Random Matrix Models and Nonparametric Method for Uncertainty Quantification

Soize, Christian

doi:10.1007/978-3-319-12385-1_5

Cited by 5 publications

(2 citation statements)

References 107 publications

(62 reference statements)

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“…Most of the time, this is applied to quantities of interest on the solutions. There are many ways to perform these uncertainty quantification and statistical methods: by applying a classical Monte Carlo approach [24] and by applying approximation techniques of the Monte Carlo method thanks to a large family of techniques, such as generative approaches where we can find, for example, the generative adversarial networks and the Bayesian approaches coupled to VAEs [25], the stochastic reduced order models such as polynomial chaos expansions, and the Gaussian process regressions coupled to data compression techniques such as the POD or with the pre-image problem for the KPOD [26], stochastic models of random matrices that appear in the nonparametric method of uncertainties [27], and the stochastic projection-based reduced-order model (SPROM) [28].…”

Section: Introductionmentioning

confidence: 99%

Data-Targeted Prior Distribution for Variational AutoEncoder

Akkari

Casenave

Daniel

et al. 2021

Fluids

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Bayesian methods were studied in this paper using deep neural networks. We are interested in variational autoencoders, where an encoder approaches the true posterior and the decoder approaches the direct probability. Specifically, we applied these autoencoders for unsteady and compressible fluid flows in aircraft engines. We used inferential methods to compute a sharp approximation of the posterior probability of these parameters with the transient dynamics of the training velocity fields and to generate plausible velocity fields. An important application is the initialization of transient numerical simulations of unsteady fluid flows and large eddy simulations in fluid dynamics. It is known by the Bayes theorem that the choice of the prior distribution is very important for the computation of the posterior probability, proportional to the product of likelihood with the prior probability. Hence, we propose a new inference model based on a new prior defined by the density estimate with the realizations of the kernel proper orthogonal decomposition coefficients of the available training data. We numerically show that this inference model improves the results obtained with the usual standard normal prior distribution. This inference model was constructed using a new algorithm improving the convergence of the parametric optimization of the encoder probability distribution that approaches the posterior. This latter probability distribution is data-targeted, similarly to the prior distribution. This new generative approach can also be seen as an improvement of the kernel proper orthogonal decomposition method, for which we do not usually have a robust technique for expressing the pre-image in the input physical space of the stochastic reduced field in the feature high-dimensional space with a kernel inner product.

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Section: Introductionmentioning

confidence: 99%

Data-Targeted Prior Distribution for Variational AutoEncoder

Akkari

Casenave

Daniel

et al. 2021

Fluids

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“…A nonparametric probabilistic approach for modeling uncertainties due to more general modeling errors was introduced in , in the context of linear structural dynamics. It is organized in two steps.…”

Section: Introductionmentioning

confidence: 99%

A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models

Soize

Farhat

2016

Numerical Meth Engineering

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International audienceA nonparametric probabilistic approach for modeling uncertainties in projection-based, nonlinear, reduced-order models is presented. When experimental data is available, this approach can also quantify uncertainties in the associated high-dimensional models. The main underlying idea is twofold. First, to substitute the deterministic Reduced-Order Basis (ROB) with a stochastic counterpart. Second, to construct the probability measure of the Stochastic Reduced-Order Basis (SROB) on a subset of a compact Stiefel manifold in order to preserve some important properties of a ROB. The stochastic modeling is performed so that the probability distribution of the constructed SROB depends on a small number of hyperparameters. These are determined by solving a reduced-order statistical inverse problem. The mathematical properties of this novel approach for quantifying model uncertainties are analyzed through theoretical developments and numerical simulations. Its potential is demonstrated through several example problems from computational structural dynamics

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Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

2022

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In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $\left (A(x)-\lambda B(x)\right )v=0$ A ( x ) − λ B ( x ) v = 0 , where A and B are symmetric matrix valued functions in $\mathbb {R}^{n\times n}$ ℝ n × n , smoothly depending on parameters $x\in {\Omega }\subset \mathbb {R}^2$ x ∈ Ω ⊂ ℝ 2 ; furthermore, B is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). Our main goal is precisely that of locating parameter values where eigenvalues are equal. We first give general theoretical results for the present generalized eigenvalue problem, and then introduce and implement numerical methods apt at detecting conical intersections. Finally, we perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where A and B are either full or banded, for several bandwidths.

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Random Matrix Models and Nonparametric Method for Uncertainty Quantification

Cited by 5 publications

References 107 publications

Data-Targeted Prior Distribution for Variational AutoEncoder

Data-Targeted Prior Distribution for Variational AutoEncoder

A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models

Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

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