无需轨迹积分估计流的长期行为：无穷小生成元方法
Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach
AI总结：
本文提出直接使用无穷小生成元而非转移算子来估计流的长期行为，避免了轨迹积分，并通过两种离散化方案（基于细胞的离散化和谱配置法）实现了更高的计算效率。
摘要：
流的轨迹的长期分布由不变密度描述，即相关转移算子的不动点。此外，全局慢混合结构，如几乎不变集（将相空间划分为几乎动态不连通的区域），也可以通过该算子的某些特征函数来识别。实际上，这些结构通常难以通过基于轨迹的暴力分析获得。在各种应用中，转移算子已被证明是分析动力系统全局行为的非常有效的工具。在构建近似转移算子时，计算成本最高的步骤是许多短期轨迹的数值积分。在本文中，我们提出直接使用无穷小生成元而不是算子，完全避免了轨迹积分。我们提出了两种不同的离散化方案：基于细胞的离散化和谱配置法。在某些情况下可以证明收敛性。我们通过数值实验表明，我们的方法比算子方法效率高得多，有时高出几个数量级。
英文摘要：
The long-term distributions of trajectories of a flow are described by invariant densities, i.e. fixed points of an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that are almost dynamically disconnected, can also be identified by certain eigenfunctions of this operator. Indeed, these structures are often hard to obtain by brute-force trajectory-based analyses. In a wide variety of applications, transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system. The computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories. In this paper, we propose to directly work with the infinitesimal generator instead of the operator, completely avoiding trajectory integration. We propose two different discretization schemes; a cell based discretization and a spectral collocation approach. Convergence can be shown in certain circumstances. We demonstrate numerically that our approach is much more efficient than the operator approach, sometimes by several orders of magnitude.
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