Une prédiction fidèle des écoulements hypersoniques à haute enthalpie est capitale pour les missions d'entrée atmosphérique. Cependant, la présence d'incertitudes est inévitable, sur les conditions de l'écoulement libre comme sur d'autres paramètres des modèles physico-chimiques. Pour cette raison, une quantification rigoureuse de l'effet de ces incertitudes est obligatoire pour évaluer la robustesse et la prédictivité des simulations numériques. De plus, une reconstruction correcte des paramètres incertains à partir des mesures en vol peut aider à réduire le niveau d'incertitude sur les sorties. Dans ce travail, nous utilisons un cadre statistique pour la propagation directe des incertitudes ainsi que pour la reconstruction inverse des conditions de l'écoulement libre dans le cas d'écoulements de rentrée atmosphérique. La possibilité d'exploiter les mesures de flux thermique au nez du véhicule pour la reconstruction des variables de l'écoulement libre et des paramètres incertains du modèle est évaluée pour les écoulements de rentrée hypersoniques. Cette reconstruction est réalisée dans un cadre bayésien, permettant la prise en compte des différentes sources d'incertitudes et des erreurs de mesure. Différentes techniques sont introduites pour améliorer les capacités de la stratégie statistique de quantification des incertitudes. Premièrement, une approche est proposée pour la génération d'un métamodèle amélioré, basée sur le couplage de Kriging et Sparse Polynomial Dimensional Decomposition. Ensuite, une méthode d'ajoute adaptatif de nouveaux points à un plan d'expériences existant est présentée dans le but d'améliorer la précision du métamodèle créé. Enfin, une manière d'exploiter les sous-espaces actifs dans les algorithmes de Markov Chain Monte Carlo pour les problèmes inverses bayésiens est également exposée. / Accurate prediction of hypersonic high-enthalpy flows is of main relevance for atmospheric entry missions. However, uncertainties are inevitable on freestream conditions and other parameters of the physico-chemical models. For this reason, a rigorous quantification of the effect of uncertainties is mandatory to assess the robustness and predictivity of numerical simulations. Furthermore, a proper reconstruction of uncertain parameters from in-flight measurements can help reducing the level of uncertainties of the output. In this work, we will use a statistical framework for direct propagation of uncertainties and inverse freestream reconstruction applied to atmospheric entry flows. We propose an assessment of the possibility of exploiting forebody heat flux measurements for the reconstruction of freestream variables and uncertain parameters of the model for hypersonic entry flows. This reconstruction is performed in a Bayesian framework, allowing to account for sources of uncertainties and measurement errors. Different techniques are introduced to enhance the capabilities of the statistical framework for quantification of uncertainties. First, an improved surrogate modeling technique is proposed, based on Kriging and Sparse Polynomial Dimensional Decomposition. Then a method is proposed to adaptively add new training points to an existing experimental design to improve the accuracy of the trained surrogate model. A way to exploit active subspaces in Markov Chain Monte Carlo algorithms for Bayesian inverse problems is also proposed.

La vitesse critique de flottement est un facteur essentiel à la conception aéronautique car elle caractérise le régime de vol au-delà duquel l’aéronef risque de subir un mécanisme de ruine. L’objectif de cette thèse est d’étudier l’impact des incertitudes d’origines aléatoires et épistémiques sur la limite de stabilité linéaire pour des configurations aéroélastiques idéalisées. Dans un premier temps, un problème de propagation directe d’incertitudes aléatoires relatives à des paramètres de fabrication d’une aile en forme de plaque en matériau composite stratifié a été considéré. La représentation du matériau par la méthode polaire lève la contrainte de grande dimensionnalité du problème stochastique initial et permet l’utilisation du Chaos Polynômial. Cependant, la corrélation introduite par cette paramétrisation nécessite une adaptation de la base polynômiale. Enfin, un algorithme d’apprentissage automatique a été employé pour traiter des discontinuités dans le comportement modal des instabilités aéroélastiques. Le second volet de la thèse concerne la quantification d’incertitudes de modélisation de caractère épistémique qui sont introduites au niveau de l’opérateur aérodynamique. Ces travaux, menés à partir d’un formalisme Bayésien, permettent non seulement d’établir des probabilités de modèle, mais aussi de calibrer les coefficients des modèles dans un contexte stochastique afin d’obtenir des prédictions robustes pour la vitesse critique. Enfin, une étude combinée des deux types d’incertitude permet d’améliorer le processus de calibration. / The critical flutter velocity is an essential factor in aeronautic design because it caracterises the flight envelope outside which the aircraft risks to be destroyed. The goal of this thesis is the study of the impact of uncertainties of aleatory and epistemic origin on the linear stability limit of idealised aeroelastic configurations. First, a direct propagation problem of aleatory uncertainties related to manufacturing parameters of a rectangular plate wing made of a laminated composite material was considered. The representation of the material through the polar method alleviates the constraint of the high number of dimensions of the initial stochastic problem, which allows the use of polynomial chaos. However, the correlation which is introduced by this parametrisation requires an adaption of the polynomial basis. Finally, a machine learning algorithm is employed for the treatment of discontinuities in the modal behaviour of the aeroelastic instabilities. The second part of the thesis is about the quantification of modelling uncertainties of epistemic nature which are introduced in the aerodynamic operator. This work, which is conducted based on a Bayesian formalism, allows not only to establish model probabilities, but also to calibrate the model coefficients in a stochastic context in order to obtain robust predictions for the critical velocity. Finally, a combined study of the two types of uncertainty allows to improve the calibration process.

Conservation laws with uncertain initial and boundary conditions are approximated using a generalized polynomial chaos expansion approach where the solution is represented as a generalized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets. The stochastic Galerkin method is used to project the governing partial differential equation onto the stochastic basis functions to obtain an extended deterministic system. The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain viscosity. We investigate well-posedness, monotonicity and stability for the stochastic Galerkin system. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability. We investigate the impact of the total spatial operator on the convergence to steady-state.  Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundary conditions. An analysis of the truncated polynomial chaos system presents a qualitative description of the development of the solution over time. An analytical solution is derived and the true polynomial chaos coefficients are shown to be smooth, while the corresponding coefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. We discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions. We present a new fully intrusive method for the Euler equations subject to uncertainty based on a Roe variable transformation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, it is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. A multiwavelet basis that can handle  discontinuities in a robust way is used. Finally, we investigate a two-phase flow problem. Based on regularity analysis of the generalized polynomial chaos coefficients, we present a hybrid method where solution regions of varying smoothness are coupled weakly through interfaces. In this way, we couple smooth solutions solved with high-order finite difference methods with non-smooth solutions solved for with shock-capturing methods.

In the first part of the thesis, we propose a new class of designs called multi-layer sliced Latin hypercube design (DSLHD) for running computer experiments. A general recursive strategy for constructing MLSLHD has been developed. Ordinary Latin hypercube designs and sliced Latin hypercube designs are special cases of MLSLHD with zero and one layer respectively. A special case of MLSLHD with two layers, doubly sliced Latin hypercube design, is studied in detail. The doubly sliced structure of DSLHD allows more flexible batch size than SLHD for collective evaluation of different computer models or batch sequential evaluation of a single computer model. Both finite-sample and asymptotical sampling properties of DSLHD are examined. Numerical experiments are provided to show the advantage of DSLHD over SLHD for both sequential evaluating a single computer model and collective evaluation of different computer models. Other applications of DSLHD include design for Gaussian process modeling with quantitative and qualitative factors, cross-validation, etc. Moreover, we also show the sliced structure, possibly combining with other criteria such as distance-based criteria, can be utilized to sequentially sample from a large spatial data set when we cannot include all the data points for modeling. A data center example is presented to illustrate the idea. The enhanced stochastic evolutionary algorithm is deployed to search for optimal design. In the second part of the thesis, we propose a new smoothing technique called completely-data-driven smoothing, intended for smoothing over irregular regions. The idea is to replace the penalty term in the smoothing splines by its estimate based on local least squares technique. A close form solution for our approach is derived. The implementation is very easy and computationally efficient. With some regularity assumptions on the input region and analytical assumptions on the true function, it can be shown that our estimator achieves the optimal convergence rate in general nonparametric regression. The algorithmic parameter that governs the trade-off between the fidelity to the data and the smoothness of the estimated function is chosen by generalized cross validation (GCV). The asymptotic optimality of GCV for choosing the algorithm parameter in our estimator is proved. Numerical experiments show that our method works well for both regular and irregular region smoothing. The third part of the thesis deals with uncertainty quantification in building energy assessment. In current practice, building simulation is routinely performed with best guesses of input parameters whose true value cannot be known exactly. These guesses affect the accuracy and reliability of the outcomes. There is an increasing need to perform uncertain analysis of those input parameters that are known to have a significant impact on the final outcome. In this part of the thesis, we focus on uncertainty quantification of two microclimate parameters: the local wind speed and the wind pressure coefficient. The idea is to compare the outcome of the standard model with that of a higher fidelity model. Statistical analysis is then conducted to build a connection between these two. The explicit form of statistical models can facilitate the improvement of the corresponding modules in the standard model.

Uncertainty quantification involves sampling the reservoir parameters correctly from a posterior probability function that is conditioned to both static and dynamic data. Rigorous sampling methods like Markov Chain Monte Carlo (MCMC) are known to sample from the distribution but can be computationally prohibitive for high resolution reservoir models. Approximate sampling methods are more efficient but less rigorous for nonlinear inverse problems. There is a need for an efficient and rigorous approach to uncertainty quantification for the nonlinear inverse problems. First, we propose a two-stage MCMC approach using sensitivities for quantifying uncertainty in history matching geological models. In the first stage, we compute the acceptance probability for a proposed change in reservoir parameters based on a linearized approximation to flow simulation in a small neighborhood of the previously computed dynamic data. In the second stage, those proposals that passed a selected criterion of the first stage are assessed by running full flow simulations to assure the rigorousness. Second, we propose a two-stage MCMC approach using response surface models for quantifying uncertainty. The formulation allows us to history match three-phase flow simultaneously. The built response exists independently of expensive flow simulation, and provides efficient samples for the reservoir simulation and MCMC in the second stage. Third, we propose a two-stage MCMC approach using upscaling and non-parametric regressions for quantifying uncertainty. A coarse grid model acts as a surrogate for the fine grid model by flow-based upscaling. The response correction of the coarse-scale model is performed by error modeling via the non-parametric regression to approximate the response of the computationally expensive fine-scale model. Our proposed two-stage sampling approaches are computationally efficient and rigorous with a significantly higher acceptance rate compared to traditional MCMC algorithms. Finally, we developed a coarsening algorithm to determine an optimal reservoir simulation grid by grouping fine scale layers in such a way that the heterogeneity measure of a defined static property is minimized within the layers. The optimal number of layers is then selected based on a statistical analysis. The power and utility of our approaches have been demonstrated using both synthetic and field examples.

We propose a variant of the stochastic finite element method, where the random elements occuring in the problem formulation are approximated by simple random elements, i.e. random elements with only a finite number of possible values.

Subsurface flow phenomena characterize many important societal issues in energy and the environment. A key feature of these problems is that subsurface properties are uncertain, due to the sparsity of direct observations of the subsurface. The Bayesian formulation of this inverse problem provides a systematic framework for inferring uncertainty in the properties given uncertainties in the data, the forward model, and prior knowledge of the properties. We address the problem: given noisy measurements of the head, the pdf describing the noise, prior information in the form of a pdf of the hydraulic conductivity, and a groundwater flow model relating the head to the hydraulic conductivity, find the posterior probability density function (pdf) of the parameters describing the hydraulic conductivity field. Unfortunately, conventional sampling of this pdf to compute statistical moments is intractable for problems governed by large-scale forward models and high-dimensional parameter spaces. We construct a Gaussian process surrogate of the posterior pdf based on Bayesian interpolation between a set of "training" points. We employ a greedy algorithm to find the training points by solving a sequence of optimization problems where each new training point is placed at the maximizer of the error in the approximation. Scalable Newton optimization methods solve this "optimal" training point problem. We tailor the Gaussian process surrogate to the curvature of the underlying posterior pdf according to the Hessian of the log posterior at a subset of training points, made computationally tractable by a low-rank approximation of the data misfit Hessian. A Gaussian mixture approximation of the posterior is extracted from the Gaussian process surrogate, and used as a proposal in a Markov chain Monte Carlo method for sampling both the surrogate as well as the true posterior. The Gaussian process surrogate is used as a first stage approximation in a two-stage delayed acceptance MCMC method. We provide evidence for the viability of the low-rank approximation of the Hessian through numerical experiments on a large scale atmospheric contaminant transport problem and analysis of an infinite dimensional model problem. We provide similar results for our groundwater problem. We then present results from the proposed MCMC algorithms. / text

In this dissertation, we focus on the uncertainty quantification problems where the goal is to sample the porous media properties given integrated responses. We first introduce a reduced order model using the level set method to characterize the channelized features of permeability fields. The sampling process is completed under Bayesian framework. We hence study the regularity of posterior distributions with respect to the prior measures. The stochastic flow equations that contain both spatial and random components must be resolved in order to sample the porous media properties. Some type of upscaling or multiscale technique is needed when solving the flow and transport through heterogeneous porous media. We propose ensemble-level multiscale finite element method and ensemble-level preconditioner technique for solving the stochastic flow equations, when the permeability fields have certain topology features. These methods can be used to accelerate the forward computations in the sampling processes. Additionally, we develop analysis-of-variance-based mixed multiscale finite element method as well as a novel adaptive version. These methods are used to study the forward uncertainty propagation of input random fields. The computational cost is saved since the high dimensional problem is decomposed into lower dimensional problems. We also work on developing efficient advanced Markov Chain Monte Carlo methods. Algorithms are proposed based on the multi-stage Markov Chain Monte Carlo and Stochastic Approximation Monte Carlo methods. The new methods have the ability to search the whole sample space for optimizations. Analysis and detailed numerical results are presented for applications of all the above methods.

Recently there has been growing interest to characterize and reduce uncertainty in stochastic dynamical systems. This drive arises out of need to manage uncertainty in complex, high dimensional physical systems. Traditional techniques of uncertainty quantification (UQ) use local linearization of dynamics and assumes Gaussian probability evolution. But several difficulties arise when these UQ models are applied to real world problems, which, generally are nonlinear in nature. Hence, to improve performance, robust algorithms, which can work efficiently in a nonlinear non-Gaussian setting are desired. The main focus of this dissertation is to develop UQ algorithms for nonlinear systems, where uncertainty evolves in a non-Gaussian manner. The algorithms developed are then applied to state estimation of real-world systems. The first part of the dissertation focuses on using polynomial chaos (PC) for uncertainty propagation, and then achieving the estimation task by the use of higher order moment updates and Bayes rule. The second part mainly deals with Frobenius-Perron (FP) operator theory, how it can be used to propagate uncertainty in dynamical systems, and then using it to estimate states by the use of Bayesian update. Finally, a method to represent the process noise in a stochastic dynamical system using a nite term Karhunen-Loeve (KL) expansion is proposed. The uncertainty in the resulting approximated system is propagated using FP operator. The performance of the PC based estimation algorithms were compared with extended Kalman filter (EKF) and unscented Kalman filter (UKF), and the FP operator based techniques were compared with particle filters, when applied to a duffing oscillator system and hypersonic reentry of a vehicle in the atmosphere of Mars. It was found that the accuracy of the PC based estimators is higher than EKF or UKF and the FP operator based estimators were computationally superior to the particle filtering algorithms.

Dans de nombreuses disciplines, les approches permettant d'étudier et de quantifier l'influence de données incertaines sont devenues une nécessité. Bien que la propagation directe d'incertitudes ait été largement étudiée, la propagation inverse d'incertitudes demeure un vaste sujet d'étude, sans méthode standardisée. Dans cette thèse, une nouvelle méthode de propagation inverse d'incertitude est présentée. Le but de cette méthode est de déterminer l'incertitude d'entrée à partir de données de sortie considérées comme incertaines. Parallèlement, les méthodes d'analyse de sensibilité sont également très utilisées pour déterminer l'influence des entrées sur la sortie lors d'un processus de modélisation. Ces approches permettent d'isoler les entrées les plus significatives, c'est à dire les plus influentes, qu'il est nécessaire de tester lors d'une analyse d'incertitudes. Dans ce travail, nous approfondirons tout d'abord la méthode d'analyse de sensibilité de Sobol, qui est l'une des méthodes d'analyse de sensibilité globale les plus efficaces. Cette méthode repose sur le calcul d'indices de sensibilité, appelés indices de Sobol, qui représentent l'effet des données d'entrées (vues comme des variables aléatoires continues) sur la sortie. Nous démontrerons ensuite que la méthode de Sobol donne des résultats fiables même lorsqu'elle est appliquée dans le cas discret. Puis, nous étendrons le cadre d'application de la méthode de Sobol afin de répondre à la problématique de propagation inverse d'incertitudes. Enfin, nous proposerons une nouvelle approche de la méthode de Sobol qui permet d'étudier la variation des indices de sensibilité par rapport à certains facteurs du modèle ou à certaines conditions expérimentales. Nous montrerons que les résultats obtenus lors de ces études permettent d'illustrer les différentes caractéristiques des données d'entrée. Pour conclure, nous exposerons comment ces résultats permettent d'indiquer les meilleures conditions expérimentales pour lesquelles l'estimation des paramètres peut être efficacement réalisée. / Approaches for studying uncertainty are of great necessity in all disciplines. While the forward propagation of uncertainty has been investigated extensively, the backward propagation is still under studied. In this thesis, a new method for backward propagation of uncertainty is presented. The aim of this method is to determine the input uncertainty starting from the given data of the uncertain output. In parallel, sensitivity analysis methods are also of great necessity in revealing the influence of the inputs on the output in any modeling process. This helps in revealing the most significant inputs to be carried in an uncertainty study. In this work, the Sobol sensitivity analysis method, which is one of the most efficient global sensitivity analysis methods, is considered and its application framework is developed. This method relies on the computation of sensitivity indexes, called Sobol indexes. These indexes give the effect of the inputs on the output. Usually inputs in Sobol method are considered to vary as continuous random variables in order to compute the corresponding indexes. In this work, the Sobol method is demonstrated to give reliable results even when applied in the discrete case. In addition, another advancement for the application of the Sobol method is done by studying the variation of these indexes with respect to some factors of the model or some experimental conditions. The consequences and conclusions derived from the study of this variation help in determining different characteristics and information about the inputs. Moreover, these inferences allow the indication of the best experimental conditions at which estimation of the inputs can be done.