The New Scramble for Faure Sequence Based on Irrational Numbers

Ali Mogharrabi O,R Farnoosh,Behrooz Fathi V,M H Behzadi,Leopoldo Greco

doi:10.1155/2021/6696895

Abstract

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.

Highlights

It is well known that Monte Carlo calculations are based on the generation of random numbers on interval (0,1)
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 generated numbers on (0,1), so that we can estimate the obtained solution to the desired unknown solution of the problem
We studied the original Faure sequence and some of its recent years introduced scrambles

Summary

Introduction

It is well known that Monte Carlo calculations are based on the generation of random numbers on interval (0,1). 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.

The Faure Sequence

Scrambling Matrix with Irrational Members

Investigation of the Uniformity of Generated Sequences

Numerical Integration

Conclusion