We consider Maxwell eigenvalue problems on uncertain shapes with perfectly conducting TESLA cavities being the driving example. Due to the shape uncertainty, the resulting eigenvalues and eigenmodes are also uncertain and it is well known that the eigenvalues may exhibit crossings or bifurcations under perturbation. We discuss how the shape uncertainties can be modelled using the domain mapping approach and how the deformation mapping can be expressed as coefficients in Maxwell's equations. Using derivatives of these coefficients and derivatives of the eigenpairs, we follow a perturbation approach to compute approximations of mean and covariance of the eigenpairs. For small perturbations, these approximations are faster and more accurate than Monte Carlo or similar sampling-based strategies. Numerical experiments for a three-dimensional 9-cell TESLA cavity are presented.

翻译：我们以完美导电TESLA腔为例，研究不确定形状下的Maxwell特征值问题。由于形状不确定性，所得特征值与特征模态同样具有不确定性，且众所周知特征值在扰动下可能发生交叉或分岔。我们讨论了如何利用区域映射方法对形状不确定性进行建模，以及如何将形变映射表示为Maxwell方程中的系数。通过利用这些系数及特征对的导数，我们采用扰动方法计算特征对均值与协方差的近似值。对于小扰动，这些近似方法比蒙特卡洛或类似采样策略更快且更精确。文中给出了三维9单元TESLA腔的数值实验结果。