Generalized Polynomial Chaos (gPC) expansions are well established for forward uncertainty propagation in many application areas. Although the associated computational effort may be reduced in comparison to Monte Carlo techniques, for instance, further convergence acceleration may be important to tackle problems with high parametric sensitivities. In this work, we propose the use of conformal maps to construct a transformed gPC basis, in order to enhance the convergence order. The proposed basis still features orthogonality properties and hence, facilitates the computation of many statistical quantities such as sensitivities and moments. The corresponding surrogate models are computed by pseudo‐spectral projection using mapped quadrature rules, which leads to an improved cost accuracy ratio. We apply the methodology to Maxwell's source problem with random input data. In particular, numerical results for a parametric finite element model of an optical grating coupler are given.

中文翻译：

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带有随机输入数据的麦克斯韦源问题的保形映射多项式混沌展开

广义多项式混沌（gPC）扩展已经很好地建立，可以在许多应用领域传播正向不确定性。例如，尽管与蒙特卡洛技术相比，可以减少相关的计算工作量，但是进一步的收敛加速对于解决具有高参数敏感性的问题可能很重要。在这项工作中，我们建议使用共形图来构建变换后的gPC基础，以增强收敛顺序。所提出的基础仍然具有正交特性，因此有助于计算许多统计量，例如灵敏度和矩。相应的替代模型是使用映射的正交规则通过伪光谱投影来计算的，从而提高了成本准确率。我们将方法论应用于麦克斯韦 随机输入数据的来源问题。特别地，给出了光栅耦合器的参数有限元模型的数值结果。