提出无谱污染的转移算子谱计算方法，适用于非线性系统分析。
Avoiding spectral pollution for transfer operators using residuals
- 基于残差检测机制，避免有限维近似引入虚假谱点。
- 在布莱施克映射与蛋白质折叠模型中验证了高精度与泛化能力。
- 适合研究复杂动力系统谱特性但缺乏精确解析解的场景。
Koopman算子理论通过将非线性动力系统的演化提升到无限维函数空间，实现其线性分析。然而，对Koopman算子和转移算子（弗罗贝尼乌斯-佩龙算子）的有限维近似容易产生谱污染，引入虚假特征值，破坏谱计算的可靠性。尽管近期已有针对Koopman算子的可证明收敛方法，但针对一般转移算子的类似工具仍十分有限。本文提出计算转移算子谱性质且无谱污染的算法，并扩展至Hardy-Hilbert空间。案例研究涵盖具有已知谱的布莱施克映射族及蛋白质折叠的分子动力学模型，验证了方法的准确性与灵活性。值得注意的是，即使对应特征函数不在所选空间中，仍可出现谱特征，揭示了定义“真实”Koopman谱时的功能分析精细之处。本方法为广泛应用场景提供了稳健的谱估计工具。
原文摘要 · Abstract (English)
Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.
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