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 Fr\'echet 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 Fr\'echet 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.

翻译：我们考虑变分形式下的广义算子特征值问题，其中双线性形式存在随机扰动。这一设定源于带随机输入数据的偏微分方程的变分形式。所考虑的特征对可能具有较高但有限的重数。我们研究了特征对的随机感兴趣量，并讨论了为何当重数大于1时，仅有特征空间的随机性质具有意义，而单个特征对的则不然。为此，我们刻画了特征对关于扰动的Fréchet导数，并为高重数特征对提供了一种新的线性刻画。作为附带结果，我们证明了特征空间的局部解析性。基于特征对的Fréchet导数，我们讨论了一种针对多重特征值的有意义的蒙特卡洛抽样策略，并发展了一种不确定性量化扰动方法。通过数值算例验证了理论结果。