f(x)wal k (x)dx. [,1) s h,i= K wal (x h, x i ), As in [3] it can be shown that K wal is a reproducing kernel. In fact, K wal is just a special case of the reproducing kernel introduced in [3]. Hence the Hilbert space H wal determined by the reproducing kernel K wal and the inner product, wal is just a special case of the weighted Hilbert space based on Walsh functions introduced in [3]. Let f wal,2 := f, f 1/2 wal. As in [3] it can be shown that the worst-case error (see (1)) for integration in H wal equipped with the norm wal,2 is given by e 2 (H wal, P N ) = Thus we have the following theorem. h,i= K wal (x h, x i ). 5

7 Theorem 1 The worst-case error e(h wal, P N ) in the space H wal equipped with the norm wal,2 using a point set P N and the diaphony F b,n of P N are related by the equation e(h wal, P N ) = (1 + b) s 1F b,n (P N ). (We remark that we can also allow a more general definition of the space H wal by allowing arbitrary values for r(k), like in Definition 5.2 in [6]. The results for this case follow with the same arguments used for the special case considered here.) Hence, apart from a scaling factor, the diaphony is just the worst-case error for integration in the Hilbert space H wal. Thus the worst-case error for integration in H wal can also be related to the so-called weighted spectral test, see [6, Definition 6.1]. We can use Theorem 1 to obtain that a point set P N is uniformly distributed if and only if lim N e(h wal, P N ) =, which follows from a similar result for the diaphony, see for example [6]. On the other hand, in [3] many results on the worst-case error e(h wal, P N ) have been obtained, which now, by using Theorem 1, also apply to the diaphony and the weighted spectral test. For example, in [3] it was shown that e(h wal, P N ) is equal to the root mean square worst-case error of integration in a weighted Sobolev space using the Owen-scrambled point set P N. Hence it follows for example that a point set P N is uniformly distributed if and only if the root mean square worst-case error of the Owenscrambled point set P N tends to zero as N goes to infinity. Further, as the worst-case error of integration in the weighted Sobolev space is equal to the discrepancy of the point set, see [9], it follows that we can also interpret the diaphony geometrically, namely as the root mean square discrepancy of the Owen-scrambled point set P N. Furthermore, Dick et. al. [2] even gave construction algorithms for certain digital nets which, in some sense, minimize the worst-case error. Hence those algorithms also yield a constructive approach to point sets with small diaphony. Note that the Hilbert space based on Walsh functions introduced in [3] yields a more general worst-case error. By using a more general definition of r(k) we can also generalize the diaphony. This gives, for example, rise to a weighted diaphony and a weighted spectral test, where one can put more attention on certain lower dimensional projections. Further one can also consider function spaces with faster decaying Walsh coefficients, again yielding generalized notions of diaphony and spectral test. Such type of distribution measures seem to be especially suitable for low discrepancy point sets. The definitions 5.2 and 6.1 in [6] on the other hand seem to be to general to be useful. As mentioned in the introduction, the spectral test corresponds to the supremums norm. By a generalization of the above approach we can also obtain a connection between the spectral test and the worst-case error. The functional I s Q N,s (P N, ) has a representer ζ, which is given by Hence we have ζ(x) = 1 1 N By using Hölders inequality we obtain h= K wal (x, x h ). I s (f) Q N,s (P N, f) = f, ζ wal. I s (f) Q N,s (P N, f) = f, ζ wal D wal,p (P N ) f wal,q, (4) 6

8 where p, q 1 with 1 p + 1 q = 1, and D wal,p (P N ) := f wal,q := It can easily be shown that D wal,p (P N ) = k N s k N s k N s k r q b r p b (k) ˆζ wal (k) p (k) ˆf wal (k) q r p b (k) 1 N h= 1/q 1/p p wal k (x h ) Remark 1 By using the same arguments as in [9] we can also show that (4) is best possible for all p, q 1 with = 1. Hence it follows that p q e q (H wal, P N ) = D wal,p (P N ), where e q (H wal, P N ) is the worst-case error for integration in the space H wal equipped with the norm wal,q using the point set P N. In the following we consider the special case p = and q = 1. We obtain { } 1 D wal, (P N ) = sup r(k) wal k (x h ) k N s N h= k and f wal,1 = r 1 (k) ˆf(k). k N s Note that D wal, (P N ) = σ N (P N ), where σ N (P N ) is the spectral test for the Walsh function system, see [6]. Hence we obtain the following theorem. Theorem 2 The worst-case error e 1 (H wal, P N ) in the space H wal equipped with the norm wal,1 using a point set P N and the spectral test σ N of the point set P N coincide, that is, e 1 (H wal, P N ) = σ N (P N ). In the following we will make use of two results from Hellekalek [7]: let b 2 and P b m be a (t, m, s)-net in base b, then σ b m(p b m) 1/b m t+1. Further he showed that for b prime and P b m a strict digital (t, m, s)-net in base b, we have σ b m(p b m) = 1/b m t+1. (For the definition of (t, m, s)-nets in base b see [12] or [13].) Hence, by using Theorem 2, we obtain the following corollary. Corollary 1 Let b 2 and P b m be a (t, m, s)-net in base b, then the worst-case error for integration in H wal equipped with the norm wal,1 satisfies e 1 (P b m) 1/b m t+1. Further, let b be prime and P b m be a strict digital (t, m, s)-net in base b, then the worst-case error for integration in H wal equipped with the norm wal,1 satisfies e 1 (H wal, P b m) = 1/b m t /p.

9 4 Dyadic diaphony, the mean square worst-case error in weighted Sobolev spaces and the weighted L 2 discrepancy In this section we establish a connection between the dyadic diaphony (that is, b = 2) and the root mean square worst-case error of integration in certain weighted Sobolev spaces using a randomly digitally shifted point set, where the digital shift is in base 2. Hence in this section, unless stated otherwise, we confine ourselves to b = 2. First we introduce a digital shift in base 2 (see also [3]). Let x, σ [, 1) with x = x x 2 + and σ = σ σ 2 + be the base 2 representations 2 2 of x and σ. Then the digitally shifted point x = x σ is given by x = x x , where x i = x i + σ i (mod 2). For vectors x [, 1) s the shift is applied component-wise and we write x σ. For a reproducing kernel K the associated digital shift invariant kernel K ds is defined by (see [3]) K ds (x, y) := K(x σ, y σ) dσ. [,1) s We can relate the dyadic diaphony to the mean square worst-case error of integration in the weighted Sobolev space H sob,s,γ considered in [3]. This Sobolev space is a reproducing kernel Hilbert space with reproducing kernel K sob,s,γ given by K sob,s,γ (x, y) = and inner product f, g sob,s,γ := s K sob,γj (x j, y j ) = j=1 u {1,...,s} j u s (1 + γ j ( 1B 2 2({x j y j }) + (x j 1)(y 2 j 1)) 2 j=1 γj 1 u f u g (x)dx u (x)dx u dx u, [,1) u [,1) x s u u [,1) x s u u where for x = (x 1,..., x s ) we use the notation x u = (x j ) j u and x u = (x j ) j {1,...,s}\u. Here the weights γ = (γ j ) j 1 are introduced to modify the importance of different projections, see [18] for more information. The norm in H sob,s,γ is defined as f sob,s,γ := f, f 1/2 sob,s,γ. As shown in [3] the digital shift invariant kernel associated to K sob,ds,s,γ is given by ( s ) K sob,ds,s,γ (x, y) = rsob 2 (γ j, k) wal k (x j ) wal k (y j ) j=1 = k N s k=1 r 2 sob (γ, k) wal k(x) wal k (y), with r sob (γ, k) := s j=1 r sob(γ j, k j ), where { 1 if k =, r sob (γ, k) = γ/12 2 a if 2 a k < 2 a+1 where a N. 8

10 The mean square worst-case error for integration in H sob,s,γ using the point set P N = {x,..., x } is then given by (see again [3]) e 2 sob,ds(h sob,s,γ, P N ) = We obtain the following theorem. h,i= k N s rsob(γ, 2 k) wal k (x h )wal k (x i ). Theorem 3 The root mean square worst case error e sob in the space H sob,s,γ with weights γ j = 12 and equipped with the norm sob,s,γ using a randomly digitally shifted point set P N and the dyadic diaphony F 2,N of P N are related by the equation e sob,ds (H sob,s,γ, P N ) = 3 s 1F 2,N (P N ). Note that, as shown in [3], the mean square worst case error can be written as e 2 sob,ds (H sob,s,γ, P N ) = h,i= j=1 s (1 + γ j φ sob,ds (x h,j, x i,j )), where { 1 if x = y, φ sob,ds (x, y) = if x 6 2 i +1 i y i and x i = y i for i = 1,..., i 1. The results in [3] were actually shown for an arbitrary base b, b 2. (Note that the digital shift can easily be generalized to an arbitrary base b 2.) Using a digital shift in base b we obtain that the mean square worst-case error can be written as e 2 sob,ds,b (H sob,s,γ, P N ) = = h,i= k N s h,i= j=1 rsob,b 2 (γ, k) bwal k (x h ) b wal k (x i ) s (1 + γ j φ sob,ds,b (x h,j, x i,j )), with r sob,b (γ, k) = s j=1 r sob,b(γ j, k j ), where { 1 if k =, r sob,b (γ, k) = γ/2 b a if b a k < b a+1, a N sin 2 (κπ/b) 3, and κ = kb a, and for x = x 1 b + x 2 + and y = y 1 + y 2 + we have b 2 b b 2 { 1 if x = y, 6 φ sob,ds,b (x, y) = 1 x i y i (b x i y i ) if x 6 b i +1 i y i and x i = y i for i = 1,..., i 1. The function φ sob,ds,b seems to be a natural extension of φ sob,ds to an arbitrary base b. Hence, by using Theorem 3 and e 2 sob,ds,b (H sob,s,γ, P N ), the dyadic diaphony could be extended in a different way to an arbitrary base b. In any case, it appears that e 2 sob,ds,b (H sob,s,γ, P N ) is by itself an appropriate measure for the irregularity of distribution 9

11 of a point set. Further note that there is also a geometrical interpretation of the worst-case error for integration in H sob,s,γ, see [17]. Further we remark that the weights γ j = 12 are rather large, see [4]. This means that the diaphony puts a very high attention on the higher dimensional projections. For many applications this choice of weights seems to be far to large. We can also establish a connection between the dyadic diaphony and the expected value of the L 2 discrepancy of randomly digitally shifted points, where the digital shift is in base 2. In the following we consider the weighted L 2,γ discrepancy. As opposed to the classical L 2 discrepancy, see (2), we also include the lower dimensional projections. We need some notation. Let γ = (γ j ) j 1, the weights, be a sequence of real numbers, D denote the index set D = {1,..., s} and for u D, u, let γ u := j u γ j. Let u denote the cardinality of u and for a vector α = (α 1,..., α s ) [, 1) s let α u = (α j ) j u. Further let dα u = j u dα j and let (α u, 1) = (a 1,..., a s ), where a j = α j if j u and a j = 1 if j u. Then the weighted L 2,γ discrepancy of a point set P N is defined as 1/2 L 2,γ (P N ) := γ u [,1) 2 ((α u, 1)) dα u. u u D u Let P N be again a point set in [, 1) s and P N be the point set obtained after applying a random digital shift in base 2 to the point set P N. From [2] we obtain E[L 2 2,γ ( P N )] = 1 h,i= r 2 (γ, k)wal k (x h )wal k (x i ), k N s k where r(γ, k) = s j=1 r(γ j, k j ) and { 1 + γ/3 if k =, r(γ, k) := γ if 2 a k < 2 a+1 with a N a. We have 3r( 4, k) = r 2 (k) 2, where r 2 (k) is given by (3). Hence we obtain the following theorem. Theorem 4 The root mean square weighted L 2 discrepancy with weights γ j = 4 and the dyadic diaphony are related by the equation E[L 2 2,γ( P N )] = (( 1) s ( 1/3) s )F 2 2,N(P N ). Normally one demands that the weights are non-negative, though most of the analysis also works if some weights are negative. This restriction is due to a sensible interpretation of the notion of weighted discrepancy. In any case, Theorem 4 yields another geometrical interpretation of the diaphony. References [1] Chrestenson, H.E.: A class of generalized Walsh functions. Pacific J. Math., 5: 17 31,

12 [2] Dick, J., Kuo, F., Pillichshammer, F., and Sloan, I.H., Construction algorithms for polynomial lattice rules for multivariate integration. Submitted. [3] Dick, J., and Pillichshammer, F., Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. Submitted. [4] Dick, J., Sloan, I.H., Wang, X. and Woźniakowski, H.: Liberating the weights. J. Complexity, to appear. [5] Grozdanov, V., Stoilova, S.: On the theory of b-adic diaphony. C. R. Acad. Bulgare Sci. 54 no. 3: 31 34, 21. [6] Hellekalek, P.: On the Assessment of Random and Quasi-Random Point Sets. In: Hellekalek, P. and Larcher, G. (eds) Random and Quasi-Random Point Sets, Lecture Notes in Statistics, vol Springer-Verlag, New York, , [7] Hellekalek, P.: Digital (t, m, s)-nets and the spectral test. Acta Arith. 15: , 22. [8] Hellekalek, P., Leeb, H.: Dyadic diaphony. Acta Arith. 8: , [9] Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comp., 67: , [1] Kuipers L., Niederreiter H., Uniform Distribution of Sequences. John Wiley, New York, [11] Niederdrenk, K., Die endliche Fourier- und Walshtransformation mit einer Einführung in die Bildverarbeitung, Vieweg, Braunschweig, [12] Niederreiter, H.: Point Sets and Sequences with Small Discrepancy. Monatsh. Math. 14: [13] Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. No. 63 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia, [14] Pirsic, G.: Schnell konvergierende Walshreihen über Gruppen. Master s Thesis, University of Salzburg, (Available at [15] Rivlin, T.J. and Saff, E.B.: Joseph L. Walsh Selected Papers. Springer Verlag, New York, 2. [16] Roth, K.F., On irregularities of distribution. Mathematika, 1, 73 79, [17] Sloan, I.H., Wang, X., and Woźniakowski, H.: Finite-order weights imply tractability of multivariate integration. In preparation. [18] Sloan, I.H. and Woźniakowski, H.: When are quasi-monte Carlo algorithms efficient for high dimensional integrals? J. Complexity, 14: 1 33, [19] Walsh, J.L.: A closed set of normal orthogonal functions. Amer. J. Math., 55: 5 24,

13 [2] Yue, R.X., and Hickernell, F.J., The discrepancy and gain coefficients of scrambled digital nets. J. Complexity, 18, , 22. Author s Addresses: Josef Dick, School of Mathematics, University of New South Wales, Sydney 252, Australia and IBM Tokyo Research Laboratory, Shimotsuruma, Yamato-shi, Kanagawa-ken , Japan. josi@maths.unsw.edu.au Friedrich Pillichshammer, Institut für Analysis, Universität Linz, Altenbergstraße 69, A- 44 Linz, Austria. friedrich.pillichshammer@jku.at 12