Optimal projection of observations in a Bayesian setting

Optimal dimensionality reduction methods are proposed for the Bayesian inference of a Gaussian linear model with additive noise in presence of overabundant data. Three different optimal projections of the observations are proposed based on information theory: the projection that minimizes the Kullback-Leibler divergence between the posterior distributions of the original and the projected models, the one that minimizes the expected Kullback-Leibler divergence between the same distributions, and the one that maximizes the mutual information between the parameter of interest and the projected observations. The first two optimization problems are formulated as the determination of an optimal subspace and therefore the solution is computed using Riemannian optimization algorithms on the Grassmann manifold. Regarding the maximization of the mutual information, it is shown that there exists an optimal subspace that minimizes the entropy of the posterior distribution of the reduced model; a basis of the subspace can be computed as the solution to a generalized eigenvalue problem; an a priori error estimate on the mutual information is available for this particular solution; and that the dimensionality of the subspace to exactly conserve the mutual information between the input and the output of the models is less than the number of parameters to be inferred. Numerical applications to linear and nonlinear models are used to assess the efficiency of the proposed approaches, and to highlight their advantages compared to standard approaches based on the principal component analysis of the observations

Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function

This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-\Loeve expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Lo\`{e}ve coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Lo\`{e}ve expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters' uncertainty in the inference formulation.Comment: 34 pages, 17 figure

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.Comment: 30 pages, 15 Figure

Méthodes spectrales pour la propagation d'incertitudes dans les écoulements

Nous présenterons une revue des décompositions spectrales stochastiques pour la propagation d'incertitudes paramétriques dans les modèles d'écoulements. L'exposé sera centrée sur les projections de Galerkin stochastiques et leurs développements récents: les décomposition spectrales généralisées et les techniques adaptatives par multi-ondelettes stochastiques. Des exemples d'application aux équations de Navier-Stokes incompressible et aux écoulements hyperboliques illustreront la présentation

Estimation des forces aérodynamiques sur un corps 3D par une analyse volumique des structures tourbillonnaires

L'étude porte sur une approche novatrice de la détermination des forces aérodynamiques sur un obstacle dans un écoulement consécutive à une nouvelle forme du bilan de quantité de mouvement exprimant les forces en un terme volumique prépondérant et un autre sur le contour du domaine. Par une analyse fine des structures au voisinage d'un corps 3D en dérapage et à l'aide de ce bilan il est ainsi possible d'estimer les forces et les contributions nettes de chaque structure à ces forces

Experimental and numerical optimizations of an upwind mainsail trimming

International audienceThis paper investigates the use of meta-models for optimizing sails trimming. A Gaussian process is used to robustly approximate the dependence of the performance with the trimming parameters to be optimized. The Gaussian process construction uses a limited number of performance observations at carefully selected trimming points, potentially enabling the optimization of complex sail systems with multiple trimming parameters. We test the optimization procedure on the (two parameters) trimming of a scaled IMOCA mainsail in upwind conditions. To assess the robustness of the Gaussian process approach, in particular its sensitivity to error and noise in the performance estimation, we contrast the direct optimization of the physical system with the optimization of its numerical model. For the physical system, the optimization procedure was fed with wind tunnel measurements , while the numerical modeling relied on a fully non-linear Fluid-Structure Interaction solver. The results show a correct agreement of the optimized trimming parameters for the physical and numerical models, despite the inherent errors in the numerical model and the measurement uncertainties. In addition, the number of performance estimations was found to be affordable and comparable in the two cases, demonstrating the effectiveness of the approach

Variance-based sensitivity analysis of oil spill predictions in the Red Sea region

To support accidental spill rapid response efforts, oil spill simulations may generally need to account for uncertainties concerning the nature and properties of the spill, which compound those inherent in model parameterizations. A full detailed account of these sources of uncertainty would however require prohibitive resources needed to sample a large dimensional space. In this work, a variance-based sensitivity analysis is conducted to explore the possibility of restricting a priori the set of uncertain parameters, at least in the context of realistic simulations of oil spills in the Red Sea region spanning a two-week period following the oil release. The evolution of the spill is described using the simulation capabilities of Modelo Hidrodinâmico, driven by high-resolution metocean fields of the Red Sea (RS) was adopted to simulate accidental oil spills in the RS. Eight spill scenarios are considered in the analysis, which are carefully selected to account for the diversity of metocean conditions in the region. Polynomial chaos expansions are employed to propagate parametric uncertainties and efficiently estimate variance-based sensitivities. Attention is focused on integral quantities characterizing the transport, deformation, evaporation and dispersion of the spill. The analysis indicates that variability in these quantities may be suitably captured by restricting the set of uncertain inputs parameters, namely the wind coefficient, interfacial tension, API gravity, and viscosity. Thus, forecast variability and confidence intervals may be reasonably estimated in the corresponding four-dimensional input space

A domain decomposition method for stochastic elliptic differential equations

In this talk I will discuss the use of a Domain Decomposition method to reduced the computational complexity of classical problems arising in Uncertainty Quantification and stochastic Partial Differential equations. The first problem concerns the determination of the Karhunen-Loeve decomposition of a stochastic process given its covariance function. We propose to solve independently the decomposition problem over a set of subdomains, each with low complexity cost, and subsequently assemble a reduced problem to determined the global problem solution. We propose error estimates to control the resulting approximation error. Second, these ideas are extended to construct an efficient sampling approach for elliptic problems with stochastic coefficients expanded in a KL form. Here, we rely on the resolution of low complexity local stochastic elliptic problems to exhibit contributions to the condensed stochastic problem for the unknown boundary values at the internal subdomain boundaries. By relying intensively on local resolutions, that can be performed independently, the proposed approaches are naturally suited to parallel implementation and we will provide scalability results.Non UBCUnreviewedAuthor affiliation: Centre National de la Recherche ScientifiqueFacult