On Numerical Approximations of the Koopman Operator
On Numerical Approximations of the Koopman Operator
复制标题
考夫曼算子的数值逼近
作者：
I. Mezić
我们研究了计算复合算子谱性质的数值方法。利用广义拉普拉斯分析给出了Banach空间中Koopman模的一个刻画。在无限维算子的有限截面理论的背景下，我们给出了动态模式分解型方法，并给出了一个有限截面方法失效的混合映射的例子。在基本动力学假设下，我们给出了有限截面近似下样本容量增加时收敛速度的第一个结果。研究了有限截面法的Krylov子空间形式的误差，证明了具有纯点谱的算子在伪谱意义下的收敛。由于基于Krylov序列的近似可以缓解维度灾难，这一结果表明它们也可能具有较低的谱误差，而所需函数的数量不会以指数形式增加。
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. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed.
影响因子：
2.1
作者：
Azencot, Omri;Yin, Wotao;Bertozzi, Andrea
通讯作者：
Bertozzi, Andrea