论文标题
关于具有较高多重性的特征值和特征空间的不确定性定量
On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity
论文作者
论文摘要
我们以各种形式的变异形式考虑了广义的操作员特征值问题，并以双线性形式进行随机扰动。此设置是由带有随机输入数据的部分微分方程的变异形式的动机。所考虑的特征木材可以具有较高但有限的多重性。我们研究了本征的随机量，并讨论了为什么以大于1的多重性，只有特征空间的随机特性是有意义的，但不是单个特征的特性。为此，我们表征了本征皮的Fréchet衍生物相对于扰动，并为较高多重性的本征素提供了新的线性表征。作为一方面的结果，我们证明了特征空间的局部分析性。基于特征款的Fréchet衍生物，我们讨论了多种特征值的有意义的蒙特卡洛采样策略，并开发不确定性定量扰动方法。提出了数值示例以说明理论结果。
We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fréchet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fréchet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results.
