A good permutation for one-dimensional diaphony

Transcription

1 Monte Carlo Methods Appl. 16 (2010), DOI /MCMA de Gruyter 2010 A good permutation for one-dimensional diaphony Florian Pausinger and Wolfgang Ch. Schmid Astract. In this article we focus on two aspects of one-dimensional diaphony F.S ;N/ of generalised van der Corput sequences in aritrary ases. First we give a permutation with the est distriution ehaviour concerning the diaphony known so far. We improve a result of Chaix and Faure from 1993 from a value of 1:31574 : : : for a permutation in ase 19 to 1:13794 : : : for our permutation in ase 57. Moreover for an infinite sequence X and its symmetric version e X, we analyse the connection etween the diaphony F.X;N/ and the L 2 -discrepancy L 2.e X;N/ using another result of Chaix and Faure. Therefore we state an idea how to get a lower ound for the diaphony of generalised van der Corput sequences in aritrary ase. Keywords. Diaphony, generalised van der Corput sequence Mathematics Suject Classification. 11K06, 11K38. 1 Introduction In the field of quasi-monte Carlo methods various notions of discrepancy occur in the evaluation of the distriution ehaviour of infinite sequences. The term low discrepancy sequence is used for sequences showing a particularly good distriution ehaviour. An example for a low discrepancy sequence is the (one-dimensional) generalised van der Corput sequence whose ehaviour and properties were studied in great detail y Henri Faure in many different papers. In this paper we consider two different types of discrepancy, namely the (classical) diaphony as introduced y Zinterhof [6] and the L 2 -discrepancy of generalised van der Corput sequences. Main motivation is the fact that in the last 15 years computer systems have ecome much faster. Together with changed and improved algorithms we are ale to provide permutations that eat the est known values for one-dimensional diaphony significantly. Moreover these new results let us also investigate the connection etween the diaphony and the L 2 -discrepancy of generalised van der Corput sequences numerically. This work is supported y the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network Analytic Cominatorics and Proailistic Numer Theory.

2 308 F. Pausinger and W. Ch. Schmid 2 Definitions and notations Let X D.x i / i1 e an infinite sequence in the half open unit interval Œ0; 1Œ. For N 1 and for reals 0 <ˇ 1 let A.Œ ; ˇŒ; N; X/ denote the numer of indices i N for which x i 2 Œ ; ˇŒ. The discrepancy function of X is defined as E.Œ ; ˇŒ; N; X/ D A.Œ ; ˇŒ; N; X/.ˇ /N. Definition 2.1. For any one-dimensional, infinite sequence X the L 2 -discrepancy of the first N points of X is defined y Z 1 1 L 2.N; X/ WD je.œ0; Œ;N;X/j 2 2 d : 0 Definition 2.2. For any one-dimensional, infinite sequence X the diaphony F of the first N points of X is defined y F.N;X/ WD 2 1X m 2 md1 1 NX ˇ nd1 exp 2imx n 1 2 ˇ Remark 2.3. Note that there is an equivalent definition for the diaphony (details in [5]): For any one-dimensional, infinite sequence X the diaphony F of the first N points of X is defined y Z 1 Z 1 1 F.N;X/ WD 2 2 je.œ ; ˇŒ; N; X/j 2 2 d dˇ : 0 0 Note that with this second definition it is easy to see that L 2 -discrepancy and diaphony are linked in a way similar to the relation of star discrepancy and extreme discrepancy. We call a one-dimensional sequence X a low discrepancy sequence if there exists a constant c such that for all N As a consequence computing F 2.N; X/ c log N: f.x/wd lim sup.f 2.N; X/= log N/ N!1 enales us to look for sequences with est distriution ehaviour. 2 :

3 A good permutation for one-dimensional diaphony 309 Throughout the paper let 2 and n 1 e integers. Let S e the set of all permutations of ¹0;1;:::; 1º. The identity in S is always denoted y id. In all examples and concrete results we will write down the permutations in the following way: For D we will write D.0;4;2;6;1;5;3;7/. Definition 2.4. For a fixed ase 2 and a permutation 2 S the generalised van der Corput sequence S is defined y 1 S.n/ D X.a j.n// j 1 ; j D0 where P 1 j D0 a j.n/ j is the -adic representation of an integer n 0. The analysis of the diaphony of S is ased on special functions which have een first introduced y Faure in [1] and which are defined as follows: Definition 2.5. For 2 S let Z D..0/=;.1/=; : : : ;. 1/=/. For h 2¹0;1;:::; 1º and x 2 Œ k 1 ; k Œ where k 2¹1;:::;º we define ² A.Œ0; h=œi ki Z ' ;h.x/ WD / hx if 0 h.k 1/;. h/x A.Œh=; 1ŒI ki Z / if.k 1/ < h < : The function ' is extended to the reals y periodicity. ;h Note that ' ;0 D 0 for any and that ' ;h.0/ D 0 for any 2 S and any h 2¹0;:::; 1º. Furthermore: Definition 2.6. Let ' ;.r/ 1 X WD hd0.' ;h /r where for r D 1 we omit the superscript, i.e., ' WD ';.1/. Chaix and Faure introduced in [4] a new class of functions ased on ' ;.r/ : Definition 2.7. WD ';.2/.' /2 :

4 310 F. Pausinger and W. Ch. Schmid Proposition 2.8 (Propriété 3:3 in [4]). The following holds: The function ' is continuous, piecewise linear on the intervals Œk=;.k C 1/= and '.0/ D '.1/. ;.2/ The functions ' and are continuous, piecewise quadratic on the intervals Œk=;.k C 1/= and ' ;.2/.0/ D ' ;.2/.1/. For our investigations the following is of great importance: Proposition 2.9 (Propriété 3:5 in [4]). For each interval Œk=;.k C 1/= the paraolic arcs of are translated versions of the paraola y D /x 2 =12. Remark Therefore if we consider two intervals of the form I i WD Œk i =;.k i C 1/=, 0 k i <, i 2¹1; 2º, with.k 1=/ D.k 2=/ and..k 1 C 1/=/ <..k 2 C 1/=/ then it follows immediately that min.i 1/ < min.i 2/. The starting point for our work are two theorems that show that the diaphony of S can e computed exactly. The first theorem states how to compute the diaphony of the first N points of S, while the second theorem estalishes a method for computing the constant which allows us to classify different permutations in aritrary ase. Theorem 2.11 (Theorem 4:2 in [4]). For all N 1, we have F 2.N; S / D X 1 42.N j /= 2 : j D1 Theorem 2.12 (Theorem 4:10 in [4]). Let WD inf sup n1 x2œ0;1 nx j D1.xj /=n ; then f.s / D lim sup.f 2.N; S /= log N/ D 42 =.2 log /: N!1

5 A good permutation for one-dimensional diaphony A good permutation In order to improve the est known result for diaphony otained y Chaix and Faure in [4] in ase 19, we will give a permutation in ase 57 and examine its dominant intervals. According to Theorem 2.12 it is necessary to compute. Therefore we introduce so called supporting functions gn ; W Œ0; 1! R, with g ; n.x/ WD 1 n 1 X n.xj /: Following [1] and [4] we examine intervals I n h WD Œh=n ;.hc 1/= n, with h 2 0;:::; n 1. Definition 3.1. The interval I n is called dominated, if there exists a set J of integers with h J such that gn ;.x/ max j 2J gn ;..x C.j h//= n /, for all h x 2 I n. Otherwise the interval is called dominant. h j D Figure 1. Dominant intervals (gray) of a -function in ase 10. Our algorithms enaled us to find many permutations with exceptionally good distriution ehaviour. The est ones we otained so far are in ase 57, where for the following we were ale to prove an exact value. Theorem 3.2. For WD 57 D (0, 24, 37, 8, 43, 18, 52, 29, 11, 48, 33, 4, 21, 40, 14, 54, 26, 45, 6, 35, 16, 50, 31, 2, 20, 39, 10, 47, 27, 55, 13, 42, 23, 3, 32, 51, 17, 36, 7, 46, 28, 56, 15, 41, 22, 5, 34, 49, 9, 25, 53, 38, 12, 30, 1, 19, 44) we have f.s57 / D log 57/ D 1:13794 : : : :

6 312 F. Pausinger and W. Ch. Schmid To simplify the proof we first show the following result for the supporting functions of our permutation 57 : Lemma 3.3. For all n 2 the functions g n WD gn 57; have two dominant intervals I n and I n, with k n k1 n k2 n 1 D 26 57n 1 C P n 2 id0 5 57i and k2 n D 26 57n 1 C 6 C P n 2 id1 5 57i.Forn D 2, max x2œ0;1 g 2.x/ D g C7 / and for n 3, 57 2 max x2œ0;1 g n.x/ D g n.k1 n=57n /. Proof. Numerical investigations show that there are two dominant intervals I25 1 and I26 1 for n D 1. This can e verified easily with Remark 2:10 and Tale 1. k k 57 / k k 57 / k k 57 / k k 57 / k k 57 / k k 57 / Tale 1. Values of 57. For n D 2 we get again two dominant intervals I and I and max x2œ0; g 2.x/ D g D 811:8481 : : : : For n D 3 we get the two dominant intervals I and I and max x2œ0; g 3.x/ D g D 1190:1108 : : : : Further numerical investigations allow us to make the following induction hypothesis: For n 3 the indices k1 n and kn 2 of the dominant intervals I n and I n k1 n k2 n

7 A good permutation for one-dimensional diaphony 313 of g n are and n 2 k1 n D 26 X 57n 1 C 5 57 i id0 n 2 k2 n D 26 X 57n 1 C 6 C 5 57 i : For the induction step let us suppose that our induction hypothesis holds for an aritrary n 3. To prove that it holds for n C 1 we have to add 57.xn / to g n on I n and I k1 n n and check that g k2 n nc1 has I nc1 and I nc1 again as dominant intervals. We verified this for all suintervals of I n and found that those k nc1 1 k nc1 2 k n two intervals are again dominant. Moreover the same numerical investigations show that the maximum of g nc1 is at position p nc1 D n C P 57 nc1 n 1 id0 5 57i /. id1 added in dominant intervals Figure 2. To determine max g nc1 it suffices to add the initial function only in the dominant intervals of g n. Proof of Theorem 3.2. According to Lemma 3:3 the position of the maximum of g n is p n D n 1 C P n 2 n id0 557i /. Due to its definition g n is the sum of n values of the -function divided y n. Therefore we can compute the maximum of the n-th approximation step y simply evaluating the -function at those positions. For interval I5 1 we have.x/ D x x C 8636, fori26 1 we have.x/ D x x C Letf.i;n/WD P n j Di 5 57 j C1 i.

8 314 F. Pausinger and W. Ch. Schmid Consequently n 1 X n g n.p n / D p n / 2 C.f.i; n 1// p n Easy calculation gives For n!1we get and therefore n 1 j D1 X f.i;n 1/ C C.n 1/ 8636: j D1 g n.p n / D C n n n n 112n 12544n C n 19 1 n 56n C n 19 3 n 3136n n 19 2 n 392n C n : lim g n.p n / D n!1 57 D ; f.s 57 / D log 57/ D 1:13794 : : : : 4 A lower ound and some estimations An infinite sequence Y D.y n / n1 is called symmetric, if for all n y 2n C y 2nC1 D 1 holds. A symmetric sequence Y is said to e created y the sequence X D.x n / n1, if for all n y 2n D x n or y 2nC1 D x n holds. We will denote this y Y WD ex. Due to Theorem 4.11 in [4] the L 2 -discrepancy of S f can e ounded y the diaphony of S as follows: l 2. f S / WD lim sup N!1.L2.N; S f //2 log N f.s / 2 :

9 A good permutation for one-dimensional diaphony 315 In [2] Faure proved that 0:089 l 2. f S id 2 / 0:103; which is the est value for the L 2 -discrepancy currently known. With our new est value for the diaphony it is interesting to investigate this connection once again. We tried to improve the value of the L 2 -discrepancy y finding a permutation for the diaphony that is good enough to eat the ound of Faure. It turned out that it is not possile (at least in small ases) to improve the value y simply applying the aove mentioned inequality. To outline our approach let us first recall the asic facts and the according remark aout -functions, which were mentioned in the introduction of this paper. Assuming we have already found a permutation in ase and know that max D s, we now want to estimate (see Theorem 2.12). So we construct a function D ;s only depending on the ase and on s. First we define on the interval Œx; y WD Œ 1 2 c; 1. 2cC1/, y.x/ D.y/ WD s and y.z/ WD Az 2 C Bz C C for z 2 Œx; y. LetA D / 12,thenB and C can e computed y simply solving the linear system.x/ D s;.y/ D s: We have now constructed a fictive dominant interval for and can introduce new functions eg n D eg n.; s/ similar to the already defined functions g n to get a lower ound for. In detail: eg n WD 1 n.z 1 / C nx j D2 z j 1 C 1 j k j ; 2 with n 2 N, z 1 WD x and for j>1, z j WD z j 1 C 1 j 2c. The fact that we always add the maximum of in the middle (D the minimum) of the fictive dominant interval gives us a lower ound for if the real function contains a dominant interval of the form Œk 1 ;k 2 with.k 1/ D.k 2/ D max. This lower ound turned out to e sufficient for all practical investigations, also if the dominant interval of is not of the aove assumed form. However for theoretical correctness we assume now that the real set of dominant intervals D contains at least one interval I with a ound whose value is smaller than max, which means that max I 2D min.i / < min.œx; y /. This implies that it is theoretically possile to find a function for which is smaller than the value of our lower ound. To e ale to guarantee that this will not happen we can compute an interval J D J l [ J r, which has to contain max. To get the ounds

10 316 F. Pausinger and W. Ch. Schmid of J l and J r we set m WD A.x C x /2 C B.x C x / C C and solve the equation m D Az 2 C B I z C C I ; (4.1) for z (see Figure 3), where B I, C I denote the coefficients of in the interval I that contains max I 2D min.i /. It means if the real maximum is added left of z 1 or right of z 2 in the real dominant interval, then the resulting value has to e greater than the value in the fictive dominant interval. This information is now transformed to fit the initial -function which is added (review Figure 2). As a result we get J as follows: h J D 0; x 1 l x C 1 2 z 1 2mi h [ y C 1 l x C 1 2 z 2 2m i ;1 ; where z 1, z 2 are the solution of equation (4.1) with z 1 <z 2. s s z1 m z2 => J1 => J2 Figure 3. Main idea of the lower ound. We can now formulate the following criterion: Lemma 4.1. Let a ase and a permutation e given and let max s. Then lim n!1 eg n.; s/ holds if there exists an interval Œk=;.k C 1/= with. k / D. kc1 / D s (it is sufficient that oth values are s) or if there exists an x 2 argmax with x 2 J. Remark 4.2. In practice it is possile that the requirements for the lemma are not fulfilled in some cases. However we did not find any permutation for which this lower ound did not hold. It turns out that the -functions of those permutations, that do not fulfill the requirements, usually have one value.z/ with z 2 J which is only a little it smaller than the maximum of, so that the lower ound still holds.

11 A good permutation for one-dimensional diaphony 317 Using various acktracking search strategies we were ale to determine lowest possile values for max D s in the ases 4 to 50. Furthermore we have good (ut due to computational complexity not necessarily optimal) values for ases 51 to s ase Figure 4. Smallest possile values for s D max. Remark 4.3. The functions eg n.; s/, which do not depend on a specific permutation, help investigating the connection etween the L 2 -discrepancy and the diaphony of S f respectively S (see Tales 2 4). According to these values it is not possile to otain a new est value for the L 2 -discrepancy in ases <51and it seems not possile for <73. Moreover it seems that the est values of for any ase <73are on a fairly constant level which lead us to the conclusion that it is not possile to improve the value of Faure for the L 2 -discrepancy stated in [2] via the diaphony for any small ase. Remark 4.4. The functions eg n.; s/ in comination with the position of the maximum of a specific function support our computational search for new est values for the diaphony of S f. It is a tedious and time consuming work to determine all dominant intervals for a specific function and to compute.as a simple heuristic rule for picking good candidates we state that the smaller the values of are near the ounds of the interval Œ0; 1 the smaller usually is. Remark 4.5. We can, of course, provide large files containing permutations for ases <73for which max D s holds.

12 318 F. Pausinger and W. Ch. Schmid s eg 1.; s/ f.eg 1.; s/; /.f.eg 1.; s/; //= 2 4 1:25 0: : : =2 1: : =3 89=30 1: : : : =7 1:6951 0: =3 1: : :25 8: : : =2 1: : =22 1: : =2 1: : :25 18:2115 1: : =3 20 1: : :25 21:9611 1: : : : =17 1: : =2 1: : :25 38:9605 1: : =3 87=2 1: : :25 46:2103 1: : : : =23 1: : : : =25 1: : Tale 2. Best values for s for 26 and the according estimations.

13 A good permutation for one-dimensional diaphony 319 s eg 1.; s/ f.eg 1.; s/; /.f.eg 1.; s/; //= =3 147=2 1: : =162 1: : =2 1: : =58 1:1903 0: : : =31 1: : = : : :25 122:21 1: : =2 1: : =70 1: : =2 1: : =74 1: : = : : :25 171:959 1: : : : = =123 1: : =2 1: : = =172 1: : =6 1: : =270 1: : : : = =141 1: : : : =49 1: : Tale 3. Best values for s for and the according estimations.

14 320 F. Pausinger and W. Ch. Schmid s eg 1.; s / f.eg 1.; s /; /.f.eg 1.; s /; //= =6 1: : =306 1: : =2 1: : = =318 1: : : : = =220 1: : = : : =171 1:1184 0: =2 1: : =118 1: : =2 1:1059 0: = =244 1: : = : : = =756 1: : : : = =260 1: : =2 1:1054 0: =134 1: : =3 1115=2 1:0918 0: =414 1: : : : = =213 1: : Tale 4. So far est (ut improvale) values for s for and the according estimations.

15 A good permutation for one-dimensional diaphony Conclusion In the first part of our work we present a permutation in ase 57 that generates a generalised van der Corput sequence with exceptionally good distriution ehaviour with respect to the diaphony. Since it is a very tedious work to determine the exact asymptotic distriution ehaviour of a sequence generated y a given permutation, the second part of our paper states a method how this computations can e approximated respectively simplified. The so computed lower ounds are then used to examine an inequality stated y Faure. We find that this inequality that links the diaphony of generalised van der Corput sequences with the L 2 - discrepancy of its symmetric version can not e used to improve the est known numerical results for the L 2 -discrepancy of these sequences. In a forthcoming work we will try to find algorithms for constructing good permutations like Faure did in [3]. Moreover we will use a different approach to get a lower ound for any generalised van der Corput sequence in aritrary ase as well as to determine and count classes of permutations that show similar distriution ehaviour with respect to the diaphony. Acknowledgments. We would like to thank Henri Faure for his valuale comments and suggestions. Biliography [1] H. Faure: Discrépance de suites associées à un système de numération (en dimension un). Bull. Soc. Math. France 109 (1981), [2] H. Faure: Discrépance quadratique de la suite de van der Corput et de sa symètrique. Acta Arith. 55 (1990), [3] H. Faure: Good Permutations for Extreme Discrepancy. Journal of Numer Theory 42 (1992), [4] H. Chaix and H. Faure: Discrépance et diaphonie en dimension un. Acta Arith. 63 (1993), [5] H. Faure: Discrepancy and diaphony of digital.0; 1/-sequences in prime ase. Acta Arith. 117 (2005), [6] P. Zinterhof: Üer einige Aschätzungen ei der Approximation von Funktionen mit Gleichverteilungsmethoden. Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 185 (1976),

16 322 F. Pausinger and W. Ch. Schmid Received Decemer 1, 2009; revised Septemer 22, Author information Florian Pausinger, Fachereich Mathematik, Universität Salzurg, Hellrunner Straße 34, A-5020 Salzurg, Austria. Wolfgang Ch. Schmid, Fachereich Mathematik, Universität Salzurg, Hellrunner Straße 34, A-5020 Salzurg, Austria.