ESTIMATING LONG-TERM BEHAVIOR OF FLOWS WITHOUT TRAJECTORY INTEGRATION: THE INFINITESIMAL GENERATOR APPROACH
ESTIMATING LONG-TERM BEHAVIOR OF FLOWS WITHOUT TRAJECTORY INTEGRATION: THE INFINITESIMAL GENERATOR APPROACH
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DOI：
10.1137/110819986
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发表时间：
2013-01-01
影响因子：
2.9
通讯作者：
Koltai, Peter
中科院分区：
文献类型：
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作者：
Froyland, Gary;Junge, Oliver;Koltai, Peter
流动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，全球缓慢混合结构，如几乎不变集，分区相空间几乎是动态断开的区域，也可以确定由该运营商的某些特征函数。事实上，这些结构通常很难通过基于蛮力概率的分析来获得。在广泛的应用中，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。在构造近似转移算子中，计算成本最高的步骤是对许多短期轨迹进行数值积分。在本文中，我们建议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方法。在某些情况下可以表现出趋同。我们数值证明，我们的方法是更有效的比运营商的方法，有时几个数量级。
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.