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

Abstract

Approximation and uncertainty quantification methods based on Lagrange interpolation are typically abandoned in cases where the probability distributions of one or more system parameters are not normal, uniform, or closely related distributions, due to the computational issues that arise when one wishes to define interpolation nodes for general distributions. This paper examines the use of the recently introduced weighted Leja nodes for that purpose. Weighted Leja interpolation rules are presented, along with a dimension-adaptive sparse interpolation algorithm, to be employed in the case of high-dimensional input uncertainty. The performance and reliability of the suggested approach is verified by four numerical experiments, where the respective models feature extreme value and truncated normal parameter distributions. Furthermore, the suggested approach is compared with a well-established polynomial chaos method and found to be either comparable or superior in terms of approximation and statistics estimation accuracy.Journal Not Specified 2020, xx, 1 2 of 21 interpolation schemes [5,6] or generalized polynomial chaos (gPC) [7], where the latter approach is typically based on least squares (LS) regression [8][9][10] or pseudo-spectral projection methods [11][12][13], or, less commonly, on interpolation [14]. Assuming a smooth input-output dependence, spectral UQ methods provide fast convergence rates, in some cases even of exponential order. However, their performance decreases rapidly for an increasing number of input parameters due to the curse of dimensionality [15]. To overcome this bottleneck, state-of-the-art approaches rely on algorithms for the adaptive construction of sparse approximations [8][9][10][16][17][18].Adaptive sparse approximation algorithms for Lagrange interpolation methods typically employ nested sequences of univariate interpolation nodes. While not strictly necessary [19], nested node sequences are helpful for the efficient construction of sparse grid interpolation algorithms. Nested node sequences are readily available for the well studied cases of uniformly or normally distributed random variables (RVs) [20,21], but not for more general probability measures. In principle, one could derive nested interpolation (or quadrature) rules tailored to an arbitrary probability density function (PDF) [22,23], however, this is a rather cumbersome task for which dedicated analyses are necessary each and every time a new PDF is considered.Diversely, the main requirement in gPC approximations, either full or sparse, is to find a complete set of polynomials that are orthogonal to each other with respect to the input parameter PDFs. In the case of arbitrary PDFs, such polynomials can be numerically constructed [24][25][26]. As a result, despite the fact that according to a number of studies Lagrange interpolation methods enjoy an advantage over gPC-based methods with respect to the error-cost ratio [27][28][29], they are abandoned whenever the input parameter PDFs are not uniform, or normal, or closely ...

Enhanced Adaptive Surrogate Models With Applications in Uncertainty Quantification for Nanoplasmonics

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An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

“…Additionally, interpolation and quadrature grids with respect to any continuous PDF can be constructed by using weighted Leja sequences. 39,[50][51][52] Moreover, due to the fact that Leja sequences are by definition nested, that is, {x i } j i=0 ⊂ {x i } j+1 i=0 , they allow for re-using readily available Leja points and model evaluations on those points in case the sequence is further expanded. Due to the nestedness property, Leja points are natural candidates for constructing sparse interpolation or quadrature grids.…”

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“…With respect to interpolation in particular, the Lebesgue constant of Leja sequence based interpolation grids is known to grow subexponentially [39][40][41] , thus resulting in comparatively stable interpolations. Additionally, interpolation and quadrature grids with respect to any continuous PDF can be constructed by using weighted Leja sequences [42][43][44][45] . Moreover, due to the fact that Leja sequences are by definition nested, i.e.…”