IMPROVEMENT ON THE DISCREPANCY. 1. Introduction

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1 t m Mathematical Publications DOI: /tmmp Tatra Mt. Math. Publ. 59 ( IMPROVEMENT ON THE DISCREPANCY OF (t es-sequences Shu Tezuka ABSTRACT. Recently a notion of (t es-sequences in base b was introduced where e =(e 1...e s is a positivnteger vector and their discrepancy bounds were obtained based on the signed splitting method. In this paper we first propose a general framework of (T E Es-sequences and present that it includes (Ts-sequences and (t es-sequences as special cases. Next we show that a careful analysis leads to an asymptotic improvement on the discrepancy bound of a (t es-sequencn an even base b. It follows that the constant in the leading term of the star discrepancy bound is given by c s = bt s! b 1 2 log b. 1. Introduction Low-discrepancy sequences form a mainstay of quasi-monte Carlo methods in scientific computing. All current constructions of s-dimensional low-discrepancy sequences yield a bound on the discrepancy low-discrepancy sequences signed splitting method (t es-sequences discrepancy DN of the form DN (log N s ( (log N s 1 c s + O (1 N N for all N > 1. Then it is the aim to obtain constructions and/or methods for bounding discrepancy that achieve a constant c s as small as possible. At present we have two types of constructions: Halton sequences and (t s-sequences. (To be precise Kronecker sequences (see e.g. [2] are known to satisfy the definition (1 only for the one-dimensional case s = 1. Recently the author [16] introduced a generalization of the theory of (t s-sequences namely the concept of (t es- -sequences with e = (e 1...e s beingans-tuple of positivntegers where c 2014 Mathematical Institute Slovak Academy of Sciences M a t h e m a t i c s Subect Classification: Primary 11K38; Secondary 11K06. K e y w o r d s: discrepancy low-discrepancy sequences signed splitting method (t es-sequences. This research was supported by JSPS KAKENHI(

2 SHU TEZUKA the special case of e =(e 1...e s =(1...1 corresponds to that of classical (t s-sequences. Then the author applied this concept to prove that a generalized Niederreiter sequence [14] [15] is a (t es-sequence with t =0where is the degree of thth base polynomial in the construction and obtained much better discrepancy bounds for generalized Niederreiter sequences by using the signed splitting method described below. Subsequently under the framework of (t es-sequences Hofer and Niederreiter [10] and Niederreiter and Y e o [12] proposed new constructions of low-discrepancy sequences with better discrepancy bounds based on global function fields. There are three methods of obtaining discrepancy bounds depending on the particular constructions. (Chinese remainder theorem In 1960 H a l t o n [9] became the first who obtained a construction of low-discrepancy sequences which satisfy the definition (1. Today his construction is called the Halton sequence. He employed the Chinese remainder theorem to analyze the discrepancy of his sequences. Although sommprovements in this direction have been done (see e.g. F a u r e [4] the constant c s in the discrepancy bound still grows super-exponentially in the dimension. (Double recursion method In 1967 S o b o l [13] invented the double recursion method to analyze the discrepancy of what is today called (t s- -sequences in base 2. However the constant c s obtained for the Sobol sequence which is a special case of (t s-sequences in base 2 still superexponentially increases in the dimension. In 1982 F a u r e [4] applied this method to the so-called Faure sequence which is a special case of (0s- -sequences in a prime base b and showed that the constant c s converges to zero as the dimension goes to infinity provided that the base b is chosen to be the least prime with b s. In 1987 N i e d e r r t e r [11] introduced anotionof(t s-sequences in base b for an arbitrary integer b 2 and established a general framework of what is today called the net theory of digital sequences [3]. (Signed splitting method In 2004 A t a n a s s o v [1] proposed the signed splitting method for the discrepancy analysis of generalized Halton sequences and showed that the constant c s for the Halton sequence converges to zero as the dimension goes to infinity. Recently the author [16] applied this method to (t es-sequences and showed that the constant c s for the Niederreiter sequencn base 2 which is a special case of (t s-sequences in base 2 converges to zero as the dimension goes to infinity. In this paper we present recent results on (t es-sequences. In Section 2 wntroduce a general concept of (T E Es-sequences in base b andshowthat (Ts-sequences and (t es-sequences are special cases of (T E Es-sequences. 28

3 IMPROVEMENT ON THE DISCREPANCY OF (t es-sequences In Section 3 we first overview the signed splitting method for (t es-sequences in base b as well as the previous results on the discrepancy of these sequences. Then wmprove the discrepancy bound for (t es-sequences in even bases which yields the smaller constant c s than the previous one [16]. In the final section we discuss interesting open questions for future research. 2. (T E Es-sequences and (t es-sequences First wntroduce the definition of discrepancy. For a point set P N = {X 0 X 1...X N 1 } of N points in [0 1] s andanintervalj [0 1] s wedefine A N (J asthenumberofn 0 n N 1 with X n J and μ(j isthe volume of J. Then the star discrepancy of P N is defined by D N =sup J A N (J N μ(j where the supremum is taken over all intervals J of the form s [0α i for 0 <α i 1. The (unanchored discrepancy D N is obtained when the supremum is taken over all intervals J of the form s [α iβ i for 0 α i <β i 1. Let b 2 be an integer. An elementary interval in base b whichisakey concept of the net theory is an interval of the form [ ai E(l; a = a i +1 b l i b l i where a i and l i arntegers with 0 a i <b l i and l i 0 for i =1...s. Denote a subset of nonnegativnteger vectors by E N s 0 where card(e =. Define a set of elementary intervals as follows: E(E = E(l l E where E(l = { E(l; a 0 a i <b l i (1 i s }. Define l = l l s for a nonnegativnteger vector l =(l 1...l s. Denote a set of nonnegativntegers by N(E = { l l =(l 1...l s E }. Remark that card ( N(E = because card(e =. We first give the definition of (t m Es-nets as follows: Definition 1. Let t and m bntegers with 0 t m such that m t N(E. A (t m Es-net in base b is a point set of b m points in [0 1] s such that A b m(e =b t for every elementary interval E E(E withμ(e =b t m. Let T E be a mapping from N 0 to N 0 where0 T E (m m such that there arnfinitely many m satisfying m T E (m N(E. Then (T E Es-sequences are defined as follows: 29

4 SHU TEZUKA Definition 2. A(T E Es-sequencn base b is an infinite sequence X = (X n n 0 ofpointsin[0 1] s such that for all integers k 0andallm T E (m satisfying m T E (m N(E the point set { [X kb m] bm...[x (k+1bm 1] bm } is a(t E (mmes-net where [X n ] bm means the coordinate-wise b-ary m-digit truncation of a point X n. It is easy to obtain the following propositions. Proposition 1. When E = N s 0 a(t E Es-sequencn base b is identical to a (Ts-sequencn base b. Proposition 2. Let e =(e 1...e s be a positivnteger vector. When the mapping T E is constant i.e. T E t ande = { l divides l i (1 i s } then a (T E Es-sequencn base b is identical to a (t es-sequencn base b. The example below gives two (0 e 1-sequences in base 2 with e =(3. Example 1. The first cass the following generator matrix of a strict (2 1- -sequencn base 2. J O O G 1 = O J O O O... where J is a 3 3 matrix defined as J = In this case we observe that 0 if m =0 (mod3 T(m = 1 if m =1 (mod3 2 if m =2 (mod3. Thus this is a (0 e 1-sequencn base 2 with e = (3 equivalently a (T E Es- -sequence with E = { l l =0 (mod3 } and T E 0. The second generator matrix of a strict (2 1-sequencn base 2 is as follows: ( J O G 2 = O I where I is an infinitdentity matrix. In this case we observe that 0 if m =0orm 3 T(m = 1 if m =1 2 if m =2. Thus this is also a (0 e 1-sequencn base 2 with e =(3. 30

5 IMPROVEMENT ON THE DISCREPANCY OF (t es-sequences 3. Signed splitting method for (t es-sequences 3.1. Overview of previous results We first overview the notion of signed splitting introduced by A t a n a s - s o v [1] and its relevant lemma. Definition 3. Consider an interval J [0 1] s. We call a signed splitting of J any collection of intervals J 1...J n and respective signs ɛ 1...ɛ n equal to ±1 such that for any finitely additive function ν on thntervals in [0 1] s wehave ν(j = n ɛ iν(j i. Lemma 1. Let J = s [0z(i be an interval in [0 1] s and let n i 0 be a given integer for i =1...s.Setz (i 0 =0z ni +1 = z (i and if n i 1 letz (i i [0 1] be arbitrary numbers for i =1...n i. Then the collection of intervals I( 1... s = [ min with signs ɛ( 1... s = s sgn( z (i is a signed splitting of thnterval J. ( z (i i z (i i +1 i +1 z(i i max ( z (i i z (i i +1 for i =0 1...n i (1 i s Take any z = ( z (1...z (s [0 1] s.eachz (i is expanded as =0 a(i b ei with a (i 0 1 a(i b and a (i + a(i +1 b 1 for 1. (The existence of such an expansion is proved in A t a n a s s o v [1]. Let n i = log b N t + 1 and define z (i 0 = 0 and z (i n i +1 = z(i. Consider the numbers z (i k = k 1 =0 a(i b ei for k =1...n i. By using the additivity of the local discrepancy and applying the above lemma we have n 1 n s ( ( ( A N (J Nμ(J = ɛ( A N I( Nμ I( = =0 s =0 whern 1 we put all vectors such that b(e +t N andin 2 the rest. In order to estimate the two sums 1 and 2 the following two lemmas [16] are needed. Lemma 2. Let b 2 be an arbitrary integer e =(e 1...e s and =( 1... s bnteger vectors with 1 and i 0 for i =1...s.LetI be an interval given by I = [ ai b i c i b i where a i and c i arntegers with 0 a i <c i b i for i =1...s.Then for the first N points of the truncated version of a (t e s-sequencn base b 31

6 we have SHU TEZUKA A N (I Nμ(I b t (c i a i for every positivnteger N anda N (I b t s (c i a i if N < b (e +t where the truncation sizs taken to be large enough depending on. By using the fact that the number of positivnteger vectors =( 1... s satisfying (e α for α>0 is bounded by 1 s! s α we obtain the following Lemma 3. Let e 1...e s be positivntegers and α > 0. Let some numbers g (i 0 be given for 0 and i =1...s satisfying g (i 0 1 and g (i f i ( for 1 wheref 1 (e 1...f s (e s are some numbers. Then g (i i 1 ( f i ( α + s. s! ( 1...s (e α Based on the above lemmas the next theorem was obtained [16]. Theorem 1. Let b 2 be an arbitrary integer. The star discrepancy for the first N>b t points of a (t es-sequencn base b is bounded as follows: ( NDN bt b (log s! e b N t+s i s 1 b t+e k+1 k ( b + (log k! b N t+k. (2 k=0 In the above theorem the first term of the righthand side of (2 is the upperbound of 1 and the second term is that of 2.Since 2 = O ( (log b N s 1 the leading constant is given as follows: c s = bt s! b log b. ( Improvement of the leading constant We now give the main result of this section. The key idea of the proof which exploits the property a (i + a(i +1 b 1 to improve the leading constant is due to A t a n a s s o v [1]. F a u r e and L e m i e u x [6] applied his idea in a slightly modified form to (t s-sequences. However their proof contains a serious error (see the corrigendum [7] for the detail. The proof given below can be viewed as a generalized and corrected version because (t s-sequences are a special case of (t es-sequences. 32

7 IMPROVEMENT ON THE DISCREPANCY OF (t es-sequences Theorem 2. Let b 2 be an arbitrary integer. The star discrepancy for the first N>b t points of a (t es-sequencn base b is bounded as follows: ( NDN bt b 1 (log s! 2e b N t+s i + bt+e s s 1 e ( b (log 2 e b N t+ b i s 1 + k=0 b t+e k+1 k! k ( b (log b N t+k. (4 P r o o f. First we divide the first sum 1 as follows: 1 = 1A + 1B wherewetake 1A over all with b(e+t Nb e and 1B over all with Nb e <b (e+t N. First the sum 1A is considered. Define S = { (e log b N e t } S = { (2 ( 1 +1 δ ( s +1 δ s ( 1... s S and (δ 1...δ s {0 1} s} and S = { ( ( ( s +1 ( 1... s S }. We definntegers c (i h = a(i 2h 1 + a(i 2h for h 1 andc(i 0 =1.Note that c(i 0 a(i 0 because a (i 0 = 0 or 1. Then we observe 1 δ 1 =0 in other words 1 δ s =0 (i ( a a (i 2h i δ i = ( 1...s i =h i (1 i s ( 1...s i =h i (1 i s 2h i 1 (i a i = c (i i + a (i 2h i = c (i h i for any positive vector where i = i/2 if i is even; otherwise ( i +1/2. Remark that if thers som with i =0wehave (i a i c (i. i Let b = b 2. For any S thereexistsone S.Thuswehave be e s s = b 2( even i i /2+ odd i ( i +1/2 b (e b e Nb t b e b e = Nb t. 33

8 Since c (i = (i a bt 1A S 2 1 SHU TEZUKA + a (i b 1andlog b b = 2 Lemmas 2 and 3 give us 2 (i a i b t S (i a i b t bt s! S c (i i ( b 1 2 (log b N t+s. Nextwe consider the sum 1B.Taking the logarithm of the condition we have log b N t e < (e log b N t. This means that we have linear relations (e =m in ( 1... s for log b N t e < m log b N t. Thus the number of nonnegativnteger vectors ( 1... s satisfying such relations is at most e s 1( (logb N t/ +1. Taking into consideration that each contributes at most b t s b to the estimate of 1B weobtain s 1 bt+es e ( b (log 2 e b N t+ b. i 1B Remark that the third term of the righthand side of (4 is the upper bound of 2 which remains the same as that obtained in Theorem 1. Since both bounds on the sums 1A and 1B arndependent of each interval J = s [ 0z (i the discrepancy bound for the truncated version is obtained. As it is shown in [16] the discrepancy bound for the untruncated version remains the same as the truncated version. The proof is complete. Since we have 1B = O( (log b N s 1 in the above theorem the leading constant for (t es-sequences in base b is given as c bt b 1 s (new = s! 2 log b. In comparison with the previous constant of (3 the new constant yields an improvement for the case of even bases. We should notice that for any odd base b the bound in (4 has no improvement on the previous bound in (2. The leading constant currently known as the best for (t s-sequences in base b which was recently obtained by F a u r e and K r i t z e r [5] based on an improvement of the double recursion method is given as ( 34 c s (t s = b 2 b t 2(b 2 1 s! 1 2 b t s! ( b 1 2log b b 1 s 2log b if b is even s otherwise.

9 IMPROVEMENT ON THE DISCREPANCY OF (t es-sequences Since (t s-sequences in base b are equivalent to (t es-sequences in base b with e=(1...1 we can compare the above constants to conclude that the new constant is slightly bigger than c s(t s by a factor of at most 2. In the most practical case of b = 2 the factor is 1.5 for the new constant whilt is s for the previous constant of (3. Therefore our improvement is significant in particular for large dimensions s. Remark 1. The new paper of F a u r e-l e m i e u x [8] uses a different approach from the onn this paper to obtain the same leading constant as c s(new. If we look at their Theorem 2 and its preceding paragraph we can easily find what is the main difference between their new bound and the bound in (2 of this paper namely the term log b N t in (2 is replaced by log b N + in their result. As for any odd base b it is obvious that their new upper bound is always bigger than the upper bound in (2. Even for any even base b whereasthe term b is reduced to b 1 the same situation happens at least if N<b bt. Furthermore in the paragraph ust after the proof of Lemma 7 of their paper the significance of their new finding N 1 in Theorem 2 is emphasized. However this paragraph is completely wrong and misleading. First we should notice that their upper bound in Theorem 2 is bigger than b t.secondallofus know the trivial upper bound of the discrepancy i.e. D (NX N. Combining these two facts we find that their Theorem 2 is meaningless unless N>b t. In conclusion their new bound gives almost no improvement compared to the previous bound of (2. 4. Discussions Generalized Niederreiter sequences [14] [15] include Sobol sequences Niederreiter sequences generalized Faure sequences polynomial Halton sequences etc. The generator matrices of this class of sequences are constructed by using rational functions which consist of numerators and denominators. In the construction numerators are commonly called direction numbers and denominators are called base polynomials. When we apply the discrepancy bounds (2 and (4 to generalized Niederreiter sequences we set the parameter to be the degree of thth base polynomial for i =1...s. Figure 1 shows numerical results of the leading constants 2 s c s(new of the unanchored discrepancy for the Sobol sequence and the Niederreiter sequencn base b = 2 up to 360 dimensions. The difference between the two sequences is as follows: the Sobol sequence uses primitive polynomials over GF (2 for the base polynomials except for the first base polynomal which is p(z =z. On the other hand the Niederreiter sequence uses irreducible polynomials for the base polynomials. In both cases the degrees of the base polynomials are 35

10 SHU TEZUKA Figure 1. The leading constants of the unanchored discrepancy for the Sobol sequence and the Niederreiter sequencn base b = 2 up to 360 dimensions. sorted in a nondecreasing order. Surprisingly enough the figure shows that the Sobol constant looks going to infinity while the Niederreiter constant looks converging to zero. Although the difference between the primitivity and thrreducibility is small the constants based on (t es-sequences clearly distinguish them. We should note that the constants obtained for Sobol and Niederreiter sequences based on (t s-sequences cannot do because both of them whichever star or unanchored discrepancy super-exponentially go to infinity. Further theoretical investigation into this phenomenon will bnteresting. It is well known that direction numbers are very important parameters for obtaining good practical performancn real world applications. As easily seen discrepancy bounds of generalized Niederreiter sequences based on (t e s- -sequences as well as (t s-sequences do not contain any information about direction numbers. A general framework of (T E Es-sequences is capable of dealing with such information by an appropriate choice of the set E. For example getting back to Example 1 if we choose E = {l l 3} the generator G 2 has T E 0 while the generator G 1 has T E 0. In order to employ the signed 36

11 IMPROVEMENT ON THE DISCREPANCY OF (t es-sequences splitting method for analyzing (T E Es-sequences the set E must be a direct product i.e. with E = E (1 E (s E (i = { } l (i 1 l(i i s where 0 l (i 1 <l (i 2 < are an increasing sequence of integers for i =1...s. This direction of research must bnteresting. Acknowledgments. This paper was presented at the Oberwolfach workshop Uniform Distribution Theory and Applications September 29 October I thank the organizers of the workshop for inviting me to Oberwolfach. REFERENCES [1] ATANASSOV E. I.: On the discrepancy of the Halton sequences Math. Balkanica (N.S. 18 ( [2] BECK J.: Probabilistic diophantine approximation I. Kronecker sequences Ann. of Math. 140 ( [3] DICK J. PILLICHSHAMMER F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration Cambridge Univ. Press Cambridge [4] FAURE H.: Discrépance de suites associées àunsystèmedenumération (en dimension s ActaArith.XLI ( [5] FAURE H. KRITZER P.: New star discrepancy bounds for (t m s-nets and (t s- -sequences Monatsh. Math. 172 ( [6] FAURE H. LEMIEUX C.: Improvements on the star discrepancy of (t s-sequences Acta Arith. 154 ( [7] FAURE H. LEMIEUX C.: Corrigendum to: Improvements on the star discrepancy of (t s-sequences ActaArith.159 ( [8] FAURE H. LEMIEUX C.: A variant of Atanassov s method for (t s-sequences and (t es-sequences J.Complexity30 ( [9] HALTON J. H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals Numer. Math. 2 ( [10] HOFER R. NIEDERREITER H.: A construction of (ts-sequences with finite-row generating matrices using global function fields Finite Fields Appl. 21 ( [11] NIEDERREITER H.: Point sets and sequences with small discrepancy Monatsh. Math. 104 ( [12] NIEDERREITER H. YEO A.: Halton-type sequences from global function fields Sci. China Ser. A 56 ( [13] SOBOL I. M.: The distribution of points in a cube and the approximate evaluation of integrals U.S.S.R. Comput. Math. Math. Phys 7 ( ; translation from Zh. Vychisl. Mat. Mat. Fiz. 7 (

12 SHU TEZUKA [14] TEZUKA S.: Polynomial arithmetic analogue of Halton sequences ACMTrans. Model. Comput. Simul. 3 ( [15] TEZUKA S.: Uniform Random Numbers: Theory and Practice Kluwer Acad. Publ. Boston [16] TEZUKA S.: On the discrepancy of generalized Niederreiter sequences J. Complexity 29 ( Received June Institute of Mathematics for Industry Kyushu University 744 Motooka Nishi-ku Fukuoka-shi Fukuoka-ken JAPAN tezuka@imi.kyushu-u.ac.p 38