computational investigations of scrambled faure sequences

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Mathematics and Computers in Simulation 81 (2010) 522–535

Computational investigations of scrambled Faure sequences

Bart Vandewoestyne a, Hongmei Chi b, Ronald Cools a,∗a Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium

b Department of Computer and Information Science, Florida A & M University, Tallahassee, FL 32307-5100, USA

Received 7 December 2007; accepted 7 September 2009Available online 20 November 2009

Abstract

The Faure sequence is one of the well-known quasi-random sequences used in quasi-Monte Carlo applications. In its originaland most basic form, the Faure sequence suffers from correlations between different dimensions. These correlations result in poorlydistributed two-dimensional projections. A standard solution to this problem is to use a randomly scrambled version of the Fauresequence. We analyze various scrambling methods and propose a new nonlinear scrambling method, which has similarities withinversive congruential methods for pseudo-random number generation. We demonstrate the usefulness of our scrambling by meansof two-dimensional projections and integration problems.© 2009 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Faure sequence; Low-discrepancy sequences; (quasi)-Monte Carlo; Linear scrambling; Nonlinear scrambling

1. Introduction

The term ‘Monte Carlo (MC) method’ is often used to refer to a well-known family of stochastic algorithmsand techniques for solving a wide variety of problems. It is well-known that the probabilistic error for these MonteCarlo methods converges as O(N−1/2) if information about regularity (or smoothness) is not used. Here, N is thenumber of sample points used. So-called ‘quasi-Monte Carlo (qMC) methods’ [21], based on deterministic point-sets or sequences, form an alternative to MC methods and lead to smaller approximation errors in many practicalsituations. While quasi-random numbers do improve the convergence of applications like numerical integration, it isby no means trivial to provide practical error estimates in qMC due to the fact that the only rigorous error bounds,provided via the Koksma–Hlawka inequality, are very hard to utilize. In fact, the common practice in MC of using apredetermined error criterion as a deterministic termination condition, is almost impossible to achieve in qMC withoutextra technology. In order to provide such dynamic error estimates for qMC methods, several researchers [27,23]proposed the use of Randomized qMC (RqMC) methods [14], where randomness can be brought to bear on quasi-random sequences through scrambling and other related randomization techniques [30,3]. One can rigorously show[16] that under relatively loose conditions each of the randomized qMC rules are statistically independent and thus canbe used to form a traditional MC error estimate using confidence intervals based on the sample variance. The core ofrandomized qMC is a fast and effective algorithm to randomize (scramble) quasi-random sequences.

∗ Corresponding author. Fax: +32 16 32 79 96.E-mail addresses: [email protected] (B. Vandewoestyne), [email protected] (H. Chi), [email protected]

(R. Cools).

0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2009.09.007

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B. Vandewoestyne et al. / Mathematics and Computers in Simulation 81 (2010) 522–535 523

When randomized qMC is used to estimate integration problems, the integration variance can depend stronglyon the scrambling methods [24]. Much of the work dealing with scrambling methods has been aimed at ways oflinear scrambling methods. In this paper, we take a close look at various scrambling methods and propose a nonlinearscrambling method for Faure sequences which we also compare with the linear scrambling methods. The nonlinearscrambling methods will focus on inversive scrambling. We compare these nonlinear scrambling methods with linearscrambling methods by two-dimensional projections, discrepancy and a set of test-integrals.

The organization of this paper is as follows: in Section 2, a brief introduction to the theory of constructing Fauresequences is given. In Section 3, we give an overview of different scrambling methods and we then introduce a nonlinearscrambling method in Section 4. Properties for the two-dimensional projections of this nonlinear scrambling methodare presented in Section 5 and L2-discrepancy computations are reported in Section 6. Numerical integration resultsare given in Section 7 and conclusions follow in Section 8.

2. The Faure Sequence

Before we begin our discussion of the various scrambling methods for the Faure sequence, it behooves us to describein detail the standard and widely accepted methods of Faure sequence generation. We start from the construction ofanother related “classical” quasi-random sequence, namely the Halton sequence.

2.1. Van der Corput and Halton sequences

Let b ≥ 2 be an integer and n a non-negative integer with:

n = nmbm + · · · + n1b + n0

its b-adic representation. Then the nth term of the Van der Corput sequence is

φb(n) = n0

b+ n1

b2 + · · · + nm

bm.

Here φb(n) is the radical inverse function in base b and n = (n0, n1, . . . , nm)T is the digit vector of the b-adic repre-sentation of n. The function φb(·) simply reverses the digit expansion of n and places it to the right of the “decimal”point. The Van der Corput sequence in s dimensions, more commonly called the Halton sequence, is one of the mostbasic quasi-random sequences and its nth point can be written in the following form:

xn = (φb1 (n), φb2 (n), . . . , φbs (n)), (1)

where the bases b1, b2, . . . , bs are pairwise coprime. Note that (1) is a special case of the more general form

xn = (φb1 (C(1)n), φb2 (C(2)n), . . . , φbs (C(s)n)), (2)

where the C(j) are called “generator matrices”. For the Halton sequence, each “generator matrix” C(j) for j = 1, . . . , s

is the identity matrix.

2.2. Faure sequences

By cleverly constructing the generator matrices in (2), one can obtain other quasi-random sequences. Faure [7] setsbj = b for j = 1, . . . , s and uses powers of the upper triangular Pascal matrix modulo b for the generator matrices.The nth element of the Faure sequence is expressed as

xn = (φb(P0n), φb(P1n), . . . , φb(Ps−1n)),

where b is a prime number greater than or equal to the dimension s and P is the Pascal matrix modulo b whose (i, j)-

element is equal to

(j − 1

i − 1

)mod b. The matrix-vector products Pjn for j = 0, . . . , s − 1 are done in modulo b

arithmetic. Fig. 1 illustrates a disadvantage of the original Faure sequence: the above construction leads to a sequencethat has correlations between its individual coordinates. This leads among others to bad two-dimensional projectionsand also has its consequences when the sequence is used for numerical integration, as will be illustrated in Section 7.

524 B. Vandewoestyne et al. / Mathematics and Computers in Simulation 81 (2010) 522–535

Fig. 1. Projection of the first 1025 points from a 40-dimensional Faure sequence in base 41.

A solution to this problem consists of randomly scrambling the Faure sequence. The next section surveys knownscrambling schemes.

3. Scrambling the Faure sequence

3.1. Owen’s general scrambling framework

Owen [22] proposed a scheme based on digit permutations to randomize qMC point-sets and sequences. Letxn = (x(1)

n , x(2)n , . . . , x(s)

n ) ∈ [0, 1)s and zn = (z(1)n , z(2)

n , . . . , z(s)n ) be the scrambled version of xn. Suppose each x(i)

n can

be represented in base b as x(i)n =∑+∞

j=1x(i)n,jb

−j , then the randomized point z(i)n =∑+∞

j=1z(i)n,jb

−j , with z(i)n,j defined in

terms of random permutations of the x(i)n,j:

z(i)n,1 = πi(x

(i)n,1)

z(i)n,2 = π

i, x(i)n,1

(x(i)n,2)

z(i)n,3 = π

i, x(i)n,1, x

(i)n,2

(x(i)n,3)

...

z(i)n,k = π

i, x(i)n,1, x

(i)n,2,...,x

(i)n,k−1

(x(i)n,k).

Each permutation π is uniformly distributed over the b! permutations of {0, 1, . . . , b − 1} and the permutations aremutually independent. The first digit in the base b expansion of x(i)

n is permuted by πi for all n. The second digit

is permuted by the permutation πi, x

(i)n,1

which depends on the value of the first digit x(i)n,1. Similarly, the permutation

applied to the kth digit x(i)n,k depends on the values of the first k − 1 digits x

(i)n,1 through x

(i)n,k−1.

The bookkeeping that is necessary for this scrambling method prevents an easy and effective practical implementa-tion [11]. Therefore, various simplified versions of Owen’s scrambling method have appeared in the literature. Linearscrambling methods [17] are the most commonly used and have the best developed theory. They are easy to imple-ment and allow a fast generation of scrambled sequences. Formulas for linear scrambling methods also show strongsimilarities with the recurrence used for linear congruential pseudo-random number generation, as can be seen in thefollowing subsections.


3.2. GFaure sequence

Tezuka [27,28] proposed the generalized Faure sequence, GFaure, with the jth dimension generator matrixC(j) = A(j)Pj−1 and where the A(j) for j = 1, . . . , s are arbitrary non-singular lower triangular matrices over Fb.An implementation of GFaure in the C programming language is given in [28].

A special case for GFaure is given in [8]. The sequences discussed there use A(j) the lower triangular matrixwith all entries equal to 1 for every j. If A(j) is diagonal only, then this sequence will be the original Faure sequence.According to [9], this choice is not satisfactory.

3.3. Faure sequence with random linear (digit) scrambling

Matousek [17] discusses several simplified versions of Owen’s scrambling in which matrices and shift vectors areused. The points of a sequence constructed with Matousek’s methods have the form

xn = (φb(A(1)P0n + g1), φb(A(2)P1n + g2), . . . , φb(A(s)Ps−1n + gs)).

The matrices A(1), . . . , A(s) are different random scrambling matrices of a certain form. The vectors g1, . . . , gs aredifferent random shift vectors. For the random linear scrambling, we have

A(j) =

⎛⎜⎜⎜⎜⎝

h1,1 0 0 0

h2,1 h2,2 0 0

h3,1 h3,2 h3,3 0...

......

. . .

⎞⎟⎟⎟⎟⎠ , gj =

⎛⎜⎜⎜⎜⎝

g1

g2

g3

...

⎞⎟⎟⎟⎟⎠ ,

where the gj’s and the hi,j’s with i > j are chosen randomly and independently from {0, 1, . . . , b − 1} and the hj,j’sare chosen randomly and independently from {1, 2, . . . , b − 1}. Note that the scrambling method used in Tezuka’sGFaure sequence is a special case of this method. The additive terms gj are omitted for GFaure.

Another special case of the above scrambling is the random linear digit scrambling where the matrices A(j) and thevectors gj have the form

A(j) =

⎛⎜⎜⎜⎜⎝

h1 0 0 0

0 h2 0 0

0 0 h3 0...

......

. . .

⎞⎟⎟⎟⎟⎠ , gj =

⎛⎜⎜⎜⎜⎝

g1

g2

g3

...

⎞⎟⎟⎟⎟⎠ ,

with the hj ∈ {1, 2, . . . , b − 1} and the gj ∈ {0, 1, . . . , b − 1} chosen uniformly and independently at random.

3.4. Faure sequence with I-binomial scrambling

A subset of the family of random linear scrambling methods is called left I-binomial scrambling [29]. Here, the A(j)

and the gj for j = 1, . . . , s are defined as

A(j) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

h1 0 0 0 0

h2 h1 0 0 0

h3 h2 h1 0 0

h4 h3 h2 h1 0

. . .. . .

. . .. . .

. . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, gj =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

g1

g2

g3

g4

...

⎞⎟⎟⎟⎟⎟⎟⎟⎠

,

where h1 is chosen randomly and independently from {1, 2, . . . , b − 1} and hr (r > 1) and gr (r ≥ 1) are chosenrandomly and independently from {0, 1, . . . , b − 1}. The term ‘I-binomial’ refers to a property of the elements in thegenerator matrices (see [29, Section 2.2]).


The diagonal element h1 of A(j) scrambles all digits of each original Faure point. The element h2 scrambles all butthe first digit of that Faure point. Most importantly, the two most significant digits of the Faure point are only scrambledby h1 and h2. Thus the choice of these two elements is crucial for producing optimally scrambled Faure sequences.In [2], Chi considers a simple form for A(j) by setting hj = 0 for j > 2. The choices for h1 and h2 are based on anexample in [18]. This example states that for prime moduli b and multipliers h which are primitive roots modulo b,the discrepancy D

(2)b−1 of a two-dimensional point-set xn = (xn, xn+1) generated with the multiplicative congruential

generator

yn+1 = hyn mod b, xn = yn

b,

is bounded by

(b − 1)D(2)b−1 ≤ 2 +

q∑i=1

ai,

where a1, . . . , aq are the partial quotients in the continued fraction expansion of h /b (with aq = 1). Thus, searching forthe primitive roots modulo b that have the smallest and second smallest sum of partial quotients will lead to multiplicativecongruential generators with the best and second best possible bound for D

(2)b−1. Based on this observation, it is then

hoped for that these multipliers will also give good results when used for scrambling the Faure sequence. We arehowever not aware of any supporting theory for this assumption.

3.5. Striped matrix scrambling

Owen [24] proposed a scrambling method with matrices A(j) and shift vectors gj of the form

A(j) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

h1 0 0 0 0

h1 h2 0 0 0

h1 h2 h3 0 0

h1 h2 h3 h4 0...

......

.... . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, gj =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

g1

g2

g3

g4

...

⎞⎟⎟⎟⎟⎟⎟⎟⎠

,

where the hi are chosen randomly and independently from {1, 2, . . . , b − 1} and the gi are chosen randomly andindependently from {0, 1, . . . , b − 1}. Owen essentially only studied this type of scrambling in one dimension andindicates that it is not clear how well this scrambling method performs for dimensions higher than one.

3.6. Optimal scrambling

In a recent work, Faure [9] proposed a method to choose subsets of Zb of various sizes for the random selection ofthe entries of the scrambling matrices A(j), which he takes to be diagonal. The idea behind this is similar to what isdone in [3]. After random multiplication of all rows of the generator matrices by good selected factors f, scrambled(0, s)-sequences are obtained whose one-dimensional coordinate projections have better properties than the originalones generated by the powers of the Pascal matrix. Note that the results in [9] are for one-dimensional sequencesonly and one cannot infer that left multiplication by scrambling matrices with the selected entries will also give goodresults in the s-dimensional case. However, the hope is that scrambling by matrices carried out from a subset of Zb

selected by this method, will perform better because it systematically avoids a subset of digits like 1 and b − 1. Wedid not implement this scrambling method because the tables in [9] only list multiplicative factors f for the bases 2, 3,5, 7 and 367 and the sequences used in our numerical experiments require information for more than only these basesbecause their dimension goes up to 40. We do however follow the suggestion given at the end of [9], namely that ofimplementing and testing random linear (digit) scrambling, with and without additive term.


4. Nonlinear scrambling

The linear scrambling methods considered so far all show strong similarities with the linear congruential methodfor generating pseudo-random numbers, first introduced by Lehmer in 1949 [15]. Nonlinear congruential methods area more recent alternative for generating pseudo-random numbers. Surveys of such nonlinear methods can be found in[19,20]. One example is the inversive congruential method [21,6,5]. For c ∈Zp, define c−1 ∈Zp by cc−1 = 1 mod p

if c /= 0 and c−1 = 0 if c = 0. Then, for parameters a, b ∈Zp with a /= 0 and an initial value y0 ∈Zp, the sequencey0, y1, . . . ∈Zp generated by the recursion

yn+1 = ay−1n + b mod p for n = 0, 1, . . .

is called an inversive congruential generator modulo p and the corresponding pseudo-random numbers xn = yn/p arecalled inversive congruential pseudo-random numbers.

Fig. 2. Six two-dimensional projections from a 40-dimensional GFaure sequence consisting of 1024 points.


In this section, we will use an idea based on this nonlinear congruential method to propose another scramblingmethod for the Faure sequence for which the jth coordinate of the nth point has the general form:

x(j)n = φb

(�−1

(A(j)�

(Pj−1n

)+ gj

)), (3)

where �(x) and �(x) are bijections that map a digit vector x to another digit-vector. The matrices A(j) and the vectorsgj for j = 1, . . . , s may be chosen as in the previously discussed scrambling methods. Note that when random lineardigit scrambling is combined with the bijections � and �, this form of scrambling becomes a special case of thegeneralized linear random scrambling as given in Definition 2.3 of [24].

Inspired by the inversive congruential method for the generation of pseudo-random numbers, we choose bijectionsthat map the non-zero digits of a digit-vector to their multiplicative inverse modulo b and which leave the zero-digitsunchanged. We will call this “inversive scrambling”. If � is applied and we take � to be the identity mapping, thenthe non-zero digits are first replaced by their inverse modulo b and a random linear (digit) scrambling is applied. We

Fig. 3. Six two-dimensional projections from a 40-dimensional Faure sequence consisting of 1024 points and with pre-inversive scrambling applied.


denote this by “pre-inversive scrambling”. If � is the identity mapping and � is applied, then first a random linear(digit) scrambling is constructed and the resulting non-zero digits are replaced by their inverse modulo b afterwards.We will call this “post-inversive scrambling”. To compute the multiplicative inverse modulo b one can use the extendedEuclidean algorithm (see for example [4]).

5. Two-dimensional projections

To get a first idea of the quality of a randomly scrambled version of the Faure sequence, one can look at itstwo-dimensional projections. To illustrate the potential power of inversive scrambling, Figs. 2 and 3 show six two-dimensional projections of 1024 points of a 40-dimensional GFaure sequence respectively pre-inversive scrambled

Fig. 4. Sixteen points of a Faure sequence (×) and a pre-inversive scrambled Faure sequence (�) in two dimensions. The elementary intervalproperties are still valid after the scrambling.


Fig. 5. Averaged L2-discrepancy for various randomized scramblings of a 20-dimensional Faure sequence.

Faure sequence. As can be seen from the figures, the GFaure sequence has a lot of structure in its different two-dimensional projections. The projections of the Faure sequence with pre-inversive scrambling do not suffer from thisproblem. Because of the large number of possible two-dimensional projections, we decided to show only the subsetsfrom Figs. 2 and 3. However, we have looked at many other possible two-dimensional projections of the GFaure andpre-inversive scrambled Faure sequence and the conclusions to be drawn were always the same.

Fig. 6. Error for integrations with a Faure sequence started at n = QS4 − 1.


If a (t, m, s)-net or (t, s)-sequence (see [21] for a definition) is scrambled within the inversive scrambling framework,then the resulting net or sequence is still a (t, m, s)-net respectively (t, s)-sequence. This is illustrated in Fig. 4 whichshows 16 points of a two-dimensional pre-inversive scrambled Faure sequence. Actually, the figure shows the pre-inversive scrambled counterpart of Figure 2 in page 1755 of [26]. It illustrates how the elementary interval propertiesare still valid when inversive scrambling is used. The reason for this is that the mapping of a non-zero digit to itsmultiplicative inverse modulo b is a bijection and thus defines a permutation of the digits.

Also, note the similarities with Niederreiter’s general construction principle in [21, Section 4.3]. Most authors focuson the elements c

(i)j,r that determine the generator matrices and the bijections are often taken as the identity mapping.

We propose to take non-trivial bijections such as the one that maps a non-zero digit to its multiplicative inverse modulothe base b.

6. Discrepancy

When further comparing scrambled versions of the Faure sequence, it is convenient to have a computable measureat hand. Discrepancy is a measure often used to compare different point-sets. One of the reasons for this is that worstcase integration error over a class of integrands can be described by different discrepancy measures. Error bounds of

Fig. 7. Relative error against number of sample points for test-integral (5) in 5 (top), 20 (middle) and 40 (bottom) dimensions.


the form

|I − I| ≤ D‖f‖,exist, where D is a certain discrepancy and ‖f‖ is a compatible norm or semi-norm on functions. Hickernell [13]obtained the following expression for the squared L2-star discrepancy of an s-dimensional point-set P:

(D∗

2(P))2 =

(4

3

)s

− 2

N

N−1∑i=0

s∏j=1

(3

2− xi,j

2

)+ 1

N2

N−1∑i=0

N−1∑k=0

s∏j=1

[2 − max(xi,j, xk,j)

]. (4)

The expectation of (4) over random xi is the mean squared L2-star discrepancy and is studied in [12,17]. It is shownthat the root mean square L2-discrepancy of (0, m, s)-nets is O(N−1(log(N))(s−1)/2). Also, many of the scramblingschemes leave the expected value of (4) unchanged [12,17]. Owen [24] illustrates this for the case s = 1.

In an attempt to compare the quality of the different scrambling schemes by using a computable measure, we havecomputed the average L2-star discrepancy for various random scramblings of a 20-dimensional Faure sequence. Keep-ing in mind the computational cost ofO(sN2), we decided to compute the averageL2-star discrepancy of 3 runs for eachscrambling method. Each run used different random scrambling matrices and (if applicable) shift vectors. The resultsare shown in Fig. 5. The figure clearly illustrates how for N large, all discrepancy curves tend to fall together. As Owenalready mentioned in [24], it is impossible to decide which scrambling method performs best based on this criterion.

Fig. 8. Relative error against number of sample points for test-integral (6) and parameters ai = 1 for i = 1, . . . , s in 5 (top), 20 (middle) and 40(bottom) dimensions.


7. Numerical integration

Because many of the scrambling schemes leave the expected value of (4) unchanged, other ways to compare thequality of the sequences have to be considered. A method to do this is to apply the sequences in high-dimensionalintegration problems [1].

Consider the following test-integrals:

I1(f ) =∫ 1

0· · ·∫ 1

0

s∏i=1

π

2(sin πxi) dx1 . . . dxs = 1, (5)

I2(f ) =∫ 1

0· · ·∫ 1

0

s∏i=1

|4xi − 2| + ai

1 + ai

dx1 . . . dxs = 1, (6)

where the ai are parameters. The integrand in (6) allows a tuning of the relative importance of the variables, as wellas of their interactions, by appropriate choices of the parameters. The key-concept used for this is effective dimension,which is also closely related to the idea of sensitivity indices [25]. We refer to [31] for more details. The effectivedimension of the function in (6) is tabulated in [31]. Two choices of the parameters will be considered:

Fig. 9. Relative error against number of sample points for test-integral (6) and parameters ai = i2 for i = 1, . . . , s in 5 (top), 20 (middle) and 40(bottom) dimensions.


(1) ai = 1 for 1 ≤ i ≤ s,(2) ai = i2 for 1 ≤ i ≤ s.

For the first choice of parameters, all variables are equally important and the truncation dimension is approximatelythe same as the nominal dimension. This is the most difficult case for numerical integration. For the second choiceof ai, the importance of the successive variables is decreasing. In general, when ai becomes bigger, the variables aredecreasing quickly in importance and the effective dimension becomes smaller.

A first thing to mention before discussing other numerical integration results is the importance of the start-index nthat is used for the Faure sequence generation. In qMC applications, it is a common practice to skip a certain amountof points and thereby hope that this will lead to better results. Fox [10] already discussed this issue and decided tostart his sequence at n = QS4 − 1, where QS is the first prime greater than or equal to the dimension s. The ‘terribleresults’ Fox mentioned can be seen in Fig. 6. The upper figure shows the relative integration error as a function ofN for the integral (6) with ai = 1 for i = 1, . . . , s and for a Faure sequence which was started at n = 1. The Fauresequence performs much worse than the different scramblings. The bottom figure uses a start-index n = QS4 − 1.Clearly, skipping the first QS4 − 2 points improves the results a lot here. In the rest of the numerical results, we alwaysstart our (scrambled) Faure sequences at n = QS4 − 1.

The relative integration errors for the test function (5) are given in Fig. 7. The different curves represent the averageintegration error over 20 repetitions for a Faure sequence randomly scrambled with a certain scrambling method. Twoimportant things can be observed. First and foremost, there is the fact that no scrambling method outperforms theothers, all scrambling methods have similar behavior. Secondly, for the given range of N the five-dimensional resultsshow an O(1/N) behavior, but with increasing dimension the convergence (for the given range of N) becomes slower.

Similar observations can be made for the second test function with ai = 1, given in Fig. 8. For this function, it isknown that the effective dimension is almost equal to the nominal dimension. All scrambling methods perform equallywell, but the rate of convergence decreases with increasing dimension.

Fig. 9, which represents the results for test function (6) with ai = i2, illustrates how a low effective dimensionis the key to success: even in 40 dimensions one still observes O(1/N) convergence for the range of N taken intoconsideration. However, again no scrambling method seems to outperform the others.

8. Conclusions

A first conclusion to be drawn from our experiments is that if one generates a Faure sequence without any skippedpoints, then the scrambled sequences perform significantly better than the original Faure sequence.

Secondly, we propose a nonlinear scrambling scheme that offers an alternative way for scrambling Faure sequences.Computational investigations show that this nonlinear scrambling achieves at least the same performance in estimatingintegrals compared to the already existing linear scrambling methods. Also, the nonlinear scrambling scheme preservesthe net-structure and due to its nonlinearity has the additional advantage that its two-dimensional projections do notshow the regular structure that is sometimes visible with other linear scrambling methods. This nonlinear scramblingmethod is suitable for other quasi-random sequences as well.

Using discrepancy computations and the evaluation of numerical integrals, we did not find any significant differencesbetween the different scrambling methods. The convergence for all methods was however strikingly better for theintegrand with low effective dimension.

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computational investigations of scrambled faure sequences

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