Computation of tail probabilities for non-classical gasdynamic phenomena

Razaaly, Nassim; Congedo, Pietro Marco

doi:10.1088/1742-6596/821/1/012006

Cited by 1 publication

(9 citation statements)

References 22 publications

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“…The accuracy of the approximation given by IS critically depends on the choice of the ISD h . In this study, the ISD is chosen as

𝒩 (0, γ^{2} I_{d})

where

γ ⩾ 1

is a parameter which is defined using a rule of thumb as discussed in Section 3.5 (or can also be tuned following Reference ). Note that a Gaussian mixture ISD with suitable empirical parameters might be used, but those empirical parameters would depend on the critical value u , which represents the unknow quantile q here.…”

Section: Basic Ingredientsmentioning

confidence: 99%

“…QeAK‐MCS is similar to AK‐MCS‐based quantile estimation, however it uses IS instead of MC (as is done in AK‐MCS), thereby extending the approach from eAK‐MCS to quantile estimation. The main steps of QeAK‐MCS can be summarized as follows: Initial DoE : An experimental design

𝒳

is generated by Latin‐Hypercube Sampling (LHS) (see Section 3.1).…”

Section: The Qeak‐mcs Algorithmmentioning

confidence: 99%

“…In this subsection, we recall the basics of the refinement selection of eAK‐MCS (see Reference for details). Given a critical value u , K p samples are selected for the refinement of the performance function surrogate

Ĝ

.…”

Section: The Qeak‐mcs Algorithmmentioning

confidence: 99%

“…The eAK‐MCS refinement strategy reads as follows: 1 sample

x_{0}^{*}

is selected among

𝒮

following the single eAK‐MCS selection:

x^{*} = \underset{x \in 𝒮}{\arg \min} U^{u} (x)

. K p −1 samples

(x_{1}^{*}, \dots, x_{K_{p} - 1}^{*})

are simultaneously selected among the set

M^{\overline{k}} (u) = {x \in 𝒮 : μ_{Ĝ} (x) - \overline{k} σ_{Ĝ} (x) < u} ∖ {x \in 𝒮 : μ_{Ĝ} (x) + \overline{k} σ_{Ĝ} (x) < u}

, using a weighted K‐means clustering technique where the weights are chosen as

P_{m}^{u} (x)

for each sample

x \in M^{\overline{k}} (u)

. If this method returns only K 1 < K p −1 samples ( K 1 =0 possibly), then the very same method is applied to the full IS population

𝒮

to provide the remaining K p −1− K 1 samples. …”

Section: The Qeak‐mcs Algorithmmentioning

confidence: 99%

“…The same comment applies to the computation of small failure probability, where AK‐MCS becomes unaffordable. eAK‐MCS extends AK‐MCS for very small failure probability, inheriting a similar refinement strategy and general properties. It requires, however, to map the input random vector to the standard space.…”

Section: Introductionmentioning

confidence: 99%

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Efficient estimation of extreme quantiles using adaptive kriging and importance sampling

Razaaly

Crommelin

Congedo

2020

Numerical Meth Engineering

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SUMMARY This study considers an efficient method for the estimation of quantiles associated to very small levels of probability (up to O(10−9)), where the scalar performance function J is complex (eg, output of an expensive‐to‐run finite element model), under a probability measure that can be recast as a multivariate standard Gaussian law using an isoprobabilistic transformation. A surrogate‐based approach (Gaussian Processes) combined with adaptive experimental designs allows to iteratively increase the accuracy of the surrogate while keeping the overall number of J evaluations low. Direct use of Monte‐Carlo simulation even on the surrogate model being too expensive, the key idea consists in using an importance sampling method based on an isotropic‐centered Gaussian with large standard deviation permitting a cheap estimation of small quantiles based on the surrogate model. Similar to AK‐MCS as presented in the work of Schöbi et al., (2016), the surrogate is adaptively refined using a parallel infill criterion of an algorithm suitable for very small failure probability estimation. Additionally, a multi‐quantile selection approach is developed, allowing to further exploit high‐performance computing architectures. We illustrate the performances of the proposed method on several two to eight‐dimensional cases. Accurate results are obtained with less than 100 evaluations of J on the considered benchmark cases.

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“…The accuracy of the approximation given by IS critically depends on the choice of the ISD h . In this study, the ISD is chosen as

𝒩 (0, γ^{2} I_{d})

where

γ ⩾ 1

Section: Basic Ingredientsmentioning

confidence: 99%

𝒳

is generated by Latin‐Hypercube Sampling (LHS) (see Section 3.1).…”

Section: The Qeak‐mcs Algorithmmentioning

confidence: 99%

Ĝ

.…”

Section: The Qeak‐mcs Algorithmmentioning

confidence: 99%

“…The eAK‐MCS refinement strategy reads as follows: 1 sample

x_{0}^{*}

is selected among

𝒮

following the single eAK‐MCS selection:

x^{*} = \underset{x \in 𝒮}{\arg \min} U^{u} (x)

. K p −1 samples

(x_{1}^{*}, \dots, x_{K_{p} - 1}^{*})

are simultaneously selected among the set

M^{\overline{k}} (u) = {x \in 𝒮 : μ_{Ĝ} (x) - \overline{k} σ_{Ĝ} (x) < u} ∖ {x \in 𝒮 : μ_{Ĝ} (x) + \overline{k} σ_{Ĝ} (x) < u}

, using a weighted K‐means clustering technique where the weights are chosen as

P_{m}^{u} (x)

for each sample

x \in M^{\overline{k}} (u)

. If this method returns only K 1 < K p −1 samples ( K 1 =0 possibly), then the very same method is applied to the full IS population

𝒮

to provide the remaining K p −1− K 1 samples. …”

Section: The Qeak‐mcs Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficient estimation of extreme quantiles using adaptive kriging and importance sampling

Razaaly

Crommelin

Congedo

2020

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

Computation of tail probabilities for non-classical gasdynamic phenomena

Cited by 1 publication

References 22 publications

Efficient estimation of extreme quantiles using adaptive kriging and importance sampling

Efficient estimation of extreme quantiles using adaptive kriging and importance sampling

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