Using permutations to reduce discrepancy

Journal of Computational and Applied Mathematics 31 (1990) 97-103 North-Holland

97

Using permutations to reduce discrepancy

H. FAURE Unit6 de Formation et de Recherche de Mathkmatiques Informatique Mkcanique *, UniversitP de Provence, 3 PIace Victor

Hugo, F-13331, Marseille CPdex 3, France

Received 16 January 1989 Revised 24 October 1989

Abstract: We present a survey of the main improvements obtained by scrambling and symmetrizing van der Corput sequences in any base. - Exact formulae for extreme and mean-square discrepancy and for diaphony in one dimension. - Asymptotic behaviour, leading to the best constants currently known. An incursion in higher dimensions is given at the end.

Keywords: Irregularities of distribution, discrepancy, Monte Carlo methods, random number generation.

1. Measures of irregularities of distribution

Let X be an infinite sequence in [0, 11, N an integer and a a point in [0, 11. We set

A(a, N, X)=card{n<N; x,E[O, a[} and E(a, N, X)=A(a, N, X)-Na.

The function E is called the remainder (or the error) to ideal repartition. Then the various discrepancies are defined by the following formulas:

D’P’(N, X) = illE(a, N, X) 1 p da) i

l/P

, p real, p>O;

D’“‘(N, X) =D*(N, X) = sup IE(a, IV, X) 1;

WC X) = 7; IN% P[; NY x) 1,

where

E([a, P[; N, X) =E(P, N, X) -E(a, N, X).

Usually II(*) is denoted by T.

* Unite associee au CNRS URA 225.

0377-0427/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

98 H. Faure / Reduction of discrepancy

Proposition (Koksma; see [9]).

T2(iv, x) = ( 5 (4 - xJ)‘+ $FZ(N, x) \n=l

with

P(N, x) =

i

2 f -$ m=l

2 l/2

1 II f . e2inmx,

n=l

F is called the “diaphony” of the sequence X [ll-151; note that

; (: -xn) =i’E(cx, N, X) da. i7=l

2. The van der Corput sequences

In the b-adic systems (b integer, b >, 2)

For n 2 1, write n - 1 = CT&aj(n)bj; let 2 = (Uj)i be a sequence of permutations of the set

(0, l,..., b - l}; then the generalized van der Corput sequence in base b is defined by

M uj(aj(n)) %w = c bit' * j=O

In the B-adic systems (variable bases)

Let B=(bj)j,o with b,= 1 and bja 2 for j>, 1; let E=(aj)j,O with ~,a permutation of the set (0, l,..., bj+l - l}; for n > 1 write n - 1 = Cy&,aj(n)Bj with Bj = FIiCObi; then the generalized van der Corput sequence in variable base B is defined by

O” Uj(ajCn)> ‘if@) = c

j=O B,

/+I

(see PI).

Symmetrization of sequences (Proinov [I I-l 51) Given a sequence X, the symmetric sequence X deduced from X is the sequence (x1, l-

Xl, x2, 1 - x2, . . . ), that is the sequence X on odd subscripts and the sequence (1 - x,), on even subscripts. Symmetrization of (noL) and of van der Corput sequences give the best L2-discrepan- ties currently known in one dimension (order of magnitude: /m).

3. Functions connected with a pair (b, a)

Let 2: = (u(O)/b, u(l)/b, . . . , a( b - 1)/b); for any integer h such that 0 < h < b - 1, the function (Pi = ~1,~ is defined as follows. Let k be an integer with 1 < k 6 b; then for

H. Faure / Reduction of discrepancy 99

x E [(k - 1)/b, k/b[ set:

cph(x) =A(;, k, 2;) -hx ifO<h<a(k-l),

(P~(x)=(~-~)x--A ;,l[; k, 2; )

if a(k-l)<h<b.

Thus the function (P;,~ is piecewise defined on [0, 11; then, it is extended to the reals by periodicity. These functions are continuous and piecewise affine; the absolute value of the coefficients involved in the affine functions are less than b - 1 (or equal to); note that ‘pa = 0. To a pair (b, a) are therefore associated b functions ‘pa = 0, (pl, (p2,. . . , (P~_.~; these functions are the keys to express the error E( (Y, N, S,“) and to obtain useful formulas for the discrepancies of the sequences Sz and $g (at present we only have partial results on $z).

The function Q& g ives rise to other functions according to the concerned discrepancy:

GE’+ = mh+&), W = mh=(-cp;,A 1c/; = 4;Tb.+ + #“b’_ ;

(these ones are positive because ‘po = 0); these functions are related to the extreme discrepancies

121; b-l

h=Q

are related to the L2-discrepancy

4. Discrepancy formulas

b-l

c d h=O

and to the diaphony.

Extreme discrepancy of the sequences Sg

D+(N, S:) = sup(E(a, N, $)) = ,+I,+( ;)p a

D-(N, S;) = sup( -+, N, $)) = J+l,-( $), a

In the series above, the end of the summation is geometric. It seems that the consideration of a variable B-adic system does not reduce the discrepancy [2].

L2-discrepancy of the sequences Sz (new result)


or

T2(N, si3 = (Fl $fT (g))’ in particular with a fixed permutation u

T2(NY %)= (~,~1~~(~))2+

+ O(log N);

O(log N).

Again the end of the summation is geometric. In the case ,S,O, it is easy to obtain a formula with finite summation in order to compute the

L2-discrepancy.

Diaphony of the sequence Sz (new result)

this formula is deduced from T2 by calculating

J ( 0

‘,q a, N, S;) dcu = E ‘V$-’ j=l b' J

Ll -discrepancy of the sequences SL (new result) (I denotes the identical permutation)

(here the error E((Y, N, S,‘) is positive and so the Ll-discrepancy is the integral of the error). At present we do not know any formula for the Ll-discrepancy of Sz and even of Sz.

L2-discrepancy of the sequences $z This family gives the best sequences currently known, with \I- for the magnitude order of

T. Its study is more difficult and we have only partial results at present.

T2(N, $) = 5 B,(N)d( 5) j=l

with cp = cp;,i and B,(N)=1-2rp(2&;(N+l))

(note that 0 =S 6’,(N) < 1) [5&l.

T2(jV, $) = E (ai(N)qi( &) + bj(N)(PlqZ( &) +yjCN)(?‘i( &))y j=O

with v1 = Q& v2 = (P:,~ and aj, pj, yj functions with the properties:

O<aj(N)<+> -$<fij(N)<O, OGyj(N)<f (new result).


In the case $‘, we expect a formula involving the ((~1,~)~ and some of the product C& x g&

with negative coefficients for the last ones. Numerical experiments with various (b, a) allow to expect smaller constants (with the best

order of magnitude \/log) than those obtained with $ and S{. Moreover, I’m expecting the upper bound

5. Asymptotic results

We set

s(x)= isi WY x) D*w, x> N*m log N ’

s*(x)= iii% N-a, log N ’

WC X) tJX) = lim log N ,

7 t2(X)= lim T2(K X>

N+CC N-C0 log N

(and also

d’P’( x) = lim Dcp'(N, X)

N-CC log N ’

so dC2) = tl).

Extreme discrepancy Let

then

4 NN, %> =G log log N+max 2, ajJ+l+k (-G and s(S,“) = log;

I b2

if b is even, &q) =J*(sb() = 4;“:” log b

4 log b if b is odd.

Vb 3a such that s(Sl) G l/log 2 = 1.44.. . (new result) (there is an algorithm to construct u for a given b).

For D, best permutations are calculated for 2 G b G 20 and the best one is obtained for b = 12; there are analogous results for D* [2].


L2-discrepancy (new results) Let

then

b2+b-2 if b is even,

f&S;) = d(P)($) = “;;‘:’ log b forlgpG2,

8b log b if b is odd,

(this result is shown independently by Proinov and Atanassov [13]). Vb 30 tl(S;) < l/log 2 (deduced from extreme discrepancy with the same permutation a).

Good permutations are known for small b.

421 6750 log 2 = 0.089 < fZ(,T?;) < 0.103 [5,6].

No definitive results for $, but great probability of reducing t, with good permutations (work on hand).

6. Incursion in higher dimensions

Hammersley, Halton, Faure and Niederreiter scrambled sequences with permutations keep their properties relatively to elementary boxes [3,4,7,8] and therefore the bounds for the discrepancy of these sequences with identical permutation are preserved. Unfortunately, at present, there are only upper bounds and the exact order of magnitude is still a conjecture.

Braaten and Weller [l] have performed calculations of LZdiscrepancy of Halton sequences, scrambled Halton sequences (not with the best permutations) and random sequences. They conclude that “scrambling shows drastic improvements over the Halton sequence in high dimensions”. Our study in one dimension, which allows to know the best permutations (or at least good permutations), strengthen their conclusions: scrambling correctly the known sequences will probably give better results in applications than using sequences with identical permutation. (See also [lo] for solving an integral equation for physics by QRS and PRS sequences.)

The proofs will appear elsewhere.

References

[l] E. Braaten and G. Weller, An improved low discrepancy sequence for multidimensional Quasi Monte Carlo integration, J. Comput. Phys. 3.3 (1979) 249-258.


[2] H. Faure, Discrepances de suites associees a un systeme de numeration en dimension un, Bull. Sot. Math. France 109 (1981) 143-182.

[3] H. Faure, Discrepances de suites associees ii un systeme de numeration en dimension s, Acta Arith. XL1 (1982) 337-351.

[4] H. Faure, On the star discrepancy of generalized Hammersley sequences in two dimensions, Monatsh. Math. 101 (1986) 291-300.

[5] H. Faure, Discrepance quadratique de suites infinies en dimension un, in: J.M. de Koninck and C. Levesque, Eds., Comptes Rendus du Collogue Znternat. de Theorie des Nombres de Quebec, Univ. Laval, 1987 (Gruyter, Berlin, 1989).

[6] H. Faure, Discrepance quadratique de la suite de van der Corput et de sa symttrique, Acta Arith., to appear. [7] H. Niederreiter, Points sets and sequences with small discrepancy, Monatsh. Math. 104 (1987) 273-337. [8] H. Niederreiter, Low-discrepancy and low-dispersion sequences, J. Number Theory 30 (1) (1988) 51-70. [9] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (Wiley, New York, 1974).

[lo] M.A. Prasad and P.K. Sarkar, A comparative study of pseudo and quasi random sequences for the solution of integrals equations, J. Comput. Phys. 68 (1987) 66-88.

[ll] P.D. Proinov, On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (9) (1986) 31-34. [12] P.D. Proinov, Symmetrization of the van der Corput generalized sequences, Proc. Japan Acad. Ser. A Math. Sci.

64 (5) (1988) 159-162. [13] P.D. Proinov and E.Y. Atanassov, On the distribution of the van der Corput generalized sequences, C.R. Acad.

Sci. Paris Ser. Z Math. 307 (1988) 895-900. [14] P.D. Proinov and V.S. Grozdanov, Symmetrization of the van der Corput-Halton sequence, C. R. Acad. Bulgare

Sci. 40 (8) (1987) 5-8. [15] P.D. Proinov and V.S. Grozdanov, On the diaphony of the van der Corput-Halton sequence, J. Number Theory

30 (1988) 94-104.

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Using permutations to reduce discrepancy