论文标题
关于Koopman操作员的数值近似
On Numerical Approximations of the Koopman Operator
论文作者
论文摘要
我们研究了组成运算符光谱特性的计算数值方法。我们使用广义Laplace分析提供了Banach空间中Koopman模式的表征。我们在无限尺寸运算符的有限截面理论的背景下施放了动态模式分解类型方法，并提供了一个混合图的示例，其中有限截面方法为其失败。在对基础动力学的假设下，我们为有限段近似的样本量增加提供了收敛速率的第一个结果。我们研究了有限部分方法的Krylov子空间版本中的误差，并证明了具有纯点光谱的操作员的伪谱意义上的收敛。该结果表明，基于Krylov序列的近似值可能会出现较低的误差，而无需指数中的数值增加近似值的函数数量。
We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode-Decomposition type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. This result indicates that Krylov sequence-based approximations can have low error without an exponential-in-dimension increase in the number of functions needed for approximation.
