Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets
Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets
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DOI：
10.1063/1.4971788
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发表时间：
2017-03-01
期刊：
影响因子：
2.9
通讯作者：
Koltai, Peter
中科院分区：
文献类型：
--
作者：
Banisch, Ralf;Koltai, Peter
许多动力系统中规则结构与混沌共存的一个方面是相干集的出现：如果我们在某个初始时间将大量被动示踪剂放置在一个相干集中，那么宏观上它们会进行集体运动并在很长一段时间内保持紧密结合在一起，而它们的周围可能会混乱地混合。自然的例子是大气或海洋流中的移动涡旋。在本文中，我们提出了一种从可能稀疏的拉格朗日轨迹数据中提取相干集的方法。这是通过在数据点上构建随机游走来实现的，该随机游走既捕获了数据固有的时间顺序，又捕获了空间接近性的概念，这是一致性的核心。在丰富的数据限制下，我们可以证明与完善的相干集功能分析框架的等价性。我们方法的输出之一是“动态坐标”，它揭示了数据内在的基于低维传输的组织。
One aspect of the coexistence of regular structures and chaos in many dynamical systems is the emergence of coherent sets: If we place a large number of passive tracers in a coherent set at some initial time, then macroscopically they perform a collective motion and stay close together for a long period of time, while their surrounding can mix chaotically. Natural examples are moving vortices in atmospheric or oceanographic flows. In this article, we propose a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. This is done by constructing a random walk on the data points that captures both the inherent time-ordering of the data and the idea of closeness in space, which is at the heart of coherence. In the rich data limit, we can show equivalence to the well-established functional-analytic framework of coherent sets. One output of our method are "dynamical coordinates,"which reveal the intrinsic low-dimensional transport-based organization of the data.