Polynomial chaos expansions for dependent random variables

Summary In this paper, we present an adaptive algorithm to construct response surface approximations of high‐fidelity models using a hierarchy of lower fidelity models. Our algorithm is based on multi‐index stochastic collocation and automatically balances physical discretization error and response surface error to construct an approximation of model outputs. This surrogate can be used for uncertainty quantification (UQ) and sensitivity analysis (SA) at a fraction of the cost of a purely high‐fidelity approach. We demonstrate the effectiveness of our algorithm on a canonical test problem from the UQ literature and a complex multiphysics model that simulates the performance of an integrated nozzle for an unmanned aerospace vehicle. We find that, when the input‐output response is sufficiently smooth, our algorithm produces approximations that can be over two orders of magnitude more accurate than single fidelity approximations for a fixed computational budget.

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“…Polynomial chaos expansions (PCE) represent the model output

{\hat{f}}_{α} (z)

as an expansion of orthonormal polynomials

\begin{align} {\overset{f}{true}}_{bold-italicα} false(bold-italicz false) & \approx {\overset{f}{true}}_{bold-italicα, normalΛ} false(bold-italicz false) = \sum_{λ \in normalΛ} η_{λ} ϕ_{λ} false(bold-italicz false), false| normalΛ false| = N . \end{align}

…”

Section: Multi‐index Sensitivity Analysismentioning

confidence: 99%

“…Alternatively, one can use sampling schemes such as multivariate Leja sequences for polynomial interpolation, which are more amenable to dimension-adaptivity. 49…”

Section: Transforming a Sparse Grid Into A Polynomial Chaos Expansionmentioning

confidence: 99%

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Jakeman

Eldred

Geraci

et al. 2019

Numerical Meth Engineering

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“…The use of localized surrogates results in small-rates of convergence. Recently, PCE tailored to the posterior distribution have been used to obtain higher rates of convergence [15,33] and extended to use multiple models of varying cost and accuracy in [48].…”

Section: Determining the Likelihoodmentioning

confidence: 99%

“…Accurate approximations can be built without tailoring the sampling and approximation strategy to the measure ω. However such approaches, called domination methods [2,14], are sub-optimal and require a larger number of evaluations of u than methods that consider ω [15]. Instead of building an approximation which minimizes the error measured by the L p ω norm, domination methods build an approximation that minimizes the error measured by another norm L p g , where g is a simpler measure.…”

Section: Introductionmentioning

confidence: 99%

“…While beyond the scope of this paper, it is worth mentioning that nested-greedy procedures, similar to those discussed here for kernel based approximation, have also been developed for other approximation methods. Discrete Leja sequences [15,41], often used for polynomial approximation, are constructed by using partially pivoted LU factorization to greedily select points from an ordered set of basis functions evaluated at a set of candidate samples. Similarly, the Empirical Interpolation Method utilizes complete pivoting to greedily select both basis functions and samples (Magic points) [42].…”

Section: Introductionmentioning

confidence: 99%

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Cholesky-Based Experimental Design for Gaussian Process and Kernel-Based Emulation and Calibration

Harbrecht¹

2021

CiCP

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Gaussian processes and other kernel-based methods are used extensively to construct approximations of multivariate data sets. The accuracy of these approximations is dependent on the data used. This paper presents a computationally efficient algorithm to greedily select training samples that minimize the weighted L p error of kernel-based approximations for a given number of data. The method successively generates nested samples, with the goal of minimizing error in high probability regions of densities specified by users. The algorithm presented is extremely simple and can be implemented using existing pivoted Cholesky factorization methods. Training samples are generated in batches which allows training data to be evaluated (labeled) in parallel. For smooth kernels, the algorithm performs comparably with the greedy integrated variance design but has significantly lower complexity. Numerical experiments demonstrate the efficacy of the approach for bounded, unbounded, multi-modal and non-tensor product densities. We also show how to use the proposed algorithm to efficiently generate surrogates for inferring unknown model parameters from data using Bayesian inference.

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A Bayesian approach to blood rheological uncertainties in aortic hemodynamics

Ranftl

Müller

Windberger

et al. 2022

Numer Methods Biomed Eng

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Computational hemodynamics has received increasing attention recently.Patient-specific simulations require questionable model assumptions, for example, for geometry, boundary conditions, and material parameters. Consequently, the credibility of these simulations is much doubted, and rightly so. Yet, the matter may be addressed by a rigorous uncertainty quantification. In this contribution, we investigated the impact of blood rheological models on wall shear stress uncertainties in aortic hemodynamics obtained in numerical simulations. Based on shear-rheometric experiments, we compare the non-Newtonian Carreau model to a simple Newtonian model and a Reynolds number-equivalent Newtonian model. Bayesian Probability Theory treats uncertainties consistently and allows to include elusive assumptions such as the comparability of flow regimes. We overcome the prohibitively high computational cost for the simulation with a surrogate model, and account for the uncertainties of the surrogate model itself, too. We have two main findings: (1) The Newtonian models mostly underestimate the uncertainties as compared to the non-Newtonian model.(2) The wall shear stresses of specific persons cannot be distinguished due to largely overlapping uncertainty bands, implying that a more precise determination of person-specific blood rheological properties is necessary for personspecific simulations. While we refrain from a general recommendation for one rheological model, we have quantified the error of the uncertainty quantification associated with these modeling choices.

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Polynomial chaos expansions for dependent random variables

Cited by 67 publications

References 65 publications

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Cholesky-Based Experimental Design for Gaussian Process and Kernel-Based Emulation and Calibration

A Bayesian approach to blood rheological uncertainties in aortic hemodynamics

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