| Abstract: |
| Let $T=\{\tau_1(x),\tau_2(x),\ldots, \tau_K(x);
p_1(x),p_2(x),\ldots,p_K(x)\}$ be a position dependent random map on $I=[0, 1]$ which posses a unique absolutely continuous invariant measure (acim) $\mu^*$ with density function $f^*.$ In this paper, first, we present a general numerical algorithm for the approximation of the density function $f^*.$ Then, we show that Ulam`s method can be derived as 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 $f^*$. In an Ulam`s method, the entries of the Ulam`s marix are calculated using inverse images of intervals under the transformations $\tau_k, k=1, 2, \dots, K$ of the randm map $T$. The main advantage of Monte-Carlo and Quasi Monte Carlo approach to Ulam`s method is that we do not need to find the inverse images of subsets under the transformations of the random map $T.$ We consider different types of random maps and compare the performances of the Monte Carlo and the Quasi Monte Carlo approached to Ulam`s method with examples. Our numerical schemes are generalizations of numerical schemes described in \cite {DW} and \cite{H} of single deterministic maps to numerical schemes for position dependent random maps. |
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