Uncertainty Quantification (UQ) has been an active area of research in recent years with a wide range of applications in data and imaging sciences. In many problems, the source of uncertainty stems from an unknown parameter in the model. In physical and engineering systems for example, the parameters of the partial differential equation (PDE) that model the observed data may be unknown or incompletely specified. In such cases, one may use a probabilistic description based on prior information and formulate a forward UQ problem of characterising the uncertainty in the PDE solution and observations in response to that in the parameters. Conversely, inverse UQ encompasses the statistical estimation of the unknown parameters from the available observations, which can be cast as a Bayesian inverse problem. The contributions of the thesis focus on examining the aforementioned forward and inverse UQ problems for the low-frequency, time-harmonic Maxwell equations, where the model uncertainty emanates from the lack of knowledge of the material conductivity parameter. The motivation comes from the Controlled-Source Electromagnetic Method (CSEM) that aims to detect and image hydrocarbon reservoirs by using electromagnetic field (EM) measurements to obtain information about the conductivity profile of the sub-seabed. Traditionally, algorithms for deterministic models have been employed to solve the inverse problem in CSEM by optimisation and regularisation methods, which aside from the image reconstruction provide no quantitative information on the credibility of its features. This work employs instead stochastic models where the conductivity is represented as a lognormal random field, with the objective of providing a more informative characterisation of the model observables and the unknown parameters. The variational formulation of these stochastic models is analysed and proved to be well-posed under suitable assumptions. For computational purposes the stochastic formulation is recast as a deterministic, parametric problem with distributed uncertainty, which leads to an infinite-dimensional integration problem with respect to the prior and posterior measure. One of the main challenges is thus the approximation of these integrals, with the standard choice being some variant of the Monte-Carlo (MC) method. However, such methods typically fail to take advantage of the intrinsic properties of the model and suffer from unsatisfactory convergence rates. Based on recently developed theory on high-dimensional approximation, this thesis advocates the use of Sparse Quadrature (SQ) to tackle the integration problem. For the models considered here and under certain assumptions, we prove that for forward UQ, Sparse Quadrature can attain dimension-independent convergence rates that out-perform MC. Typical CSEM models are large-scale and thus additional effort is made in this work to reduce the cost of obtaining forward solutions for each sampling parameter by utilising the weighted Reduced Basis method (RB) and the Empirical Interpolation Method (EIM). The proposed variant of a combined SQ-EIM-RB algorithm is based on an adaptive selection of training sets and a primal-dual, goal-oriented formulation for the EIM-RB approximation. Numerical examples show that the suggested computational framework can alleviate the computational costs associated with forward UQ for the pertinent large-scale models, thus providing a viable methodology for practical applications.

Cette thèse aborde la problématique de la modélisation multiphysique en génie électrique, avec une application à l’étude des vibrations d’origine électromagnétique des cages de développantes. Cette étude comporte quatre parties : la construction de la densité de courant, le calcul des forces locales, le transfert de solutions entre maillages et la résolution des problèmes couplés. Un premier enjeu est de correctement représenter les courants, cette opération est effectuée en deux étapes : la construction de la densité de courant et l’annulation de la divergence. Si des structures complexes sont utilisées, l’imposition du courant ne peut pas toujours être réalisée à l’aide de méthodes analytiques. Une méthode basée sur une résolution électrocinétique ainsi qu’une méthode purement géométrique sont testées. Cette dernière donne des résultats plus proches de la densité de courant réelle. Parmi les nombreuses méthodes de calcul de forces, les méthodes des travaux virtuels et des forces de Laplace, considérées par la littérature comme les plus adaptées au calcul des forces locales, ont été étudiées. Nos travaux ont montré que bien que les forces de Laplace sont particulièrement précises, elles ne sont pas valables si la perméabilité n’est plus homogène. Ainsi, la méthode des travaux virtuels, applicable de manière universelle, est préférée. Afin de modéliser des problèmes multi-physiques complexes à l’aide de plusieurs codes de calculs dédiés, des méthodes de transferts entre maillages non conformes ont été développées. Les procédures d’interpolations, les méthodes localement conservatives et les projections orthogonales sont comparées. Les méthodes d’interpolations sont réputées rapides mais très diffusives tandis que les méthodes de projections sont considérées comme les plus précises. La méthode localement conservative peut être vue comme produisant des résultats comparables aux méthodes de projections, mais évite l’assemblage et la résolution de systèmes linéaires. La modélisation des problèmes multi-physiques est abordée à l’aide des méthodes de transferts de solutions. Pour une classe de problème donnée, l’assemblage d’un schéma de couplage n’est pas unique. Des tests sur des cas analytiques sont réalisés afin de déterminer, pour plusieurs types de couplages, les stratégies les plus appropriées.Ces travaux ont permis une application à la modélisation magnéto-mécanique des cages de développantes est présentée. / The modeling of multi-phycics problems in electrical engineering is presented, with an application to the numerical computation of vibrations within the end windings of large turbo-generators. This study is divided into four parts: the impositions of current density, the computation of local forces, the transfer of data between disconnected meshes, and the computation of multi-physics problems using weak coupling, Firstly, the representation of current density within numerical models is presented. The process is decomposed into two stages: the construction of the initial current density, and the determination of a divergence-free field. The representation of complex geometries makes the use of analytical methods impossible. A method based on an electrokinetical problem is used and a fully geometrical method are tested. The geometrical method produces results closer to the real current density than the electrokinetical problem. Methods to compute forces are numerous, and this study focuses on the virtual work principle and the Laplace force considering the recommendations of the literature. Laplace force is highly accurate but is applicable only if the permeability is uniform. The virtual work principle is finally preferred as it appears as the most general way to compute local forces. Mesh-to-mesh data transfer methods are developed to compute multi-physics models using multiples meshes adapted to the subproblems and multiple computational software. The interpolation method, a locally conservative projection, and an orthogonal projection are compared. Interpolation method is said to be fast but highly diffusive, and the orthogonal projections are highly accurate. The locally conservative method produces results similar to the orthogonal projection but avoid the assembly of linear systems. The numerical computation of multi-physical problems using multiple meshes and projections is then presented. However for a given class of problems, there is not an unique coupling scheme possible. Analytical tests are used to determine, for different class of problems, the most accurate scheme. Finally, numerical computations applied to the structure of end-windings is presented.