Spectral Galerkin methods for transfer operators in uniformly expanding dynamics
Spectral Galerkin methods for transfer operators in uniformly expanding dynamics
复制标题
DOI：
10.1007/s00211-019-01031-z
复制
发表时间：
2019-06-01
影响因子：
2.1
通讯作者：
Wormell, Caroline
中科院分区：
文献类型：
--
作者：
Wormell, Caroline
马尔可夫扩展映射是一类简单的混沌系统，通常被用作混沌动力学的模型，但现有的数值方法来研究长时间的统计特性，如不变的措施有一个穷人之间的权衡计算工作量和精度。我们开发了一个谱Galerkin方法，这些映射的传输运营商，估计统计量使用有限子矩阵的传输运营商的无限傅立叶或切比雪夫基系数矩阵。这些估计的收敛速度是通过对全传递算子矩阵项的定量界获得的;我们发现该方法仅在多项式大的计算时间内就能达到统计特性的指数精度估计。为了实现这些结果，我们建议和证明两个算法：一个严格验证的算法，和一个快速，更方便的自适应算法。使用第一种算法，我们证明严格的界限上的一些样本量，大大比以前更准确。我们表明，自适应算法可以产生双浮点精度估计在一秒钟的一小部分在个人电脑上。
Markov expanding maps, a class of simple chaotic systems, are commonly used as models for chaotic dynamics, but existing numerical methods to study long-time statistical properties such as invariant measures have a poor trade-off between computational effort and accuracy. We develop a spectral Galerkin method for these maps' transfer operators, estimating statistical quantities using finite submatrices of the transfer operators' infinite Fourier or Chebyshev basis coefficient matrices. Rates of convergence of these estimates are obtained via quantitative bounds on the full transfer operator matrix entries; we find the method furnishes up to exponentially accurate estimates of statistical properties in only a polynomially large computational time. To implement these results we suggest and demonstrate two algorithms: a rigorously-validated algorithm, and a fast, more convenient adaptive algorithm. Using the first algorithm we prove rigorous bounds on some exemplar quantities that are substantially more accurate than previous. We show that the adaptive algorithm can produce double floating-point accuracy estimates in a fraction of a second on a personal computer.