Discretization of the Frobenius-Perron Operator Using a Sparse Haar Tensor Basis: The Sparse Ulam Method
Discretization of the Frobenius-Perron Operator Using a Sparse Haar Tensor Basis: The Sparse Ulam Method
复制标题
使用稀疏 Haar 张量基的 Frobenius-Perron 算子的离散化：稀疏 Ulam 方法
DOI：
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发表时间：
2009
影响因子：
2.9
通讯作者：
P. Koltai
中科院分区：
文献类型：
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作者：
O. Junge;P. Koltai
动力系统的全局宏观行为被编码在相关的Frobenius-Perron算子的本征函数中。对于具有低维长期动力学的系统，存在用于最重要的本征函数的数值近似的有效技术;参见。[M. Dellnitz和O. Junge，SIAM J. Numer.分析：36（1999），pp. 491-515]。它们基于算子到吸引子-乌拉姆方法附近支持的分段常数函数空间的投影。在本文中，我们开发了一种数值技术，使乌拉姆的方法适用于系统的高维长期动态。它是基于使用稀疏网格处理高维偏微分方程的思想[C。Zenger，稀疏网格，偏微分方程的并行算法（基尔，1990），Vieweg，布伦瑞克，1991，pp. 241-251; H.- Bungartz和M. Griebel，Acta Numer.，13（2004），pp. 147-269]。在这里，我们使用稀疏Haar张量基作为底层近似空间。我们开发的技术，建立其复杂性和收敛性的声明，并提出两个数值例子。
The global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated Frobenius-Perron operator. For systems with low dimensional long term dynamics, efficient techniques exist for a numerical approximation of the most important eigenfunctions; cf. [M. Dellnitz and O. Junge, SIAM J. Numer. Anal., 36 (1999), pp. 491-515]. They are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor—Ulam's method. In this paper we develop a numerical technique which makes Ulam's approach applicable to systems with higher dimensional long term dynamics. It is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids [C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations (Kiel, 1990), Vieweg, Braunschweig, 1991, pp. 241-251; H.-J. Bungartz and M. Griebel, Acta Numer., 13 (2004), pp. 147-269]. Here, we use a sparse Haar tensor basis as the underlying approximation space. We develop the technique, establish statements about its complexity and convergence, and present two numerical examples.