结构化扰动下的瞬态动力学：桥接非结构化与结构化伪谱
Transient dynamics under structured perturbations: bridging unstructured and structured pseudospectra
AI总结：
本文提出结构化 ε-稳定性半径概念，将非结构化与结构化伪谱统一到联合伪谱框架中，并设计了基于梯度系统秩1矩阵微分方程的高效算法，计算成本与现有秩1算法相当。
摘要：
引入结构化 $\varepsilon$-稳定性半径，用于评估线性微分方程解在矩阵结构化扰动下瞬态界的鲁棒性。该概念适用于一般线性结构，例如具有给定稀疏模式或受限值域与余值域的复矩阵或实矩阵，以及 Toeplitz 矩阵等特殊矩阵类。该概念在联合伪谱中从概念上结合了非结构化伪谱和结构化伪谱，从而既能像非结构化伪谱一样使用预解式界，又能像结构化伪谱一样处理结构化扰动。我们提出并研究了一种计算结构化 $\varepsilon$-稳定性半径的算法。该算法通过适当离散化的秩 1 矩阵微分方程求解特征值优化问题，这些微分方程源自一个梯度系统。所提算法的计算成本与已知的用于计算非结构化和结构化稳定性半径的秩 1 算法基本相同。数值实验展示了该算法的行为。
英文摘要：
The structured $\varepsilon$-stability radius is introduced as a quantity to assess the robustness of transient bounds of solutions to linear differential equations under structured perturbations of the matrix. This applies to general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz matrices. The notion conceptually combines unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. We propose and study an algorithm for computing the structured $\varepsilon$-stability radius. This algorithm solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations that originate from a gradient system. The proposed algorithm has essentially the same computational cost as the known rank-1 algorithms for computing unstructured and structured stability radii. Numerical experiments illustrate the behavior of the algorithm.
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