Integrating hyper-parameter uncertainties in a multi-fidelity Bayesian model for the estimation of a probability of failure

Stroh, Rémi; Bect, Julien; Demeyer, Séverine; Fischer, Nicolas; Vázquez, Emmanuel

doi:10.1142/9789813274303_0035

“…In both cases, the numerical model under consideration will be assumed to be deterministic. (The second approach can also deal with stochastic simulators; see, e.g., the work of Stroh and co-authors [17,18]. )…”

Section: Discretization Uncertainty: Two Paradigmsmentioning

confidence: 99%

On the Quantification of Discretization Uncertainty: Comparison of Two Paradigms

Bect¹,

Zio²,

Perrin³

et al. 2021

14th WCCM-ECCOMAS Congress

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The use of simulation has spread to all areas of engineering and science, and the use of numerical models based on partial differential equations has thus multiplied. The resolution of these models is generally based on the discretization of the space in which the solutions to the equations under consideration are sought. The finite differences method or the finite elements method are two examples of such a discretization. This discretization simplifies the solving but implies a form of uncertainty on the value of any quantity of interest. To quantify this discretization uncertainty, the grid convergence index (GCI), based on the Richardson extrapolation technique, is now standard in the Verification and Validation (V&V) literature. But alternative approaches were also proposed in the statistical literature, such as Bayesian approaches with Gaussian process models. The objective of this work is to compare on a standard test case from the literature (Timoshenko's beam) the well-established GCI-based approach to the-younger-Bayesian approach for the quantification of discretization uncertainty.

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“…(The second approach can also deal with stochastic simulators; see, e.g., the work of Stroh and co-authors [17,18]. )…”

Section: Discretization Uncertainty: Two Paradigmsmentioning

confidence: 99%

On the quantification of discretization uncertainty: comparison of two paradigms

Bect,

Zio,

Perrin

et al. 2021

Preprint

0

View full text Add to dashboard Cite

The use of simulation has spread to all areas of engineering and science, and the use of numerical models based on partial differential equations has thus multiplied. The resolution of these models is generally based on the discretization of the space in which the solutions to the equations under consideration are sought. The finite differences method or the finite elements method are two examples of such a discretization. This discretization simplifies the solving but implies a form of uncertainty on the value of any quantity of interest. To quantify this discretization uncertainty, the grid convergence index (GCI), based on the Richardson extrapolation technique, is now standard in the Verification and Validation (V&V) literature. But alternative approaches were also proposed in the statistical literature, such as Bayesian approaches with Gaussian process models. The objective of this work is to compare on a standard test case from the literature (Timoshenko's beam) the well-established GCI-based approach to the-younger-Bayesian approach for the quantification of discretization uncertainty.

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“…For the numerical experiments of this article (Sections 4.3 and 4.4) we will take a simpler route, assuming that the variance λ depends only on the fidelity level δ-which is approximately true in the two examples we shall consider. In this setting, as long as the number of fidelity levels of interest is not too large, the value of the variance at these levels can be simply estimated jointly with the other hyper-parameters of the model; a general-purpose log-normal prior for the vector of variances is proposed by Stroh et al (2016Stroh et al ( , 2017b.…”

Section: Extension To Stochastic Simulatorsmentioning

confidence: 99%

“…The mean function ξ is modeled by the additive Gaussian process model ( 2)-(4)of Section 2.2, where the variance λ is log-Gaussian as in Section 2.3 and the prior distributions for the hyper-parameters are set as in Stroh et al (2017b). The posterior mean…”

Section: Random Damped Harmonic Oscillatormentioning

confidence: 99%

Sequential design of multi-fidelity computer experiments: maximizing the rate of stepwise uncertainty reduction

Stroh,

Bect,

Demeyer

et al. 2020

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This article deals with the sequential design of experiments for (deterministic or stochastic) multi-fidelity numerical simulators, that is, simulators that offer control over the accuracy of simulation of the physical phenomenon or system under study. Very often, accurate simulations correspond to high computational efforts whereas coarse simulations can be obtained at a smaller cost. In this setting, simulation results obtained at several levels of fidelity can be combined in order to estimate quantities of interest (the optimal value of the output, the probability that the output exceeds a given threshold. . . ) in an efficient manner. To do so, we propose a new Bayesian sequential strategy called Maximal Rate of Stepwise Uncertainty Reduction (MR-SUR), that selects additional simulations to be performed by maximizing the ratio between the expected reduction of uncertainty and the cost of simulation. This generic strategy unifies several existing methods, and provides a principled approach to develop new ones. We assess its performance on several examples, including a computationally intensive problem of fire safety analysis where the quantity of interest is the probability of exceeding a tenability threshold during a building fire.

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Integrating hyper-parameter uncertainties in a multi-fidelity Bayesian model for the estimation of a probability of failure

Cited by 4 publications

References 8 publications

On the Quantification of Discretization Uncertainty: Comparison of Two Paradigms

On the Quantification of Discretization Uncertainty: Comparison of Two Paradigms

On the quantification of discretization uncertainty: comparison of two paradigms

Sequential design of multi-fidelity computer experiments: maximizing the rate of stepwise uncertainty reduction

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