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

Authors : Md Shafiqul ISLAM

Pages : 173-185

Doi:10.33434/cams.725619

View : 15 | Download : 9

Publication Date : 2020-12-22

Article Type : Research Paper

Abstract :We consider position random maps $T=\{\tau_1insert ignore into journalissuearticles values(x);,\tau_2insert ignore into journalissuearticles values(x);,\ldots, \tau_Kinsert ignore into journalissuearticles values(x);; p_1insert ignore into journalissuearticles values(x);,p_2insert ignore into journalissuearticles values(x);,\ldots,p_Kinsert ignore into journalissuearticles values(x);\}$ on $I=[0, 1],$ where $\tau_k, k=1, 2, \dots, K$ is non-singular map on $[0,1]$ into $[0, 1]$ and $\{p_1insert ignore into journalissuearticles values(x);,p_2insert ignore into journalissuearticles values(x);,\ldots,p_Kinsert ignore into journalissuearticles values(x);\}$ 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 insert ignore into journalissuearticles values(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