Assessing the Performance of Leja and Clenshaw-Curtis Collocation for Computational Electromagnetics With Random Input Data

Loukrezis, Dimitrios; Gersem, Herbert De

doi:10.1615/int.j.uncertaintyquantification.2018025234

Cited by 17 publications

(35 citation statements)

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“…We apply the methods for yield estimation and optimization discussed in the previous sections to a benchmark problem in the context of electromagnetic field simulation. In particular, we employ the model of a rectangular waveguide with a dielectric inset, similarly to the one used in [36]. This model is well suited for validation purposes, as a closedform solution is available [37].…”

Section: Numerical Resultsmentioning

confidence: 99%

Yield Optimization Based on Adaptive Newton-Monte Carlo and Polynomial Surrogates

Fuhrländer

Georg

Schöps

2020

Int. J. UncertaintyQuantification

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In this paper we present an algorithm for yield estimation and optimization consisting of Hessian-based optimization methods, an adaptive Monte Carlo (MC) strategy, polynomial surrogates, and several error indicators. Yield estimation is used to quantify the impact of uncertainty in a manufacturing process. Since computational efficiency is one main issue in uncertainty quantification, we propose a hybrid method, where a large part of a MC sample is evaluated with a surrogate model, and only a small subset of the sample is reevaluated with a high-fidelity finite element model. In order to determine this critical fraction of the sample, an adjoint error indicator is used for both the surrogate error and the finite element error. For yield optimization we propose an adaptive Newton-MC method. We reduce computational effort and control the MC error by adaptively increasing the sample size. The proposed method minimizes the impact of uncertainty by optimizing the yield. It allows one to control the finite element error, surrogate error, and MC error. At the same time it is much more efficient than standard MC approaches combined with standard Newton algorithms.

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Section: Numerical Resultsmentioning

confidence: 99%

Yield Optimization Based on Adaptive Newton-Monte Carlo and Polynomial Surrogates

Fuhrländer

Georg

Schöps

2020

Int. J. UncertaintyQuantification

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“…We will base the adaptive construction of anisotropic sparse interpolations on the dimension-adaptive algorithm first presented in [47] for quadrature purposes. Variants of this algorithm appear in a number of later works [16,30,36,37,48]. In this work, we assume the all underlying univariate interpolation rules employ Leja nodes, presented in Section 2.3, along with the level-to-nodes function m(i) = i + 1.…”

Section: Adaptive Anisotropic Leja Interpolationmentioning

confidence: 99%

“…Moreover, it was found to have a clear edge over the well-established LAR-gPC method. [37] and with a degree-adaptive LAR-gPC algorithm [9,50]. The size of the validation sample is Q = 10 5 .…”

Section: Meromorphic Functionmentioning

confidence: 99%

“…We note that this paper focuses on continuous PDFs of arbitrary shapes and does not consider data-driven approaches which define a PDF through its moments without any further assumptions about it [31,32].Due to a number of desirable properties, e.g., granularity, interpolation, and quadrature stability, and nestedness, weighted Leja nodes have been getting increasing attention in the context of approximation and UQ [16,30,[33][34][35][36]. Nonetheless, with the exceptions of [34,37,38] where beta distributions are considered for the parameters, the use of Leja nodes has been limited to uniform and log-normal parameter distributions. The contribution of this paper is exactly to start filling this gap with the application of weighted Leja interpolation for possibly high-dimensional systems with arbitrary parameter PDFs.To that end, we present here the use of weighted Leja rules in the context of sparse Lagrange interpolation for approximation and UQ purposes in a comprehensive and self-consistent way.…”

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

Loukrezis

Gersem

2020

Algorithms

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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 ...

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“…A wide range of domains have seen application of such dimension-adaptive samplers, e.g. computational electromagnetism [25], finance [21,16] or natural convection problems [13], to name just a few. We only perform a limited validation study, by examining the ability of the predicted output distribution to envelop the observed COVID-19 death count, conditional on a predefined intervention scenario.…”

Section: Introductionmentioning

confidence: 99%

Model uncertainty and decision making: Predicting the Impact of COVID-19 Using the CovidSim Epidemiological Code

Edeling¹,

Sinclair²,

Suleimenova³

et al. 2020

Preprint

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The severe acute respiratory syndrome coronavirus 2 (SARS-CoV2) virus has rapidly spread worldwide since December 2019, and early modelling work of this pandemic has assisted in identifying effective government interventions. The UK government relied in part on the CovidSim model developed by the MRC Centre for Global Infectious Disease Analysis at Imperial College London, to model various non-pharmaceutical intervention strategies, and guide its government policy in seeking to deal with the rapid spread of the COVID-19 pandemic during March and April 2020. CovidSim is subject to different sources of uncertainty, namely parametric uncertainty in the inputs, model structure uncertainty (i.e. missing epidemiological processes) and scenario uncertainty, which relates to uncertainty in the set of conditions under which the model is applied. We have undertaken an extensive parametric sensitivity analysis and uncertainty quantification of the current CovidSim code. From the over 900 parameters that are provided as input to CovidSim, we identified a key subset of 20 parameters to which the code output is most sensitive. We find that the uncertainty in the code is substantial, in the sense that imperfect knowledge in these inputs will be magnified to the outputs, up to the extent of ca. 300%. Most of this uncertainty can be traced back to the sensitivity of three parameters. Compounding this, the model can display significant bias with respect to observed data, such that the output variance does not capture this validation data with high probability. We conclude that quantifying the parametric input uncertainty is not sufficient, and that the effect of model structure and scenario uncertainty cannot be ignored when validating the model in a probabilistic sense.

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Assessing the Performance of Leja and Clenshaw-Curtis Collocation for Computational Electromagnetics With Random Input Data

Cited by 17 publications

References 43 publications

Yield Optimization Based on Adaptive Newton-Monte Carlo and Polynomial Surrogates

Yield Optimization Based on Adaptive Newton-Monte Carlo and Polynomial Surrogates

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

Model uncertainty and decision making: Predicting the Impact of COVID-19 Using the CovidSim Epidemiological Code

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