Reproducing kernel Hilbert space compactification of unitary evolution groups
Reproducing kernel Hilbert space compactification of unitary evolution groups
复制标题
DOI：
10.1016/j.acha.2021.02.004
复制
发表时间：
2021-03-22
影响因子：
2.5
通讯作者：
Slawinska, Joanna
中科院分区：
文献类型：
--
作者：
Das, Suddhasattwa;Giannakis, Dimitrios;Slawinska, Joanna
提出了一种用于原子谱分量和连续谱分量的保测度遍历动力系统的相干模式提取和观测量预测的框架。这个框架是基于一个近似的生成系统的一个紧凑的运营商W-tau的再生核希尔伯特空间（RKHS）。算子W-tau是斜伴随的，因此可以用一个投影值测度来表示，通过紧性离散，并带有一个相关的本征函数的正交基。这些本征函数是根据狄利克雷能量排序的，并提供了一个在动力学下的相干观测量的概念，类似于与光谱的原子部分相关的库普曼本征函数。此外，W-tau在RKHS上生成一个酉演化群{e（tW tau）}t <$R，它近似于系统的酉Koopman群。我们建立收敛结果的频谱和Borel功能演算的W-tau为tau -> 0（+），以及相关的数据驱动的配方利用时间序列数据。数值应用遍历系统的原子和连续光谱，即环面旋转，洛伦兹63系统，和罗斯勒系统，提出。(C)2021年，任作者。爱思唯尔公司出版
A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator W-tau on a reproducing kernel Hilbert space (RKHS). The operator W-tau is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, W-tau generates a unitary evolution group {e(tW tau)}t epsilon R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of W-tau as tau -> 0(+), as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rossler system, are presented. (C) 2021 The Author(s). Published by Elsevier Inc.