Research Article

Monte Carlo and Quasi Monte Carlo Approach to Ulam's Method for Position Dependent Random Maps

Volume: 3 Number: 4 December 22, 2020
Md Shafiqul Islam *
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Communications in Advanced Mathematical Sciences Cover Communications in Advanced Mathematical Sciences
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Monte Carlo and Quasi Monte Carlo Approach to Ulam's Method for Position Dependent Random Maps

Abstract

We consider position random maps $T=\{\tau_1(x),\tau_2(x),\ldots, \tau_K(x); p_1(x),p_2(x),\ldots,p_K(x)\}$ on $I=[0, 1],$ where $\tau_k, k=1, 2, \dots, K$ is non-singular map on $[0,1]$ into $[0, 1]$ and $\{p_1(x),p_2(x),\ldots,p_K(x)\}$ is a set of position dependent probabilities on $[0, 1]$. We assume that the random map $T$ posses a density function $f^*$ of the unique absolutely continuous invariant measure (acim) $\mu^*$. In this paper, first, we present a general numerical algorithm for the approximation of the density function $f^*.$ Moreover, we show that Ulam's method is a special case of the general method. Finally, we describe a Monte-Carlo and a Quasi Monte Carlo implementations of Ulam's method for the approximation of $f^*$. The main advantage of these methods is that we do not need to find the inverse images of subsets under the transformations of the random map $T$.

Keywords

Dynamicalsystems, Invariant measures, Invariant density, Position dependent random maps, Monte Carlo approach, Quasi Monte Carlo approach, Ulam's method

References

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  7. [7] M.S. Islam, Existence, approximation and properties of absolutely continuous invariant measures for random maps, PhD thesis, Concordia University, 2004.
  8. [8] S. Pelikan, Invariant densities for random maps of the interval, Proc. Amer. Math. Soc., 281 (1984), 813 – 825.
  9. [9] A. Lasota, J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973) , 481– 488.
  10. [10] S.M. Ulam, A collection of Mathematical problems Interscience Tracts in pure and Applied Math., 8, Interscience, New York, 1960.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Publication Date

December 22, 2020

Submission Date

April 22, 2020

Acceptance Date

October 15, 2020

Published in Issue

Year 2020 Volume: 3 Number: 4

DOI

https://doi.org/10.33434/cams.725619

IZ

https://izlik.org/JA83RW82ED

Cite

RIS / Bibtex
APA
Islam, M. S. (2020). Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps. Communications in Advanced Mathematical Sciences, 3(4), 173-185. https://doi.org/10.33434/cams.725619
AMA
1.Islam MS. Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps. Communications in Advanced Mathematical Sciences. 2020;3(4):173-185. doi:10.33434/cams.725619
Chicago
Islam, Md Shafiqul. 2020. “Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps”. Communications in Advanced Mathematical Sciences 3 (4): 173-85. https://doi.org/10.33434/cams.725619.
EndNote
Islam MS (December 1, 2020) Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps. Communications in Advanced Mathematical Sciences 3 4 173–185.
IEEE
[1]M. S. Islam, “Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps”, Communications in Advanced Mathematical Sciences, vol. 3, no. 4, pp. 173–185, Dec. 2020, doi: 10.33434/cams.725619.
ISNAD
Islam, Md Shafiqul. “Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps”. Communications in Advanced Mathematical Sciences 3/4 (December 1, 2020): 173-185. https://doi.org/10.33434/cams.725619.
JAMA
1.Islam MS. Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps. Communications in Advanced Mathematical Sciences. 2020;3:173–185.
MLA
Islam, Md Shafiqul. “Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps”. Communications in Advanced Mathematical Sciences, vol. 3, no. 4, Dec. 2020, pp. 173-85, doi:10.33434/cams.725619.
Vancouver
1.Md Shafiqul Islam. Monte Carlo and Quasi Monte Carlo Approach to Ulam’s Method for Position Dependent Random Maps. Communications in Advanced Mathematical Sciences. 2020 Dec. 1;3(4):173-85. doi:10.33434/cams.725619

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