International Journal for Uncertainty Quantification

Journal: International Journal for Uncertainty Quantification

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Begell House

ISSN

2152-5080
2152-5099

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Publications 1 - 10 of 11

Sensitivity analysis for stochastic simulators using differential entropy
Azzi, Soumaya; Sudret, Bruno; Wiart, Joe (2020)
International Journal for Uncertainty Quantification
Surrogate modeling of stochastic functions-application to computational electromagnetic dosimetry
Azzi, Soumaya; Huang, Yuanyuan; Sudret, Bruno; et al. (2019)
International Journal for Uncertainty Quantification
This paper is dedicated to the surrogate modeling of a particular type of computational model called stochastic simulators, which inherently contain some source of randomness. In this particular case the output of the simulator in a given point is a probability density function. In this paper, the stochastic simulator is represented as a stochastic process and the surrogate model is built using the Karhunen-Loève expansion. In a first approach, the stochastic process covariance was surrogated using polynomial chaos expansion; meanwhile in a second approach the eigenvectors were interpolated. The performance of the method is illustrated on a toy example and then on an electromagnetic dosimetry example. We then provide metrics to measure the accuracy of the surrogate.
Global Sensitivity Analysis using Derivative-Based Sparse Poincaré Chaos Expansions
Lüthen, Nora; Roustant, Olivier; Gamboa, Fabrice; et al. (2023)
International Journal for Uncertainty Quantification
Variance-based global sensitivity analysis, in particular Sobol' analysis, is widely used for determining the importance of input variables to a computational model. Sobol' indices can be computed cheaply based on spectral methods like polynomial chaos expansions (PCE). Another choice are the recently developed Poincare chaos expansions (PoinCE), whose orthonormal tensor-product basis is generated from the eigenfunctions of one-dimensional Poincaré differential operators. In this paper, we show that the Poincaré basis is the unique orthonormal basis with the property that partial derivatives of the basis again form an orthogonal basis with respect to the same measure as the original basis. This special property makes PoinCE ideally suited for incorporating derivative information into the surrogate modeling process. Assuming that partial derivative evaluations of the computational model are available, we compute spectral expansions in terms of Poincaré basis functions or basis partial derivatives, respectively, by sparse regression. We show on two numerical examples that the derivative-based expansions provide accurate estimates for Sobol' indices, even outperforming PCE in terms of bias and variance. In addition, we derive an analytical expression based on the PoinCE coefficients for a second popular sensitivity index, the derivative-based sensitivity measure (DGSM), and explore its performance as upper bound to the corresponding total Sobol' indices.
Surrogate modeling of indoor down-link human exposure based on sparse polynomial chaos expansion
Liu, Zicheng; Lesselier, Dominique; Sudret, Bruno; et al. (2020)
International Journal for Uncertainty Quantification
Human exposure induced by wireless communication systems increasingly draws the public attention. Here, an indoor down-link scenario is concerned and the exposure level is statistically analyzed. The electromagnetic field emitted by a WiFi box is measured and electromagnetic dosimetry features are evaluated from the whole-body specific absorption rate as computed with a finite-difference time-domain (a.k.a. FDTD) code. Due to computational cost, a statistical analysis is performed based on a surrogate model, which is constructed by means of so-called sparse polynomial chaos expansion, where the inner cross validation (ICV) is used to select the optimal hyperparameters during the model construction and assess the model performance. However, the model assessment based on ICV tends to be overly optimistic with small data sets. The method of cross-model validation is used and outer cross validation is carried out for the model assessment. The effects of the data preprocessing are investigated as well. On the basis of the surrogate model, the global sensitivity of the exposure to input parameters is analyzed from Sobol' indices.
Stochastic spectral embedding
Marelli, Stefano; Wagner, Paul-Remo; Lataniotis, Christos; et al. (2021)
International Journal for Uncertainty Quantification
Constructing approximations that can accurately mimic the behavior of complex models at reduced computational costs is an important aspect of uncertainty quantification. Despite their flexibility and efficiency, classical surrogate models such as kriging or polynomial chaos expansions tend to struggle with highly nonlinear, localized, or nonstationary computational models. We hereby propose a novel sequential adaptive surrogate modeling method based on recursively embedding locally spectral expansions. It is achieved by means of disjoint recursive partitioning of the input domain, which consists in sequentially splitting the latter into smaller subdomains, and constructing simpler local spectral expansions in each, exploiting the trade-off complexity vs. locality. The resulting expansion, which we refer to as "stochastic spectral embedding" (SSE), is a piecewise continuous approximation of the model response that shows promising approximation capabilities, and good scaling with both the problem dimension and the size of the training set. We finally show how the method compares favorably against state-of-the-art sparse polynomial chaos expansions on a set of models with different complexity and input dimension. © 2021 by Begell House, Inc.
Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applications
Lüthen, Nora; Marelli, Stefano; Sudret, Bruno (2022)
International Journal for Uncertainty Quantification
Sparse polynomial chaos expansions (PCE) are an efficient and widely used surrogate modeling method in uncertainty quantification for engineering problems with computationally expensive models. To make use of the available information in the most efficient way, several approaches for so-called basis-adaptive sparse PCE have been proposed to determine the set of polynomial regressors (“basis”) for PCE adaptively. The goal of this paper is to help practitioners identify the most suitable methods for constructing a surrogate PCE for their model. We describe three state-of-the-art basis-adaptive approaches from the recent sparse PCE literature and conduct an extensive benchmark in terms of global approximation accuracy on a large set of computational models. Investigating the synergies between sparse regression solvers and basis adaptivity schemes, we find that the choice of the proper solver and basis-adaptive scheme is very important, as it can result in more than one order of magnitude difference in performance. No single method significantly outperforms the others, but dividing the analysis into classes (regarding input dimension and experimental design size), we are able to identify specific sparse solver and basis-adaptivity combinations for each class that show comparatively good performance. To further improve on these findings, we introduce a novel solver and basis adaptivity selection scheme guided by cross-validation error. We demonstrate that this automatic selection procedure provides close-to-optimal results in terms of accuracy, and significantly more robust solutions, while being more general than the case-by-case recommendations obtained by the benchmark.
Replication-based emulation of the response distribution of stochastic simulators using generalized lambda distributions
Zhu, Xujia; Sudret, Bruno (2020)
International Journal for Uncertainty Quantification
Extending classical surrogate modeling to high dimensions through supervised dimensionality reduction: A data-driven approach
Lataniotis, Christos; Marelli, Stefano; Sudret, Bruno (2020)
International Journal for Uncertainty Quantification
Polynomial-chaos-based Kriging
Schöbi, Roland; Sudret, Bruno; Wiart, Joe (2015)
International Journal for Uncertainty Quantification
Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. Optimization and uncertainty quantification problems typically require a large number of runs of the computational model at hand, which may not be feasible with high-fidelity models directly. Thus surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade. Polynomial chaos expansions (PCE) and Kriging are two popular nonintrusive meta-modeling techniques. PCE surrogates the computational model with a series of orthonormal polynomials in the input variables where polynomials are chosen in coherency with the probability distributions of those input variables. A least-square minimization technique may be used to determine the coefficients of the PCE. Kriging assumes that the computer model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer runs, i.e., input vectors and response values. These two techniques have been developed more or less in parallel so far with little interaction between the researchers in the two fields. In this paper, PC-Kriging is derived as a new nonintrusive meta-modeling approach combining PCE and Kriging. A sparse set of orthonormal polynomials (PCE) approximates the global behavior of the computational model whereas Kriging manages the local variability of the model output. An adaptive algorithm similar to the least angle regression algorithm determines the optimal sparse set of polynomials. PC-Kriging is validated on various benchmark analytical functions which are easy to sample for reference results. From the numerical investigations it is concluded that PC-Kriging performs better than or at least as good as the two distinct meta-modeling techniques. A larger gain in accuracy is obtained when the experimental design has a limited size, which is an asset when dealing with demanding computational models.
Stochastic polynomial chaos expansions to emulate stochastic simulators
Zhu, Xujia; Sudret, Bruno (2022)
International Journal for Uncertainty Quantification
In the context of uncertainty quantification, computational models are required to be repeatedly evaluated. This task is intractable for costly numerical models. Such a problem turns out to be even more severe for stochastic simulators, the output of which is a random variable for a given set of input parameters. To alleviate the computational burden, surrogate models are usually constructed and evaluated instead. However, due to the random nature of the model response, classical surrogate models cannot be applied directly to the emulation of stochastic simulators. To efficiently represent the probability distribution of the model output for any given input values, we develop a new stochastic surrogate model called stochastic polynomial chaos expansions. To this aim, we introduce a latent variable and an additional noise variable, on top of the well-defined input variables, to reproduce the stochasticity. As a result, for a given set of input parameters, the model output is given by a function of the latent variable with an additive noise, thus a random variable. Because the latent variable is purely artificial and does not have physical meanings, conventional methods (pseudo-spectral projections, collocation, regression, etc.) cannot be used to build such a model. In this paper, we propose an adaptive algorithm that does not require repeated runs of the simulator for the same input parameters. The performance of the proposed method is compared to the generalized lambda model and a state-of-the-art kernel estimator on two case studies in mathematical finance and epidemiology and on an analytical example whose response distribution is bimodal. The results show that the proposed method is able to accurately represent general response distributions, i.e., not only normal or unimodal ones. In terms of accuracy, it generally outperforms both the generalized lambda model and the kernel density estimator.

Publications 1 - 10 of 11

Journal: International Journal for Uncertainty Quantification

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