Monte Carlo and Quasi Monte Carlo Approach to Ulam's Method for Position Dependent Random Maps
Monte Carlo and Quasi Monte Carlo Approach to Ulam's Method for Position Dependent Random Maps
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,
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- ISSN: 2651-4001
- Yayın Aralığı: Yılda 4 Sayı
- Başlangıç: 2018
- Yayıncı: Emrah Evren KARA
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