Robust adaptive least squares polynomial chaos expansions in high‐frequency applications

Abstract

We present an algorithm for computing sparse, least squares-based polynomial chaos expansions, incorporating both adaptive polynomial bases and sequential experimental designs. The algorithm is employed to approximate stochastic high-frequency electromagnetic models in a black-box way, in particular, given only a dataset of random parameter realizations and the corresponding observations regarding a quantity of interest, typically a scattering parameter. The construction of the polynomial basis is based on a greedy, adaptive, sensitivity-related method. The sequential expansion of the experimental design employs different optimality criteria, with respect to the algebraic form of the least squares problem. We investigate how different conditions affect the robustness of the derived surrogate models, that is, how much the approximation accuracy varies given different experimental designs. It is found that relatively optimistic criteria perform on average better than stricter ones, yielding superior approximation accuracies for equal dataset sizes. However, the results of strict criteria are significantly more robust, as reduced variations regarding the approximation accuracy are obtained, over a range of experimental designs. Two criteria are proposed for a good accuracy-robustness trade-off.keywordspolynomial chaos, surrogate modeling, high-frequency electromagnetic devices, least squares regression, adaptive basis, sequential experimental design

Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models

“…, i.e., it scales polynomially with the input dimension k and the maximum degree s max . For the case of high-dimensional input random variables X, several sparse PCE algorithms have been proposed in the literature for the construction of Λ such that the impact of the curse of dimensionality is mitigated [14,10,12,13,69,70].…”

“…Once the multi-index set Λ is fixed, the only thing remaining to complete the PCE is to compute the coefficients. Several approaches are suggested in the literature for computing the PCE coefficients, e.g., pseudo-spectral projection [71,72,73,74], interpolation [75,76], and, most commonly, regression [10,14,13,70,4,69,77,78]. The latter option is employed in this work also, such that the PCE coefficients are obtained by solving the penalized least squares problem [79] arg min…”

“…In such cases however, a suitable model selection criterion must be employed for tuning the hyperparameters that are used to obtain the optimal model. Several techniques exist to adaptively identify the optimal polynomial basis and associated sparse solution, which aim to keep the size of the basis small by controlling which functions are added to the basis [12,13,10,14]. Despite its advantages, in cases where very high-dimensional outputs are considered, a PCE surrogate can be both computationally intractable to construct and incapable of accurately performing out-of-sample predictions.…”

Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

“…We consider a rectangular waveguide with a dispersive dielectric inset, similar to the one investigated in [53]. A 2D illustration of the waveguide is depicted in Figure 5, where w denotes its width, is the length of the dielectric material, and d is a vacuum offset.…”

Editorial for the special issue on advances in forward and inverse surrogate modeling for high‐frequency design

“…Koziel and Pietrenko‐Dabrowska 2 describe the developments of performance‐driven surrogate modeling methods, which is one of the approaches recently proposed to address the dimensionality and parameter range issues in high‐frequency modeling. Loukreziz et al 3 propose a novel algorithm for sparse least squares‐based polynomial chaos expansion models involving sequential experimental designs, whereas Georg and Römer 4 discuss the utilization of conformal maps to construct basis functions for generalized polynomial chaos (gPC) as a way of enhancing its convergence properties. The advantages of the method are demonstrated using optical components.…”