On étudie les déformations des applications harmoniques $f$ tordues par rapport à une représentation. Après avoir construit une application harmonique tordue "universelle", on donne une construction de toute déformations du premier ordre de $f$ en termes de la théorie de Hodge ; on applique ce résultat à l'espace de modules des représentations réductives d'un groupe de Kähler, pour démontrer que les points critiques de la fonctionnelle de l'énergie $E$ coïncident avec les représentations de monodromie des variations complexes de structures de Hodge. Ensuite, on procède aux déformations du second ordre, où des obstructions surviennent ; on enquête sur l'existence de ces déformations et on donne une méthode pour le construire. En appliquant ce résultat à la fonctionnelle de l'énergie comme ci-dessus, on démontre (pour n'importe quel groupe de présentation finie) que la fonctionnelle de l'énergie est strictement pluri sous-harmonique sur l'espace des modules des représentations. En assumant de plus que le groupe soit de Kähler, on étudie les valeurs propres de la matrice hessienne de $E$ dans les points critiques. / We study the deformations of twisted harmonic maps $f$ with respect to a representation. After constructing a continuous ``universal'' twisted harmonic map, we give a construction of every first order deformation of $f$ in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a K\"ahler group, to show that the critical points of the energy functional $E$ coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is strictly pluri sub-harmonic on the moduli space of representations; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of $E$ at critical points.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-06T10:34:36Z No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-06T10:35:12Z (GMT) No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-09-06T10:35:12Z (GMT). No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-08-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study a singular partial differential problem in a bounded domain with smoth boundary. We have two main cases, one superlinear with weak singularity, and the other one sublinear with strong songularity. We use Variational Methods, such as the Ekeland Variational Principle and the Nehari Manifolds, to solve this problem, finding weak solutions and proving the multiplicity of solutions in one of the cases. / Neste trabalho estudaremos um problema diferencial parcial singular em um domínio limitado com bordo suave. Temos dois casos principais, um superlinear com singularidade fraca e um sublinear com singularidade forte. Usaremos Métodos Variacionais, como o Princípio Variacional de Ekeland e as Variedades de Nehari, para resolver este problema, encontrando soluções fracas e provando a multiplicidade das mesmas em um dos casos.

<p>The need to carry out Uncertainty quantification (UQ) is ubiquitous in science and engineering. However, carrying out UQ for real-world problems is not straightforward and they require a lot of computational budget and resources. The objective of this thesis is to develop computationally efficient approaches based on machine learning to carry out UQ. Specifically, we addressed two problems.</p> <p><br></p> <p>The first problem is, it is difficult to carry out Uncertainty propagation (UP) in systems governed by elliptic PDEs with spatially varying uncertain fields in coefficients and boundary conditions. Here as we have functional uncertainties, the number of uncertain parameters is large. Unfortunately, in these situations to carry out UP we need to solve the PDE a large number of times to obtain convergent statistics of the quantity governed by the PDE. However, solving the PDE by a numerical solver repeatedly leads to a computational burden. To address this we proposed to learn the surrogate of the solution of the PDE in a data-free manner by utilizing the physics available in the form of the PDE. We represented the solution of the PDE as a deep neural network parameterized function in space and uncertain parameters. We introduced a physics-informed loss function derived from variational principles to learn the parameters of the network. The accuracy of the learned surrogate is validated against the corresponding ground truth estimate from the numerical solver. We demonstrated the merit of using our approach by solving UP problems and inverse problems faster than by using a standard numerical solver.</p> <p><br></p> <p>The second problem we focused on in this thesis is related to inverse problems. State of the art approach to solving inverse problems involves posing the inverse problem as a Bayesian inference task and estimating the distribution of input parameters conditioned on the observed data (posterior). Markov Chain Monte Carlo (MCMC) methods and variational inference methods provides us ways to estimate the posterior. However, these inference techniques need to be re-run whenever a new set of observed data is given leading to a computational burden. To address this, we proposed to learn a Bayesian inverse map i.e., the map from the observed data to the posterior. This map enables us to do on-the-fly inference. We demonstrated our approach by solving various examples and we validated the posteriors learned from our approach against corresponding ground truth posteriors from the MCMC method.</p>

Le but de ce travail est l'étude d'un problème de transmission en dynamique de population entre deux habitats juxtaposés. Dans chacun des habitats, on considère une équation aux dérivées partielles, modélisant la dispersion généralisée, formée par une combinaison linéaire du laplacien et du bilaplacien. On commence d'abord par étudier et résoudre la même équation avec diverses conditions aux limites posée dans un seul habitat. Cette étude est effectuée grâce à une formulation opérationnelle du problème: on réécrit cette EDP sous forme d'équation différentielle, posée dans un espace de Banach construit sur les espaces Lp avec 1 < p < +∞, où les coefficients sont des opérateurs linéaires non bornés. Grâce au calcul fonctionnel, à la théorie des semi-groupes analytiques et à la théorie de l'interpolation, on obtient des résultats optimaux d'existence, d'unicité et de régularité maximale de la solution classique si et seulement si les données sont dans certains espaces d'interpolation. / The aim of this work is the study of a transmission problem in population dynamics between two juxtaposed habitats. In each habitat, we consider a partial differential equation, modeling the generalized dispersion, made up of a linear combination of Laplacian and Bilaplacian operators. We begin by studying and solving the same equation with various boundary conditions in a single habitat. This study is carried out using an operational formulation of the problem: we rewrite this PDE as a differential equation, set in a Banach space built on the spaces Lp with 1 < p < +∞, where the coefficients are unbounded linear operators. Thanks to functional calculus, analytic semigroup theory and interpolation theory, we obtain optimal results of existence, uniqueness and maximum regularity of the classical solution if and only if the data are in some interpolation spaces.