This paper introduces a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics. To alleviate the potential problem of ill-conditioned eigenvectors in the existing implementations of the Dynamic Mode Decomposition (DMD) and the Extended Dynamic Mode Decomposition (EDMD), the new method introduces a Koopman-Schur decomposition that is entirely based on unitary transformations. The analysis in terms of the eigenvectors as modes of a Koopman operator compression is replaced with a modal decomposition in terms of a flag of invariant subspaces that correspond to selected eigenvalues. The main computational tool from the numerical linear algebra is the partial ordered Schur decomposition that provides convenient orthonormal bases for these subspaces. In the case of real data, a real Schur form is used and the computation is based on real orthogonal transformations. The new computational scheme is presented in the framework of the Extended DMD and the kernel trick is used.

翻译：本文提出了一种用于非线性动力学数据驱动Koopman模态分析的新理论与计算框架。为缓解现有动态模态分解（DMD）与扩展动态模态分解（EDMD）实现中可能存在的特征向量病态问题，新方法引入了完全基于酉变换的Koopman-Schur分解。该方法将以Koopman算子压缩特征向量作为模态的分析方式，替换为基于对应选定特征值的嵌套不变子空间族的模态分解。数值线性代数中的主要计算工具是偏序Schur分解，其为这些子空间提供了便利的标准正交基。对于实数数据情形，采用实Schur形式且计算基于实正交变换。新计算方案在扩展DMD框架下提出，并运用了核技巧。