Convergence Rate Analysis

Abstract

After showing the convergence of the two numerical methods for Frobenius-Perron operators in the previous chapter, we further investigate the convergence rate problem for them. Keller’s stochastic stability result for a class of Markov operators will be studied first, which leads to his first proof of the L¹-norm convergence rate O(ln n/n) for Ulam’s method applied to the Lasota-Yorke class of mappings. Then we introduce Murrary’s work on an explicit upper bound of the convergence rate for Ulam’s method. The convergence rate analysis for the piecewise linear Markov method under the BV -norm will be presented in the last section.