An Elementary Approach to Rigorous Approximation of Invariant Measures
An Elementary Approach to Rigorous Approximation of Invariant Measures
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DOI：
10.1137/130911044
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发表时间：
2014-01-01
影响因子：
2.1
通讯作者：
Nisoli, Isaia
中科院分区：
文献类型：
--
作者：
Galatolo, Stefano;Nisoli, Isaia
我们描述了一个框架，在这个框架中，可以开发和实现具有给定近似误差界的动力系统不变测度的近似算法。我们的方法是基于对赋范向量空间之间的算子不动点近似的一般陈述，允许对误差进行显式估计。我们通过将其应用于分段展开的地图和具有不动点的地图来展示我们方法的灵活性。我们展示了如何实现所需的估计来计算L-1或l -∞距离中给定误差的不变密度。我们还展示了如何使用它来计算这些系统的熵的具有认证误差的估计。我们展示了如何解决几个相关的计算和数值问题，以获得在一些一维映射上的工作实现和实验结果。
We describe a framework in which it is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general statement on the approximation of fixed points for operators between normed vector spaces, allowing an explicit estimation of the error. We show the flexibility of our approach by applying it to piecewise expanding maps and to maps with indifferent fixed points. We show how the required estimations can be implemented to compute invariant densities up to a given error in the L-1 or L-infinity distance. We also show how to use this to compute an estimation with certified error for the entropy of those systems. We show how several related computational and numerical issues can be solved to obtain working implementations and experimental results on some one dimensional maps.