Improvements on the star discrepancy of (t,s)-sequences Henri Faure - - PowerPoint PPT Presentation

Improvements on the star discrepancy of (t,s)-sequences

Henri Faure

Institut de math´ ematiques de Luminy, Marseille, France faure@iml.univ-mrs.fr Joint work with Christiane Lemieux

MCQMC2012, Sydney, February 13–17, 2012

Outline

• 1. Reminders on discrepancy and known bounds, p. 3–9
• 2. Applying Atanassov’s method to (t, s)-sequences, p. 10–13
• 3. Improvement in the case of an even base, p. 14–16
• 4. Comparison of the constants, p. 17–19
• 5. New results on star discrepancy of (t, 1)-sequences, p. 20–22

2

Discrepancy

For a point set PN = {X1, X2, ..., XN} in Is = [0, 1)s and a subinterval J of Is, we define the discrepancy function as E(J; PN) = A(J; PN) − NV (J) where A(J; PN) = #{n; 1 ≤ n ≤ N, Xn ∈ J} and V (J) is the volume of J. Then, the star (extreme) discrepancy D∗, respectively the (extreme) discrepancy D of PN are defined as D∗(PN) = supJ∗|E(J∗; PN)| and D(PN) = supJ|E(J; PN)| where J∗ (resp. J) is of the form s

i=1[0, yi[ (resp. s i=1[yi, zi[).

3

For an infinite sequence X, D(N, X) and D∗(N, X) denote the discrep- ancies of its first N points. Similarly, for the discrepancy function, we write E(J; N; X) = A(J; N; X) − NV (J). A sequence satisfying D∗(N, X) ∈ O((log N)s) is typically considered to be a low-discrepancy sequence. But the constant hidden in the O notation needs to be made explicit to make comparisons possible across sequences. This is achieved with an inequality of the form D∗(N, X) ≤ cs(log N)s + O((log N)s−1). (1) But this is still unsatisfactory and the constants hidden in the com- plementary term can be so huge that the leading term can lose any significance in applications. Nevertheless, improving the leading term cs is still of interest from a number theory point of view.

4

Low-discrepancy sequences

Well known low-discrepancy sequences are Halton sequences and (t, s)-sequences built with one-dimensional van der Corput sequences. A generalized van der Corput sequence SΣ

b

in base b associated with a sequence Σ = (σr)r≥0 of permutations of Zb = {0, 1, . . . , b − 1} is defined as (n ≥ 1) SΣ

b (n) = ∞

• r=0

σr

• ar(n)
• br+1

with n − 1 =

∞

• r=0

ar(n) br( b-adic expansion). Original v.d.Corput sequences are obtained with σr = id for all r ≥ 0. Generalized Halton sequences are obtained with SΣ

b

sequences in co- prime bases on each coordinate: (SΣ1

b1 , . . . , SΣs bs ).

5

The concept of (t,s)-sequences was introduced by Niederreiter to give a general framework for various constructions using generating ma- trices applied to v.d.Corput sequences (Sobol’, Faure, Niederreiter). – An elementary interval E in Is is defined as (ai, di are integers) E =

s

• i=1

[aib−di, (ai + 1)b−di) with 0 ≤ ai ≤ bdi for 1 ≤ i ≤ s. – Given integers t, m with 0 ≤ t ≤ m, a (t, m, s)−net in base b is an s-dimensional set with bm points such that any elementary interval in base b with volume bt−m contains exactly bt points of the set. – An s-dimensional sequence X in Is is a (t, s)-sequence in base b if the subset {X(n) : kbm < n ≤ (k + 1)bm} is a (t, m, s)−net in base b for all integers k ≥ 0 and m ≥ t.

6

However, to give sense to new important constructions, Tezuka and then Niederreiter and Xing in a general form needed a new definition. Truncation : Let x = ∞

i=1 xib−i be a b-adic expansion of x ∈ [0, 1],

with the possibility that xi = b − 1 for all but finitely many i. For every integer m ≥ 1, we define the m-truncation of x by [x]b,m = m

i=1 xib−i

If X ∈ Is, [X]b,m means an m-truncation is applied to each coordinate. An s-dimensional sequence (Xn)n≥1, with prescribed b−adic expan- sions for each coordinate, is a (t, s)-sequence (in the broad sense) if the subset {[Xn]b,m; kbm < n ≤ (k + 1)bm} is a (t, m, s)-net in base b for all integers k ≥ 0 and m ≥ t ≥ 0. The former (t, s)-sequences are now called (t, s)-sequences in the nar- row sense and the others (t, s)-sequences (except possible confusion).

7

Bounds for the discrepancy of (t, s)-sequences

Various constants cs below refer to inequality (1) for low-disc.seq. For s ≥ 1, cNi

s

= bt s! b − 1 2⌊ b

2⌋

  ⌊ b

2⌋

log b

 

s

(Niederreiter, 1987). (2) For s ≥ 2, cKr

s

= bt s! b − 1 2(b + 1)

• b

2 log b

s

if b is an even base (3) and cKr

s

= bt s! 1 2

b − 1

2 log b

s

if b is an odd base (Kritzer, 2006), (4) hence improving cNi

s

by a factor 1

2 for odd b and b 2(b+1) for even b

in case of dimension s ≥ 2. For short, we leave out special cases.

8

Further Kritzer stated the conjecture that for s ≥ 2 and even b, cconj

s

= bt s! b2 2(b2 − 1)

b − 1

2 log b

s

. (5) As for (t, s)-sequences in the broad sense, Niederreiter–Xing (1996) showed that constant cNi

s

in (2) is still valid, but Kritzer did not take into account this generalization in his proofs. Our aim is to closely approach (5) for (t, s)-sequences in the broad

• sense. To this end, we deepen the process from a preceding study

which consists in using Atanassov’s method for Halton sequences to

• btain discrepancy bounds for (t, s)-sequences (proc. MCQMC 2010).

If time permits, we will also provide a new result for the discrepancy

• f (t, 1)-sequences for which the approach of Kritzer does not work .

9

Atanassov’s method applied to (t, s)-seq.

We have been able to adapt Atanassov’s proof for Halton sequences to a single base b and to (t, s)-sequences in the broad sense, getting Theorem 1 For any (t, s)-sequence X (in the broad sense) we have D∗(N, X) ≤ bt s!

b

2

log N

log b + s

s

+ bt

s−1

• k=0

b k!

b

2

log N

log b + k

k

. Corollary 1 The leading constant cs in (1) satisfies cs = bt s!

b − 1

2 log b

s

if b is odd, and cs = bt s!

• b

2 log b

s

if b is even. For odd b, cs = cNi

s

but for even b, it is larger by a factor

b b−1.

10

Idea of the proof

Note PN(X) the set of the first N points of X and set n :=

log N

log b

• .

Let [PN(X)] := {([X(1)

k

]b,n+1, . . . , [X(s)

k

]b,n+1), 1 ≤ k ≤ N} where [.] is the truncation operator and (X(1)

k

, . . . , X(s)

k

) is the kth term of X. We first prove Theorem 1 for this truncated version of X. (i) We use numeration systems in base b with signed digits: z ∈ [0, 1) is written as z =

∞

• j=0

ajb−j

  

with |aj| ≤ b−1

2

if b is odd with |aj| ≤ b

2 and |aj| + |aj+1| ≤ b − 1 if b is even.

(ii) We use signed splittings of an interval J ∈ Is, i.e any collection of intervals J1, . . . , Jn and respective signs ǫ1, . . . , ǫn (±1), such that for any additive function ν on intervals in Is, we have ν(J) =

n

• i=1

ǫiν(Ji).

11

(iii) We use signed splittings I(j) = I(j1, . . . , js) deduced from (i) to get the discrepancy function: E(J; [PN(X)]) =

n

• j1=0

· · ·

n

• js=0

ǫ(j)E (I(j); [PN(X)]) =: Σ1 + Σ2 where j ∈ Σ1 ⇔ bj1 . . . bjs ≤ N and Σ2 is the complementary sum. (iv) Two main ingredients to bound Σ1 and Σ2:

• |E(J; [PN(X)])| ≤ bt s

i=1(ci − bi) where J = s i=1[bib−di, cib−di)

(fundamental property of (t, s)-sequences in the broad sense).

• #
• j; bj1 . . . bjk ≤ N
• ≤ 1

k!

log N

log b

k

(from diophantine geometry). Σ1 gives the leading part of the bound ∈ (log N)s and Σ2 the other part ∈ (log N)s−1 in the statement of Theorem 1.

12

• This proves Theorem 1 for the truncated version of the sequence.
• But, as shown by Niederreiter and ¨

Ozbudak in 2007 (Lemma 4.2, Acta Arith. 130, 79–97), when we go from the truncated to the untruncated version of the sequence, the bound for the discrepancy remains exactly the same.

• Thus if a bound of the form (1) applies to the truncated version
• f a (t,s)-sequence, it applies to the untruncated version as well

with the same constants.

• In the case of an even base, a refinement of the method result-

ing from a deeper investigation of Σ1 allowed us to substantially improve the bound cKr

s

• f Kritzer, near to his conjecture cconj

s

:

13

Improvement in the case of an even base

Theorem 2 For any (t, s)-sequence X (in the broad sense) in an even base b and for any N ≥ bs, we have D∗(N, X) ≤ bt s!

• (b − 1) log N

2 log b + s

s

+ sbt

b

2

s log N

log b

s−1

+bt

s−1

• k=0

b k!

b log N

2 log b + k

k

. Sketch of the proof. At the outset, the proof is the same as for Theorem 1 until the discrepancy function E(J; [PN(X)]) is split up into Σ1 and Σ2. The end of the proof, concerning Σ2, is also the same and gives the last term in the bound of Theorem 2. And the transition from the truncated to the untruncated version is the same

• too. Hence, what remains to be done is to deal with Σ1.

14

To this end, we split up {j = (j1, . . . , js) ; bj1 · · · bjs ≤ N} in two parts: S′ =

• j ; bj1 · · · bjs ≤ N

bs

• and S′′ =
• j ; N

bs < bj1 · · · bjs ≤ N

• .

S′′ gives the second term ∈ O((log N)s−1) in Theorem 2 as it can be shown by taking the logarithm and counting the number of s-tuples (i.e. sns−1) such that n − s <

s

• i=1

ji ≤ n with n =

log N

log b

• .

S′ provides the biggest contribution and hence gives the first term in theorem 2: bt s!

• (b − 1) log N

2 log b + s

s

.

15

This is the finer part of the proof, based on the specific properties of numeration systems with signed digits in an even base. It results from the following lemma already used in the proof of The-

• rem 1 (with the main ingredient from diophantine geometry).

Lemma Let b ≥ 2, N ≥ 1 and k ≥ 1 be integers. For integers j ≥ 0 and 1 ≤ i ≤ k, let c(i)

j

≥ 0 be given numbers satisfying c(i) ≤ 1 and c(i)

j

≤ c for j ≥ 1, where c is some fixed number. Then

• {(j1,...,jk) ; bj1···bjk≤N}

k

• i=1

c(i)

ji

≤ 1 k!

• clog N

log b + k

k

. Here, this lemma is applied with k = s,˜ b = b2 and c = √˜ b − 1 = b − 1. We refer to our paper for details.

16

Comparing the constants

Corollary 2 The discrepancy of a (t, s)-sequence X in an even base b satisfies (1) with cFL

s

= bt s!

b − 1

2 log b

s

. Hence, comparing with the previous best constant: cFL

s

cKr

s

= 2(b + 1) b

b − 1

b

s−1

that is in base 2: 3 2s−1. Besides, cFL

s

is very close to the conjecture of Kritzer: cconj

s

= b2 2(b2 − 1)cFL

s

. Filling up this small gap is a challenge for future research...

17

Numerical results

Here we are interested in smallest possible constants cs obtained with sequences taking into account the best possible values of t from MinT. In the following table, the best value of cs is indicated in boldface. For cFL

s

the base is always b = 2 and for cKr

s

the base is always b = 3 except for s = 2 where b = 2. We performed calculations for all s between 2 and 50, and observed that there are only 5 cases (in dimension 3, 6, 27, 49 and 50) where base 3 wins over base 2. Note that if Kritzer’s conjecture is true, base 2 also wins for s = 6 but not for s = 3, 27, 49 and 50.

18

s best cFL

s

t best cKr

s

t s best cFL

s

t best cKr

s

t 2 2.60e-1 0.173 15 1.87e-10 15 2.04e-10 7 3 1.25e-1 1 6.28e-2 16 8.42e-12 15 3.48e-11 8 4 2.26e-2 1 4.29e-2 1 17 2.86e-12 18 5.59e-12 9 5 6.51e-3 2 7.81e-3 1 18 2.29e-13 19 2.83e-13 9 6 1.57e-3 3 1.18e-3 1 19 8.69e-15 19 4.06e-14 10 7 3.23e-4 4 4.62e-4 2 20 1.25e-15 21 1.66e-14 12 8 5.82e-5 5 1.58e-4 3 25 3.93e-20 31 4.41e-20 15 9 9.33e-6 6 1.60e-5 3 30 1.15e-25 39 1.30e-25 19 10 2.69e-6 8 4.36e-6 4 35 1.48e-31 47 1.52e-30 25 11 3.53e-7 9 3.61e-7 4 40 4.67e-38 54 9.77e-37 29 12 4.24e-8 10 8.21e-8 5 45 1.28e-43 65 3.37e-43 33 13 4.71e-9 11 1.72e-8 6 50 4.01e-49 77 6.72e-50 37 14 9.71e-10 13 3.36e-9 7

19

Star discrepancy of (t, 1)-sequences

Extending previous results of Kritzer (2005) and F (2007), we obtain the following theorem for (t, 1)-sequences in the broad sense: Theorem 3 For any base b, the original van der Corput sequences S id

b

are the worst distributed with respect to the star discrepancy among all (t, 1)-sequences Xt

b (in the broad sense) that is, for all N ≥ 1

D∗(btN, Xt

b) ≤ btD∗(N, S id b ) = btD(N, S id b ).

The proof closely follows the proof in F (2007) with the sequence S id,t

b

consisting of bt copies of S id

b

instead of the sequence S id

b .

20

Corollary 3 For all (t, 1)-sequences Xt

b (in the broad sense) and for

all N ≥ 1 we have D∗(btN, Xt

b) ≤ bt

b − 1

4 log b log N + b − 1 4 + 2

• if b is odd and

D∗(btN, Xt

b) ≤ bt

• b2

4(b + 1) log b log N + b2 4(b + 1) + 2

• if b is even.

This corollary results from the behavior of sequences S id

b

• btained in

a former study (F’81), where α id

b

are the constants above: lim sup

N→∞

D∗(N, S id

b )

log N = α id

b

log b and D∗(N, S id

b ) ≤ α id b

log b log N + α id

b + 2.

21

Comparing these constants for dimension s = 1 to the constants of Kritzer and its conjecture in dimension s ≥ 2, we find the following:

• The constants cKr

s

for odd bases and the conjecture are still valid for s = 1.

• The constants cKr

s

for even bases are not valid for s = 1. Indeed cKr

1

= bt(b2 − b) 4(b + 1) log b while cconj

1

= bt+2 4(b + 1) log b is reached with S id,t

b

(bt copies of S id

b ). Since cKr 1

< cconj

1

and S id,t

b

is a (t, 1)-sequence, we have a contradiction.

• Finally, note that cconj

2

= cKr

2

and hence the conjecture is true. Until now this is the only dimension for which it is known to hold. Thank you!

22

Some references

– E. Atanassov, On the discrepancy of the Halton sequences, Math. Balkanica, New Series 18.1-2 (2004), 15–32. – H. Faure and C. Lemieux, Improvements on the star discrepancy of (t, s)-sequences, to appear in Acta Arith., 2012. – H. Faure, C. Lemieux, X. Wang, Extensions of Atanassov’s meth-

• ds for Halton sequences, to appear in Monte Carlo and Quasi-

Monte Carlo Methods 2010, H. Wozniakowski and L. Plaskota (eds.), Springer. – P. Kritzer, Improved upper bounds on the star discrepancy of (t, m, s)−nets and (t, s)−sequences, J. Comp. 22 (2006), 336–347. – H. Niederreiter and F. ¨ Ozbudak, Low-discrepancy sequences using duality and global function fields, Acta Arith., 130 (2007), 79–97.

23