Numerical Comparison of Leja and Clenshaw-Curtis Dimension-Adaptive Collocation for Stochastic Parametric Electromagnetic Field Problems
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We consider the problem of approximating the output of a parametric electromagnetic field model in the presence of a large number of uncertain input parameters. Given a sufficiently smooth output with respect to the input parameters, such problems are often tackled with interpolation-based approaches, such as the stochastic collocation method on tensor-product or isotropic sparse grids. Due to the so-called curse of dimensionality, those approaches result in increased or even forbidding computational costs. In order to reduce the growth in complexity with the number of dimensions, we employ a dimension-adaptive, hierarchical interpolation scheme, based on nested univariate interpolation nodes. Clenshaw-Curtis and Leja nodes satisfy the nestedness property and have been found to provide accurate interpolations when the parameters follow uniform distributions. The dimension-adaptive algorithm constructs the approximation based on the observation that not all parameters or interactions among them are equally important regarding their impact on the model's output. Our goal is to exploit this anisotropy in order to construct accurate polynomial surrogate models at a reduced computational cost compared to isotropic sparse grids. We apply the stochastic collocation method to two electromagnetic field models with medium- to high-dimensional input uncertainty. The performances of isotropic and adaptively constructed, anisotropic sparse grids based on both Clenshaw-Curtis and Leja interpolation nodes are examined. All considered approaches are compared with one another regarding the surrogate models' approximation accuracies using a cross-validation error metric.
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