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This article intends to review quasirandom sequences, especially the Faure sequence to introduce a new version of scrambled of this sequence based on irrational numbers, as follows to prove the success of this version of the random number sequence generator and use it in future calculations. We introduce this scramble of the Faure sequence and show the performance of this sequence in employed numerical codes to obtain successful test integrals. Here, we define a scrambling matrix so that its elements are irrational numbers. In addition, a new form of radical inverse function has been defined, which by combining it with our new matrix, we will have a sequence that not only has a better close uniform distribution than the previous sequences but also is a more accurate and efficient tool in estimating test integrals.

It is well known that Monte Carlo calculations are based on the generation of random numbers on interval (0,1). Therefore, the generation of random numbers that have more uniformity on (0,1) guarantees better approximations in these calculations. In recent years, some researchers have employed quasirandom sequences instead of random numbers to aim producing extra uniformity of the randomly generated numbers on (0,1). Due to the breadth and complexity of some problems that are mostly unsolvable by classical mathematical methods or solving them with classical methods is associated with more time and computational cost, the stochastic solving of such cases with numerical methods and using the Monte Carlo method plays a key role. The quasirandom sequences are common in Monte Carlo calculations such as Faure, Halton, Niederreiter, and Sobol sequences, but due to the lack of complete success of these sequences in Monte Carlo computation, we use scrambled versions of them, all of which are designed to increase the uniformity of randomly (quasirandom) generated numbers on (0,1), so that we can estimate the obtained solution to the desired unknown solution of the problem.

To resolve this problem, researchers are competing on the use of scrambled quasirandom generators based on their version of random number generation to provide more accurate results in Monte Carlo calculations.

Today, Monte Carlo and quasi-Monte Carlo methods are widely used to solve the computations of physical and mathematical problems. Quasi-Monte Carlo (QMC) methods play an alternative role for Monte Carlo methods. The advantage of these methods is that they use numbers to provide extera uniformity on unit hypercube. This feature has led to the use of these methods to estimate high-dimensional integrals (Niederreiter, 1992; Spanier and Maize, 1994) [

So far, several quasirandom sequences (or low discrepancy sequences) have been introduced for the QMC method. Such as the Faure sequence, the Halton sequence, and the Sobol sequence. Despite the fact that among these three sequences, the Faure sequence has better features in terms of discrepancy bound, but in practice, it is less used. Because, the convergence rate of this class of sequences is not so good compared with the other sequences [

512 points from the original Faure sequence in several bases.

In the next section, the structure of the original Faure sequence is given. We then briefly list the scramblers that have already been introduced in Section

Suppose

For the Faure sequence, we define a different generator matrix for each dimension. If

Thus, let

Since the introduction of the Faure sequence, several methods were proposed to scramble it. In this section, we give an overview of some of such scrambles.

Tezuka [

After reviewing different versions of the Owen’s method, Matoušek introduced a scramble matrix and a transfer vector for various dimensions [

For the random linear scrambling, the matrices

Random linear digit method is the basis of other scrambles that will follow. Even the GFaure method is a subset of this method in which the members of the shift vectors are all zero.

A subset of the family of random linear scrambling methods is called left I-binomial scrambling [

The scrambling matrix

When we use the I-binomial method to scramble the Faure sequence, the value of (number)

So by cleverly selecting these two members, we can achieve better Faure sequences.

In [

Fathi and Eskandari [

Based on nonlinear congruential method, they proposed another scrambling method for the Faure sequence for which the

In this section, corresponding to the method of random linear digits, we introduce a scrambling matrix that its members are a function of square root of base

After testing many functions, we found the following function that has the most performance:

In (

We call this as

So, for example, we denote the 40-dimensional Faure sequence generated on the base 41 by the scrambled matrix

In the following sections, we have examined (studied) the quality of this sequence along with its performance compared to other sequences.

The first step in evaluating the performance of a sequence is to see how the points in the 2D projections are distributed. From Figure

1000 points from the original Faure sequence.

1000 points from our scrambled Faure sequence.

These figures show that two-dimensional projections of

One way to measure the quality of a sequence is to calculate its discrepancy [

Figure

Another way to compare the quality of sequences is to use them to solve high-dimensional integration problems with numerical methods. Consider the following test integrals:

Note that the most difficult case is when

In numerical solution of problems with qMC methods, an accepted procedure is to omit the starting points of the sequence. For example, Fox [

Therefore, we skip the first 41 points and start

Now, we compare the numerical results of different scrambled Faure sequences presented in this paper. The estimated values for the test functions are given in Figures

Estimates of the integral

Estimates of the integral

Estimates of the integral

Estimates of the integral

An observation is that estimated values by the matrix Aj-41rev very close to the actual value. This can be seen in Tables

Estimates of

Generator | ||||||
---|---|---|---|---|---|---|

Faure | 500 | 1.2175 | 2.2301 | 13.402 | 2.6464 | 0.0001 |

IB | 500 | 0.0017 | 0.0016 | 0.0014 | 0.0006 | 0.0002 |

RLD | 500 | 0.0013 | 0.0013 | 0.0011 | 0.0006 | 0.0001 |

Aj-41rev | 500 | 1.0392 | 1.0439 | 1.5286 | 0.8095 | 1.1542 |

Faure | 5000 | 0.9803 | 1.0128 | 3.1585 | 0.7848 | 0.0314 |

IB | 5000 | 0.0340 | 0.0397 | 0.0443 | 0.0402 | 0.0306 |

RLD | 5000 | 0.0469 | 0.0475 | 0.0548 | 0.0737 | 0.0341 |

Aj-41rev | 5000 | 1.0015 | 1.0154 | 1.0219 | 1.1401 | 0.6935 |

Faure | 10000 | 0.9584 | 0.9445 | 1.9919 | 0.6583 | 0.0464 |

IB | 10000 | 0.0569 | 0.0564 | 0.0593 | 0.0774 | 0.0496 |

RLD | 10000 | 0.0539 | 0.0541 | 0.0566 | 0.0758 | 0.0418 |

Aj-41rev | 10000 | 0.9992 | 0.9959 | 1.0182 | 0.8780 | 0.5630 |

Faure | 20000 | 0.9943 | 1.0522 | 1.4909 | 0.5440 | 0.0575 |

IB | 20000 | 0.0610 | 0.0607 | 0.0628 | 0.0791 | 0.0561 |

RLD | 20000 | 0.0583 | 0.0585 | 0.0592 | 0.0670 | 0.0637 |

Aj-41rev | 20000 | 0.9999 | 1.0021 | 1.0614 | 0.9301 | 0.7249 |

Faure | 50000 | 0.9965 | 1.0127 | 0.9972 | 0.5159 | 0.0674 |

IB | 50000 | 0.0617 | 0.0618 | 0.0621 | 0.0683 | 0.0628 |

RLD | 50000 | 0.0604 | 0.0605 | 0.0582 | 0.0657 | 0.0587 |

Aj-41rev | 50000 | 0.9984 | 0.9983 | 1.0241 | 0.9901 | 0.9173 |

Faure | 70000 | 0.9964 | 1.0058 | 0.9680 | 0.5029 | 0.0669 |

IB | 70000 | 0.0809 | 0.0806 | 0.0877 | 0.2000 | 2.3467 |

RLD | 70000 | 0.0711 | 0.0713 | 0.0671 | 0.0702 | 0.0583 |

Aj-41rev | 70000 | 0.9985 | 0.9954 | 1.0260 | 1.0841 | 0.9623 |

Faure | 100000 | 1.0008 | 1.0248 | 1.0459 | 0.5466 | 0.1665 |

IB | 100000 | 0.4610 | 0.4616 | 0.4667 | 0.4694 | 1.7654 |

RLD | 100000 | 0.2742 | 0.2746 | 0.2772 | 0.2529 | 0.1366 |

Aj-41rev | 100000 | 0.9979 | 1.0007 | 1.0215 | 1.1251 | 1.3969 |

Estimates of

Generator | ||||||
---|---|---|---|---|---|---|

Faure | 500 | 1.0759 | 1.2370 | 1.7994 | 0.6913 | 0.1727 |

IB | 500 | 1.5015 | 1.5000 | 1.5572 | 1.3445 | 1.1510 |

RLD | 500 | 1.5123 | 1.4820 | 1.5456 | 1.4661 | 1.2191 |

Aj-41rev | 500 | 0.9879 | 0.9014 | 0.9347 | 0.8930 | 0.9629 |

Faure | 5000 | 0.9963 | 0.9908 | 1.0342 | 0.7406 | 0.4475 |

IB | 5000 | 1.4836 | 1.4785 | 1.4787 | 1.4212 | 1.2841 |

RLD | 5000 | 1.4807 | 1.4770 | 1.4840 | 1.4455 | 1.2824 |

Aj-41rev | 5000 | 0.9976 | 0.9840 | 0.9964 | 0.9898 | 1.0902 |

Faure | 10000 | 0.9935 | 0.9864 | 1.0062 | 0.7860 | 0.5359 |

IB | 10000 | 1.4769 | 1.4750 | 1.4697 | 1.4370 | 1.3296 |

RLD | 10000 | 1.4779 | 1.4759 | 1.4784 | 1.4438 | 1.3261 |

Aj-41rev | 10000 | 0.9990 | 0.9865 | 0.9925 | 0.9968 | 1.0920 |

Faure | 20000 | 0.9945 | 0.9954 | 0.9861 | 0.8140 | 0.5897 |

IB | 20000 | 1.4753 | 1.4755 | 1.4706 | 1.4244 | 1.3306 |

RLD | 20000 | 1.4763 | 1.4752 | 1.4640 | 1.4288 | 1.3271 |

Aj-41rev | 20000 | 1.0000 | 1.0002 | 1.0151 | 1.0336 | 1.0965 |

Faure | 50000 | 0.9992 | 0.9969 | 0.9609 | 0.8397 | 0.6360 |

IB | 50000 | 1.4750 | 1.4744 | 1.4697 | 1.4397 | 1.3342 |

RLD | 50000 | 1.4755 | 1.4751 | 1.4725 | 1.4398 | 1.3372 |

Aj-41rev | 50000 | 1.0015 | 1.0039 | 1.0192 | 1.0222 | 1.0509 |

Faure | 70000 | 0.9983 | 0.9935 | 0.9613 | 0.8428 | 0.6458 |

IB | 70000 | 1.4654 | 1.4650 | 1.4582 | 1.4249 | 1.3238 |

RLD | 70000 | 1.4710 | 1.4705 | 1.4653 | 1.4362 | 1.3491 |

Aj-41rev | 70000 | 1.0020 | 1.0040 | 1.0237 | 1.0418 | 1.1168 |

Faure | 100000 | 0.9997 | 1.0003 | 0.9798 | 0.8856 | 0.7312 |

IB | 100000 | 1.2785 | 1.2783 | 1.2727 | 1.2476 | 1.1649 |

RLD | 100000 | 1.3848 | 1.3844 | 1.3821 | 1.3508 | 1.2698 |

Aj-41rev | 100000 | 1.0027 | 1.0050 | 1.0233 | 1.0371 | 1.1048 |

Estimates of

Generator | ||||||
---|---|---|---|---|---|---|

Faure | 500 | 1.0243 | 1.0200 | 1.0077 | 0.9945 | 0.9817 |

IB | 500 | 1.5010 | 1.5022 | 1.5031 | 1.5030 | 1.5023 |

RLD | 500 | 1.5055 | 1.5029 | 1.5043 | 1.5051 | 1.5045 |

Aj-41rev | 500 | 0.9951 | 0.9746 | 0.9768 | 0.9761 | 0.9770 |

Faure | 5000 | 0.9972 | 0.9944 | 0.9908 | 0.9877 | 0.9851 |

IB | 5000 | 1.4837 | 1.4835 | 1.4833 | 1.4833 | 1.4832 |

RLD | 5000 | 1.4810 | 1.4809 | 1.4807 | 1.4806 | 1.4804 |

Aj-41rev | 5000 | 0.9990 | 0.9957 | 0.9963 | 0.9968 | 0.9974 |

Faure | 10000 | 0.9977 | 0.9964 | 0.9946 | 0.9929 | 0.9915 |

IB | 10000 | 1.4770 | 1.4769 | 1.4768 | 1.4768 | 1.4766 |

RLD | 10000 | 1.4781 | 1.4780 | 1.4779 | 1.4778 | 1.4777 |

Aj-41rev | 10000 | 0.9998 | 0.9971 | 0.9973 | 0.9977 | 0.9984 |

Faure | 20000 | 0.9979 | 0.9977 | 0.9965 | 0.9956 | 0.9949 |

IB | 20000 | 1.4752 | 1.4753 | 1.4753 | 1.4752 | 1.4752 |

RLD | 20000 | 1.4764 | 1.4763 | 1.4763 | 1.4762 | 1.4762 |

Aj-41rev | 20000 | 1.0002 | 1.0002 | 1.0007 | 1.0012 | 1.0017 |

Faure | 50000 | 0.9999 | 0.9996 | 0.9991 | 0.9987 | 0.9983 |

IB | 50000 | 1.4750 | 1.4750 | 1.4750 | 1.4750 | 1.4749 |

RLD | 50000 | 1.4755 | 1.4755 | 1.4755 | 1.4755 | 1.4755 |

Aj-41rev | 50000 | 1.0009 | 1.0012 | 1.0019 | 1.0024 | 1.0027 |

Faure | 70000 | 0.9993 | 0.9989 | 0.9986 | 0.9983 | 0.9980 |

IB | 70000 | 1.4654 | 1.4654 | 1.4654 | 1.4654 | 1.4653 |

RLD | 70000 | 1.4710 | 1.4710 | 1.4709 | 1.4709 | 1.4709 |

Aj-41rev | 70000 | 1.0011 | 1.0013 | 1.0021 | 1.0026 | 1.0029 |

Faure | 100000 | 0.9997 | 0.9996 | 0.9993 | 0.9990 | 0.9989 |

IB | 100000 | 1.2785 | 1.2785 | 1.2785 | 1.2785 | 1.2784 |

RLD | 100000 | 1.3848 | 1.3848 | 1.3848 | 1.3848 | 1.3848 |

Aj-41rev | 100000 | 1.0013 | 1.0016 | 1.0024 | 1.0028 | 1.0032 |

Estimates of

Generator | ||||||
---|---|---|---|---|---|---|

Faure | 500 | 1.0125 | 1.0115 | 1.0110 | 1.0108 | 1.0106 |

IB | 500 | 1.5003 | 1.5005 | 1.5006 | 1.5006 | 1.5006 |

RLD | 500 | 1.5017 | 1.5015 | 1.5015 | 1.5015 | 1.5015 |

Aj-41rev | 500 | 0.9966 | 0.9936 | 0.9936 | 0.9936 | 0.9936 |

Faure | 5000 | 0.9988 | 0.9986 | 0.9985 | 0.9984 | 0.9984 |

IB | 5000 | 1.4838 | 1.4838 | 1.4838 | 1.4838 | 1.4838 |

RLD | 5000 | 1.4810 | 1.4810 | 1.4810 | 1.4810 | 1.4810 |

Aj-41rev | 5000 | 0.9993 | 0.9989 | 0.9989 | 0.9989 | 0.9989 |

Faure | 10000 | 0.9991 | 0.9990 | 0.9989 | 0.9989 | 0.9988 |

IB | 10000 | 1.4770 | 1.4770 | 1.4770 | 1.4770 | 1.4770 |

RLD | 10000 | 1.4781 | 1.4781 | 1.4781 | 1.4781 | 1.4781 |

Aj-41rev | 10000 | 1.0000 | 0.9996 | 0.9996 | 0.9996 | 0.9996 |

Faure | 20000 | 0.9992 | 0.9991 | 0.9991 | 0.9991 | 0.9991 |

IB | 20000 | 1.4753 | 1.4753 | 1.4753 | 1.4753 | 1.4753 |

RLD | 20000 | 1.4764 | 1.4764 | 1.4764 | 1.4764 | 1.4764 |

Aj-41rev | 20000 | 1.0003 | 1.0002 | 1.0003 | 1.0003 | 1.0003 |

Faure | 50000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

IB | 50000 | 1.4750 | 1.4750 | 1.4750 | 1.4750 | 1.4750 |

RLD | 50000 | 1.4755 | 1.4755 | 1.4755 | 1.4755 | 1.4755 |

Aj-41rev | 50000 | 1.0006 | 1.0006 | 1.0007 | 1.0007 | 1.0007 |

Faure | 70000 | 0.9998 | 0.9997 | 0.9997 | 0.9997 | 0.9997 |

IB | 70000 | 1.4655 | 1.4655 | 1.4655 | 1.4655 | 1.4655 |

RLD | 70000 | 1.4710 | 1.4710 | 1.4710 | 1.4710 | 1.4710 |

Aj-41rev | 70000 | 1.0006 | 1.0006 | 1.0007 | 1.0007 | 1.0007 |

Faure | 100000 | 0.9998 | 0.9998 | 0.9998 | 0.9998 | 0.9998 |

IB | 100000 | 1.2785 | 1.2785 | 1.2785 | 1.2785 | 1.2785 |

RLD | 100000 | 1.3848 | 1.3848 | 1.3848 | 1.3848 | 1.3848 |

Aj-41rev | 100000 | 1.0007 | 1.0007 | 1.0007 | 1.0008 | 1.0008 |

We studied the original Faure sequence and some of its recent years introduced scrambles. Then, we introduced a new scrambling matrix based on irrational numbers that its elements are function of square root of base

The data is contained in the article itself.

The authors declares that there is no conflict of interest regarding the publication of this paper.