Bayesian inference occurs when prior knowledge about uncertain parameters in mathematical models is merged with new observational data related to the model outcome. In this thesis we focus on models given by partial differential equations where the uncertain parameters are coefficient functions belonging to infinite dimensional function spaces. The result of the Bayesian inference is then a well-defined posterior probability measure on a function space describing the updated knowledge about the uncertain coefficient. For decision making and post-processing it is often required to sample or integrate wit resprect to the posterior measure. This calls for sampling or numerical methods which are suitable for infinite dimensional spaces. In this work we focus on Kalman filter techniques based on ensembles or polynomial chaos expansions as well as Markov chain Monte Carlo methods. We analyze the Kalman filters by proving convergence and discussing their applicability in the context of Bayesian inference. Moreover, we develop and study an improved dimension-independent Metropolis-Hastings algorithm. Here, we show geometric ergodicity of the new method by a spectral gap approach using a novel comparison result for spectral gaps. Besides that, we observe and further analyze the robustness of the proposed algorithm with respect to decreasing observational noise. This robustness is another desirable property of numerical methods for Bayesian inference. The work concludes with the application of the discussed methods to a real-world groundwater flow problem illustrating, in particular, the Bayesian approach for uncertainty quantification in practice. / Bayessche Inferenz besteht daraus, vorhandenes a-priori Wissen über unsichere Parameter in mathematischen Modellen mit neuen Beobachtungen messbarer Modellgrößen zusammenzuführen. In dieser Dissertation beschäftigen wir uns mit Modellen, die durch partielle Differentialgleichungen beschrieben sind. Die unbekannten Parameter sind dabei Koeffizientenfunktionen, die aus einem unendlich dimensionalen Funktionenraum kommen. Das Resultat der Bayesschen Inferenz ist dann eine wohldefinierte a-posteriori Wahrscheinlichkeitsverteilung auf diesem Funktionenraum, welche das aktualisierte Wissen über den unsicheren Koeffizienten beschreibt. Für Entscheidungsverfahren oder Postprocessing ist es oft notwendig die a-posteriori Verteilung zu simulieren oder bzgl. dieser zu integrieren. Dies verlangt nach numerischen Verfahren, welche sich zur Simulation in unendlich dimensionalen Räumen eignen. In dieser Arbeit betrachten wir Kalmanfiltertechniken, die auf Ensembles oder polynomiellen Chaosentwicklungen basieren, sowie Markowketten-Monte-Carlo-Methoden. Wir analysieren die erwähnte Kalmanfilter, indem wir deren Konvergenz zeigen und ihre Anwendbarkeit im Kontext Bayesscher Inferenz diskutieren. Weiterhin entwickeln und studieren wir einen verbesserten dimensionsunabhängigen Metropolis-Hastings-Algorithmus. Hierbei weisen wir geometrische Ergodizität mit Hilfe eines neuen Resultates zum Vergleich der Spektrallücken von Markowketten nach. Zusätzlich beobachten und analysieren wir die Robustheit der neuen Methode bzgl. eines fallenden Beobachtungsfehlers. Diese Robustheit ist eine weitere wünschenswerte Eigenschaft numerischer Methoden für Bayessche Inferenz. Den Abschluss der Arbeit bildet die Anwendung der diskutierten Methoden auf ein reales Grundwasserproblem, was insbesondere den Bayesschen Zugang zur Unsicherheitsquantifizierung in der Praxis illustriert.

Dans cette thèse, nous proposons une approche progressive (forward) pour la représentation probabiliste d'Equations aux Dérivées Partielles (EDP) nonlinéaires et nonconservatives, permettant ainsi de développer un algorithme particulaire afin d'en estimer numériquement les solutions. Les Equations Différentielles Stochastiques Nonlinéaires de type McKean (NLSDE) étudiées dans la littérature constituent une formulation microscopique d'un phénomène modélisé macroscopiquement par une EDP conservative. Une solution d'une telle NLSDE est la donnée d'un couple $(Y,u)$ où $Y$ est une solution d' équation différentielle stochastique (EDS) dont les coefficients dépendent de $u$ et de $t$ telle que $u(t,cdot)$ est la densité de $Y_t$. La principale contribution de cette thèse est de considérer des EDP nonconservatives, c'est-à- dire des EDP conservatives perturbées par un terme nonlinéaire de la forme $Lambda(u,nabla u)u$. Ceci implique qu'un couple $(Y,u)$ sera solution de la représentation probabiliste associée si $Y$ est un encore un processus stochastique et la relation entre $Y$ et la fonction $u$ sera alors plus complexe. Etant donnée la loi de $Y$, l'existence et l'unicité de $u$ sont démontrées par un argument de type point fixe via une formulation originale de type Feynmann-Kac. / This thesis performs forward probabilistic representations of nonlinear and nonconservative Partial Differential Equations (PDEs), which allowto numerically estimate the corresponding solutions via an interacting particle system algorithm, mixing Monte-Carlo methods and non-parametric density estimates.In the literature, McKean typeNonlinear Stochastic Differential Equations (NLSDEs) constitute the microscopic modelof a class of PDEs which are conservative. The solution of a NLSDEis generally a couple $(Y,u)$ where $Y$ is a stochastic process solving a stochastic differential equation whose coefficients depend on $u$ and at each time $t$, $u(t,cdot)$ is the law density of the random variable $Y_t$.The main idea of this thesis is to consider this time a non-conservative PDE which is the result of a conservative PDE perturbed by a term of the type $Lambda(u, nabla u) u$. In this case, the solution of the corresponding NLSDE is again a couple $(Y,u)$, where again $Y$ is a stochastic processbut where the link between the function $u$ and $Y$ is more complicated and once fixed the law of $Y$, $u$ is determined by a fixed pointargument via an innovating Feynmann-Kac type formula.

Ce mémoire présente une méthode de reconstruction de la surface d’un pare-brise à partir d’une image observée au travers de celui-ci. Cette image est déformée, car les rayons lumineux traversant le pare-brise subissent deux réfractions : une de chaque côté du verre. La déformation de l’image est dépendante de la forme du pare-brise, c’est donc cette donnée qui est utilisée pour résoudre le problème. La première étape est la construction d’un champ de vecteurs dans l’espace ambiant à partir des déviations des rayons lumineux passant par le pare-brise. Elle repose sur la loi de la réfraction de Snell-Descartes et sur des hypothèses simplificatrices au sujet de la courbure et de l’épaisseur du pare-brise. Le vecteur en un point de ce champ correspond à une prédiction du vecteur normal à la surface, sous l’hypothèse que celle-ci passe par le point en question. La deuxième étape est de trouver une surface compatible avec le champ de vecteurs obtenu. Pour y arriver, on formule un problème de minimisation où la donnée minimisée est la différence entre les vecteurs normaux à la surface et ceux construits à partir des mesures du système d’inspection. Il en résulte une équation d’Euler-Lagrange non linéaire à laquelle on impose des conditions de Dirichlet. Le graphe de la solution à ce problème est alors la surface recherchée. La troisième étape est une méthode de point fixe pour résoudre l’équation d’Euler-Lagrange. Elle donne une suite d’équations de Poisson linéaires dont la limite des solutions respecte l’équation non linéaire étudiée. On utilise le théorème du point fixe de Banach pour obtenir des conditions suffisantes d’existence et d’unicité de la solution, qui sont aussi des conditions suffisantes pour lesquelles la méthode de point fixe converge. / This Master’s thesis presents a method for the reconstruction of a windshield surface using an image observed through it. This image is distorted because the light rays passing through the windshield undergo two refractions : one on each side of the glass. The distortion depends on the windshield shape and therefore this data is used to solve the problem. The first step is the construction of a vector field in the ambient space, from the deviations of the light rays passing through the windshield. This step relies on the Snell-Descartes refraction law and on simplifying assumptions regarding the curvature and thickness of a windshield. A vector at a point of this field corresponds to a prediction of the surface normal vector at this point, under the hypothesis that this point lies on the surface. The second step is to find a surface that is compatible with the obtained vector field. For this purpose, a minimisation problem is formulated for which the minimized variable is the difference between the surface normal vector and the one deduced from the system’s measurements. This leads to a nonlinear Euler- Lagrange equation for which the Dirichlet boundary conditions are imposed. The graph of the solution is the desired surface. The third step is a fixed-point method to solve the Euler- Lagrange equation. At the center of this method is a sequence of linear Poisson equations, each giving an approximating solution. It is shown that the limit of this sequence of solutions respects the original nonlinear equation. The Banach fixed-point theorem is used to get sufficient existence and uniqueness conditions, that are also sufficient conditions under which the proposed fixed-point method converges.

<p>Nuclear engineering hosts diverse domains including, but not limited to, power plant automation, human-machine interfacing, detection and identification of special nuclear materials, modeling of reactor kinetics and dynamics that most frequently are described by systems of differential equations (DEs), either ordinary (ODEs) or partial ones (PDEs). In this work we study multiple problems related to safety and Special Nuclear Material detection, and numerical solutions for partial differential equations using neural networks. More specifically, this work is divided in six chapters. Chapter 1 is the introduction, in Chapter</p> <p>2 we discuss the development of a gamma-ray radionuclide library for the characterization</p> <p>of gamma-spectra. In Chapter 3, we present a new approach, the ”Variance Counterbalancing”, for stochastic</p> <p>large-scale learning. In Chapter 4, we introduce a systematic approach for constructing proper trial solutions to partial differential equations (PDEs) of up to second order, using neural forms that satisfy prescribed initial, boundary and interface conditions. Chapter 5 is about an alternative, less imposing development of neural-form trial solutions for PDEs, inside rectangular and non-rectangular convex boundaries. Chapter 6 presents an ensemble method that avoids the multicollinearity issue and provides</p> <p>enhanced generalization performance that could be suitable for handling ”few-shots”- problems frequently appearing in nuclear engineering.</p>

Tesis por compendio / [ES] La idea de seguir una secuencia de pasos para lograr un resultado deseado es inherente a la naturaleza humana: desde que empezamos a andar, siguiendo una receta de cocina o aprendiendo un nuevo juego de cartas. Desde la antigüedad se ha seguido este esquema para organizar leyes, corregir escritos, e incluso asignar diagnósticos. En matemáticas a esta forma de pensar se la denomina 'algoritmo'. Formalmente, un algoritmo es un conjunto de instrucciones definidas y no-ambiguas, ordenadas y finitas, que permite solucionar un problema. Desde pequeños nos enfrentamos a ellos cuando aprendemos a multiplicar o dividir, y a medida que crecemos, estas estructuras nos permiten resolver diferentes problemas cada vez más complejos: sistemas lineales, ecuaciones diferenciales, problemas de optimización, etcétera. Hay multitud de algoritmos que nos permiten hacer frente a este tipo de problemas, como métodos iterativos, donde encontramos el famoso Método de Newton para buscar raíces; algoritmos de búsqueda para localizar un elemento con ciertas propiedades en un conjunto mayor; o descomposiciones matriciales, como la descomposición LU para resolver sistemas lineales. Sin embargo, estos enfoques clásicos presentan limitaciones cuando se enfrentan a problemas de grandes dimensiones, problema que se conoce como `la maldición de la dimensionalidad'. El avance de la tecnología, el uso de redes sociales y, en general, los nuevos problemas que han aparecido con el desarrollo de la Inteligencia Artificial, ha puesto de manifiesto la necesidad de manejar grandes cantidades de datos, lo que requiere el diseño de nuevos mecanismos que permitan su manipulación. En la comunidad científica, este hecho ha despertado el interés por las estructuras tensoriales, ya que éstas permiten trabajar eficazmente con problemas de grandes dimensiones. Sin embargo, la mayoría de métodos clásicos no están pensados para ser empleados junto a estas operaciones, por lo que se requieren herramientas específicas que permitan su tratamiento, lo que motiva un proyecto como este. El presente trabajo se divide de la siguiente manera: tras revisar algunas definiciones necesarias para su comprensión, en el Capítulo 3, se desarrolla la teoría de una nueva descomposición tensorial para matrices cuadradas. A continuación, en el Capítulo 4, se muestra una aplicación de dicha descomposición a grafos regulares y redes de mundo pequeño. En el Capítulo 5, se plantea una implementación eficiente del algoritmo que proporciona la nueva descomposición matricial, y se estudian como aplicación algunas EDP de orden dos. Por último, en los Capítulos 6 y 7 se exponen unas breves conclusiones y se enumeran algunas de las referencias consultadas, respectivamente. / [CA] La idea de seguir una seqüència de passos per a aconseguir un resultat desitjat és inherent a la naturalesa humana: des que comencem a caminar, seguint una recepta de cuina o aprenent un nou joc de cartes. Des de l'antiguitat s'ha seguit aquest esquema per a organitzar lleis, corregir escrits, i fins i tot assignar diagnòstics. En matemàtiques a aquesta manera de pensar se la denomina algorisme. Formalment, un algorisme és un conjunt d'instruccions definides i no-ambigües, ordenades i finites, que permet solucionar un problema. Des de xicotets ens enfrontem a ells quan aprenem a multiplicar o dividir, i a mesura que creixem, aquestes estructures ens permeten resoldre diferents problemes cada vegada més complexos: sistemes lineals, equacions diferencials, problemes d'optimització, etcètera. Hi ha multitud d'algorismes que ens permeten fer front a aquesta mena de problemes, com a mètodes iteratius, on trobem el famós Mètode de Newton per a buscar arrels; algorismes de cerca per a localitzar un element amb unes certes propietats en un conjunt major; o descomposicions matricials, com la descomposició DL. per a resoldre sistemes lineals. No obstant això, aquests enfocaments clàssics presenten limitacions quan s'enfronten a problemes de grans dimensions, problema que es coneix com `la maledicció de la dimensionalitat'. L'avanç de la tecnologia, l'ús de xarxes socials i, en general, els nous problemes que han aparegut amb el desenvolupament de la Intel·ligència Artificial, ha posat de manifest la necessitat de manejar grans quantitats de dades, la qual cosa requereix el disseny de nous mecanismes que permeten la seua manipulació. En la comunitat científica, aquest fet ha despertat l'interés per les estructures tensorials, ja que aquestes permeten treballar eficaçment amb problemes de grans dimensions. No obstant això, la majoria de mètodes clàssics no estan pensats per a ser emprats al costat d'aquestes operacions, per la qual cosa es requereixen eines específiques que permeten el seu tractament, la qual cosa motiva un projecte com aquest. El present treball es divideix de la següent manera: després de revisar algunes definicions necessàries per a la seua comprensió, en el Capítol 3, es desenvolupa la teoria d'una nova descomposició tensorial per a matrius quadrades. A continuació, en el Capítol 4, es mostra una aplicació d'aquesta descomposició a grafs regulars i xarxes de món xicotet. En el Capítol 5, es planteja una implementació eficient de l'algorisme que proporciona la nova descomposició matricial, i s'estudien com a aplicació algunes EDP d'ordre dos. Finalment, en els Capítols 6 i 7 s'exposen unes breus conclusions i s'enumeren algunes de les referències consultades, respectivament. / [EN] The idea of following a sequence of steps to achieve a desired result is inherent in human nature: from the moment we start walking, following a cooking recipe or learning a new card game. Since ancient times, this scheme has been followed to organize laws, correct writings, and even assign diagnoses. In mathematics, this way of thinking is called an algorithm. Formally, an algorithm is a set of defined and unambiguous instructions, ordered and finite, that allows for solving a problem. From childhood, we face them when we learn to multiply or divide, and as we grow, these structures will enable us to solve different increasingly complex problems: linear systems, differential equations, optimization problems, etc. There is a multitude of algorithms that allow us to deal with this type of problem, such as iterative methods, where we find the famous Newton Method to find roots; search algorithms to locate an element with specific properties in a more extensive set; or matrix decompositions, such as the LU decomposition to solve some linear systems. However, these classical approaches have limitations when faced with large-dimensional problems, a problem known as the `curse of dimensionality'. The advancement of technology, the use of social networks and, in general, the new problems that have appeared with the development of Artificial Intelligence, have revealed the need to handle large amounts of data, which requires the design of new mechanisms that allow its manipulation. This fact has aroused interest in the scientific community in tensor structures since they allow us to work efficiently with large-dimensional problems. However, most of the classic methods are not designed to be used together with these operations, so specific tools are required to allow their treatment, which motivates work like this. This work is divided as follows: after reviewing some definitions necessary for its understanding, in Chapter 3, the theory of a new tensor decomposition for square matrices is developed. Next, Chapter 4 shows an application of said decomposition to regular graphs and small-world networks. In Chapter 5, an efficient implementation of the algorithm provided by the new matrix decomposition is proposed, and some order two PDEs are studied as an application. Finally, Chapters 6 and 7 present some brief conclusions and list some of the references consulted. / María Mora Jiménez acknowledges funding from grant (ACIF/2020/269) funded by the Generalitat Valenciana and the European Social Found / Mora Jiménez, M. (2023). Numerical approximations with tensor-based techniques for high-dimensional problems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/202604 / Compendio

<p dir="ltr">Neural operator learning is emerging as a prominent technique in scientific machine learn- ing for modeling complex nonlinear systems with multi-physics and multi-scale applications. A common drawback of such operators is that they are data-hungry and the results are highly dependent on the quality and quantity of the training data provided to the models. Moreover, obtaining high-quality data in sufficient quantity can be computationally prohibitive. Faster surrogate models are required to overcome this drawback which can be learned from datasets of variable fidelity and also quantify the uncertainty. In this work, we propose a Multi-Level Stacked Deep Operator Network (MLSDON) which can learn from datasets of different fidelity and is not dependent on the input function. Through various experiments, we demonstrate that the MLSDON can approximate the high-fidelity solution operator with better accuracy compared to a Vanilla DeepONet when sufficient high-fidelity data is unavailable. We also extend MLSDON to build robust confidence intervals by making conformalized predictions. This technique guarantees trajectory coverage of the predictions irrespective of the input distribution. Various numerical experiments are conducted to demonstrate the applicability of MLSDON to multi-fidelity, multi-scale, and multi-physics problems.</p>

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This thesis details the development, verification and validation of an unsteady unstructured high order (≥ 3) h/p Discontinuous Galerkin - Fourier solver for the incompressible Navier-Stokes equations on static and rotating meshes in two and three dimensions. This general purpose solver is used to provide insight into cross-flow (wind or tidal) turbine physical phenomena. Simulation of this type of turbine for renewable energy generation needs to account for the rotational motion of the blades with respect to the fixed environment. This rotational motion implies azimuthal changes in blade aero/hydro-dynamics that result in complex flow phenomena such as stalled flows, vortex shedding and blade-vortex interactions. Simulation of these flow features necessitates the use of a high order code exhibiting low numerical errors. This thesis presents the development of such a high order solver, which has been conceived and implemented from scratch by the author during his doctoral work. To account for the relative mesh motion, the incompressible Navier-Stokes equations are written in arbitrary Lagrangian-Eulerian form and a non-conformal Discontinuous Galerkin (DG) formulation (i.e. Symmetric Interior Penalty Galerkin) is used for spatial discretisation. The DG method, together with a novel sliding mesh technique, allows direct linking of rotating and static meshes through the numerical fluxes. This technique shows spectral accuracy and no degradation of temporal convergence rates if rotational motion is applied to a region of the mesh. In addition, analytical mappings are introduced to account for curved external boundaries representing circular shapes and NACA foils. To simulate 3D flows, the 2D DG solver is parallelised and extended using Fourier series. This extension allows for laminar and turbulent regimes to be simulated through Direct Numerical Simulation and Large Eddy Simulation (LES) type approaches. Two LES methodologies are proposed. Various 2D and 3D cases are presented for laminar and turbulent regimes. Among others, solutions for: Stokes flows, the Taylor vortex problem, flows around square and circular cylinders, flows around static and rotating NACA foils and flows through rotating cross-flow turbines, are presented.

Les objets d’étude de cette thèse sont les systèmes d’équations quasilinéaires du premier ordre. Dans une première partie, on fait une analyse du point de vue du groupe de Lie classique des symétries ponctuelles d’un modèle de la plasticité idéale. Les écoulements planaires dans les cas stationnaire et non-stationnaire sont étudiés. Deux nouveaux champs de vecteurs ont été obtenus, complétant ainsi l’algèbre de Lie du cas stationnaire dont les sous-algèbres sont classifiées en classes de conjugaison sous l’action du groupe. Dans le cas non-stationnaire, une classification des algèbres de Lie admissibles selon la force choisie est effectuée. Pour chaque type de force, les champs de vecteurs sont présentés. L’algèbre ayant la dimension la plus élevée possible a été obtenues en considérant les forces monogéniques et elle a été classifiée en classes de conjugaison. La méthode de réduction par symétrie est appliquée pour obtenir des solutions explicites et implicites de plusieurs types parmi lesquelles certaines s’expriment en termes d’une ou deux fonctions arbitraires d’une variable et d’autres en termes de fonctions elliptiques de Jacobi. Plusieurs solutions sont interprétées physiquement pour en déduire la forme de filières d’extrusion réalisables. Dans la seconde partie, on s’intéresse aux solutions s’exprimant en fonction d’invariants de Riemann pour les systèmes quasilinéaires du premier ordre. La méthode des caractéristiques généralisées ainsi qu’une méthode basée sur les symétries conditionnelles pour les invariants de Riemann sont étendues pour être applicables à des systèmes dans leurs régions elliptiques. Leur applicabilité est démontrée par des exemples de la plasticité idéale non-stationnaire pour un flot irrotationnel ainsi que les équations de la mécanique des fluides. Une nouvelle approche basée sur l’introduction de matrices de rotation satisfaisant certaines conditions algébriques est développée. Elle est applicable directement à des systèmes non-homogènes et non-autonomes sans avoir besoin de transformations préalables. Son efficacité est illustrée par des exemples comprenant un système qui régit l’interaction non-linéaire d’ondes et de particules. La solution générale est construite de façon explicite. / The objects under consideration in this thesis are systems of first-order quasilinear equations. In the first part of the thesis, a study is made of an ideal plasticity model from the point of view of the classical Lie point symmetry group. Planar flows are investigated in both the stationary and non-stationary cases. Two new vector fields are obtained. They complete the Lie algebra of the stationary case, and the subalgebras are classified into conjugacy classes under the action of the group. In the non-stationary case, a classification of the Lie algebras admissible under the chosen force is performed. For each type of force, the vector fields are presented. For monogenic forces, the algebra is of the highest possible dimension. Its classification into conjugacy classes is made. The symmetry reduction method is used to obtain explicit and implicit solutions of several types. Some of them can be expressed in terms of one or two arbitrary functions of one variable. Others can be expressed in terms of Jacobi elliptic functions. Many solutions are interpreted physically in order to determine the shape of realistic extrusion dies. In the second part of the thesis, we examine solutions expressed in terms of Riemann invariants for first-order quasilinear systems. The generalized method of characteristics, along with a method based on conditional symmetries for Riemann invariants are extended so as to be applicable to systems in their elliptic regions. The applicability of the methods is illustrated by examples such as non-stationary ideal plasticity for an irrotational flow as well as fluid mechanics equations. A new approach is developed, based on the introduction of rotation matrices which satisfy certain algebraic conditions. It is directly applicable to non-homogeneous and non-autonomous systems. Its efficiency is illustrated by examples which include a system governing the non-linear superposition of waves and particles. The general solution is constructed in explicit form.