Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

Moustapha, Maliki; Sudret, Bruno; Bourinet, Jean‐Marc; Guillaume, Benoît

doi:10.1007/s00158-016-1504-4

Cited by 99 publications

(56 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Future works could also consider the adaptation of the proposed SoGP framework to task-oriented problems, such as solving optimization problems by Efficient Global Optimization strategies [49], or the simulation of rare events [50]. Such developments will require objective oriented active learning strategies to improve the SoGP predictive capabilities for the input values of interest.…”

Section: Resultsmentioning

confidence: 99%

Systems of Gaussian process models for directed chains of solvers

Sanson

Maître

Congedo

2019

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

The simulation of complex multi-physics phenomena often relies on a System of Solvers (SoS), which we define here as a set of interdependent solvers where the output of an upstream solver is the input of downstream solvers. Constructing a surrogate model of a SoS presents a clear interest when multiple evaluations of the system are needed, for instance to perform uncertainty quantification and global sensitivity analyses, the resolution of optimization or control problems, and generally any task based on fast query evaluations. In this work, we develop an original mathematical framework, based on Gaussian Process (GP) models, to construct a global surrogate model of the directed SoS, (i.e., only featuring one-way dependencies between solvers). The two central ideas of the proposed approach are, first, to determine a local GP model for each solver constituting the SoS and, second, to define the prediction as the composition of the individual GP models constituting a system of GP models (SoGP). We further propose different adaptive sampling strategies for the construction of the SoGP. These strategies use the decomposition of the SoGP prediction variance into individual contributions of the constitutive GP models and on extensions to SoGP of the Maximum Mean Square Predictive Error criterion. We finally assess the performance of the SoGP framework on several SoS involving different numbers of solvers and structures of input dependencies. The results show that the SoGP framework is very flexible and can handle different types of SoS, with a significantly reduced construction cost (measured by the number of training samples) compared to constructing a unique GP model of the SoS. , systems of solvers have received interest from the community trying to develop efficient UQ methods for SoS. In [11], the authors proposed a method based on importance sampling to decouple the uncertainty propagation process of individual solvers in order to gain flexibility. Other recent works focused on adapting Global Polynomial Chaos (gPC) based methods to SoS, with the challenge of deriving efficient quadrature rules on intermediate inputs with unknown distributions. Using the structure of SoS, the authors of [12] proposed a method for propagating uncertainty in a composite function by adapting the quadrature rule of intermediate inputs in the SoS, thus limiting the number of quadrature points compared to a global black box approach. This work used the recursive formula for orthogonal polynomials and Lanczos algorithms. The same authors generalized this idea in [13] to a full SoS. Their approach relies on Galerkin projection methods at intermediate layers of the SoS. By solving an optimization problem, they proposed a quadrature rule for latent variables, regularized in order to promote sparsity in the weights, thus reducing the number of quadrature points. In [14], the authors tackled the problem of strongly coupled systems. Their main idea is that the dimension of the coupling variables and the amount of information transferred from ...

show abstract

Section: Resultsmentioning

confidence: 99%

Systems of Gaussian process models for directed chains of solvers

Sanson

Maître

Congedo

2019

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

“…kernels), being aware that only functions that yield symmetric positive semidefinite covariance matrices are valid choices [25]. The selection of the covariance function encodes assumptions such as the degree of regularity of the underlying process [31]. A general expression for the covariance between any pair of scalar input points x,x ∈ R ds is given by…”

Section: Gaussian Process Metamodeling Of Scalar-input Codesmentioning

confidence: 99%

Gaussian process metamodeling of functional-input code for coastal flood hazard assessment

Betancourt

Bachoc

Klein

et al. 2020

Reliability Engineering & System Safety

View full text Add to dashboard Cite

et al.. Gaussian process metamodeling of functional-input code for coastal flood hazard assessment. Reliability Engineering and System Safety, Elsevier, 2020, 198, Abstract This paper investigates the construction of a metamodel for coastal flooding early warning at the peninsula of Gâvres, France. The code under study is an hydrodynamic model which receives time-varying maritime conditions as inputs. We concentrate on Gaussian pocess metamodels to emulate the behavior of the code. To model the inputs we make a projection of them onto a space of lower dimension. This setting gives rise to a model selection methodology which we use to calibrate four characteristics of our functional-input metamodel: (i) the family of basis functions to project the inputs; (ii) the projection dimension; (iii) the distance to measure similarity between functional input points; and (iv) the set of functional predictors to keep active. The proposed methodology seeks to optimize these parameters for metamodel predictability, at an affordable computational cost. A comparison to a dimensionality reduction approach based on the projection error of the input functions only showed that the latter may lead to unnecessarily large projection dimensions. We also assessed the adaptability of our methodology to changes in the number of training and validation points. The methodology proved its robustness by finding the optimal solution for most of the instances, while being computationally efficient.

show abstract

“…Given a critical value u , K p samples are selected for the refinement of the performance function surrogate

Ĝ

. For each sample

x \in R^{d}

, the so‐called probability of misclassification P m ( x ) is defined as Reference :

P_{m}^{u} (x) = φ (- \frac{| μ_{Ĝ} (x) - u |}{σ_{Ĝ} (x)}) .

…”

Section: The Qeak‐mcs Algorithmmentioning

confidence: 99%

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

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.

show abstract

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

Cited by 99 publications

References 47 publications

Systems of Gaussian process models for directed chains of solvers

Systems of Gaussian process models for directed chains of solvers

Gaussian process metamodeling of functional-input code for coastal flood hazard assessment

Efficient estimation of extreme quantiles using adaptive kriging and importance sampling

Contact Info

Product

Resources

About