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In this thesis, two distinct problems in data-driven computational science are considered. The main problem of interest is the multiobjective optimization problem, where the tradeoff surface (called the Pareto front) between multiple conflicting objectives must be approximated in order to identify designs that balance real-world tradeoffs. In order to solve multiobjective optimization problems that are derived from computationally expensive blackbox functions, such as engineering design optimization problems, several methodologies are combined, including surrogate modeling, trust region methods, and adaptive weighting. The result is a numerical software package that finds approximately Pareto optimal solutions that are evenly distributed across the Pareto front, using minimal cost function evaluations. The second problem of interest is the closely related problem of multivariate interpolation, where an unknown response surface representing an underlying phenomenon is approximated by finding a function that exactly matches available data. To solve the interpolation problem, a novel algorithm is proposed for computing only a sparse subset of the elements in the Delaunay triangulation, as needed to compute the Delaunay interpolant. For high-dimensional data, this reduces the time and space complexity of Delaunay interpolation from exponential time to polynomial time in practice. For each of the above problems, both serial and parallel implementations are described. Additionally, both solutions are demonstrated on real-world problems in computer system performance modeling. / Doctor of Philosophy / Science and engineering are full of multiobjective tradeoff problems. For example, a portfolio manager may seek to build a financial portfolio with low risk, high return rates, and minimal transaction fees; an aircraft engineer may seek a design that maximizes lift, minimizes drag force, and minimizes aircraft weight; a chemist may seek a catalyst with low viscosity, low production costs, and high effective yield; or a computational scientist may seek to fit a numerical model that minimizes the fit error while also minimizing a regularization term that leverages domain knowledge. Often, these criteria are conflicting, meaning that improved performance by one criterion must be at the expense of decreased performance in another criterion. The solution to a multiobjective optimization problem allows decision makers to balance the inherent tradeoff between conflicting objectives. A related problem is the multivariate interpolation problem, where the goal is to predict the outcome of an event based on a database of past observations, while exactly matching all observations in that database. Multivariate interpolation problems are equally as prevalent and impactful as multiobjective optimization problems. For example, a pharmaceutical company may seek a prediction for the costs and effects of a proposed drug; an aerospace engineer may seek a prediction for the lift and drag of a new aircraft design; or a search engine may seek a prediction for the classification of an unlabeled image. Delaunay interpolation offers a unique solution to this problem, backed by decades of rigorous theory and analytical error bounds, but does not scale to high-dimensional "big data" problems. In this thesis, novel algorithms and software are proposed for solving both of these extremely difficult problems.

La surface de Bolza est la surface hyperbolique orientable compacte la plus symétrique de genre 2. Pour tout genre supérieur à 2, il existe une surface orientable compacte construite de manière similaire à la surface de Bolza et ayant le même type de symétries. Nous appelons ces surfaces des surfaces hyperboliques symétriques. Cette thèse porte sur le calcul des triangulations de Delaunay (TD) de surfaces hyperboliques symétriques. Les TD de surfaces compactes peuvent être considérées comme des TD périodiques de leur revêtement universel (dans notre cas, le plan hyperbolique). Une TD est pour nous un complexe simplicial. Cependant, les ensembles de points ne définissent pas tous une décomposition simpliciale d'une surface hyperbolique symétrique. Dans la littérature, un algorithme a été proposé pour traiter ce problème avec l'utilisation de points factices : initialement une TD de la surface est construite avec un ensemble de points connu, puis des points d'entrée sont insérés avec le célèbre algorithme incrémental de Bowyer, et enfin les points factices sont supprimés, si la triangulation reste toujours un complexe simplicial. Pour la surface de Bolza, les points factices sont spécifiés. L'algorithme existant calcule une DT de la surface de Bolza comme une DT périodique du plan hyperbolique, ce qui nécessite de travailler dans un sous-ensemble approprié du plan hyperbolique. Nous étudions les propriétés des TD de la surface de Bolza définies par des ensembles de points contenants l'ensemble proposé de points factices, et nous décrivons en détail une implémentation de l'algorithme incrémentiel pour cette surface. Nous commençons par définir un représentant canonique unique qui est contenu dans un sous-ensemble borné du plan hyperbolique pour chaque face d'une TD de la surface. Nous donnons une structure de données pour représenter une TD de la surface de Bolza via les représentants canoniques de ses faces. Nous détaillons les étapes de la construction d'une telle triangulation et les opérations supplémentaires qui permettent de localiser les points et de retirer des sommets. Nous présentons également les résultats sur le degré algébrique des prédicats nécessaires pour toutes les opérations. Nous fournissons une implémentation entièrement dynamique pour la surface de Bolza, en offrant l'insertion de nouveaux points, la suppression des sommets existants, la localisation des points, et la construction d'objets duaux. Notre implémentation est basée sur la bibliothèque CGAL (Computational Geometry Algorithms Library), et est actuellement en cours de révision pour être intégrée dans la bibliothèque. L'intégration de notre code dans CGAL nécessite que tous les objets que nous introduisons soient compatibles avec le cadre existant et conformes aux standards adoptés par la bibliothèque. Nous donnons une description détaillée des classes utilisées pour représenter et traiter les triangulations hyperboliques périodiques et les objets associés. Des analyses comparatives et des tests sont effectués pour évaluer notre implémentation, et une application simple est donnée sous la forme d'une démonstration CGAL. Nous discutons une extension de notre implémentation à des surfaces hyperboliques symétriques de genre supérieur à 2. Nous proposons trois méthodes pour engendrer des ensembles de points factices pour chaque surface et présentons les avantages et les inconvénients de chaque méthode. Nous définissons un représentant canonique contenu dans un sous-ensemble borné du plan hyperbolique pour chaque face d'une TD de la surface. Nous décrivons une structure de données pour représenter une telle triangulation via les représentants canoniques de ses faces, et donnons des algorithmes pour l'initialisation de la triangulation. Enfin, nous discutons une implémentation préliminaire dans laquelle nous examinons les difficultés d'avoir des prédicats exacts efficaces pour la construction de TD de surfaces hyperboliques symétriques / The Bolza surface is the most symmetric compact orientable hyperbolic surface of genus 2. For any genus higher than 2, there exists one compact orientable surface constructed in a similar way as the Bolza surface having the same kind of symmetry. We refer to this family of surfaces as symmetric hyperbolic surfaces. This thesis deals with the computation of Delaunay triangulations of symmetric hyperbolic surfaces. Delaunay triangulations of compact surfaces can be seen as periodic Delaunay triangulations of their universal cover (in our case, the hyperbolic plane). A Delaunay triangulation is for us a simplicial complex. However, not all sets of points define a simplicial decomposition of a symmetric hyperbolic surface. In the literature, an algorithm has been proposed to deal with this issue by using so-called dummy points: initially a triangulation of the surface is constructed with a set of dummy points that defines a Delaunay triangulation of the surface, then input points are inserted with the well-known incremental algorithm by Bowyer, and finally the dummy points are removed, if the triangulation remains a simplicial complex after their removal. For the Bolza surface, the set of dummy points to initialize the triangulation is given. The existing algorithm computes a triangulation of the Bolza surface as a periodic triangulation of the hyperbolic plane and requires to identify a suitable subset of the hyperbolic plane in which to work. We study the properties of Delaunay triangulations of the Bolza surface defined by sets of points containing the proposed set of dummy points, and we describe in detail an implementation of the incremental algorithm for it. We begin by identifying a subset of the hyperbolic plane that contains at least one representative for each face of a Delaunay triangulation of the surface, which enables us to define a unique canonical representative in the hyperbolic plane for each face on the surface. We give a data structure to represent a Delaunay triangulation of the Bolza surface via the canonical representatives of its faces in the hyperbolic plane. We detail the construction of such a triangulation and additional operations that enable the location of points and the removal of vertices. We also report results on the algebraic degree of predicates needed for all operations. We provide a fully dynamic implementation for the Bolza surface, supporting insertion of new points, removal of existing vertices, point location, and construction of dual objects. Our implementation is based on CGAL, the Computational Geometry Algorithms Library, and is currently under revision for integration in the library. To incorporate our code into CGAL, all the objects that we introduce must be compatible with the existing framework and comply with the standards adopted by the library. We give a detailed description of the classes used to represent and handle periodic hyperbolic triangulations and related objects. Benchmarks and tests are performed to evaluate our implementation, and a simple application is given in the form of a CGAL demo. We discuss an extension of our implementation to symmetric hyperbolic surfaces of genus higher than 2. We propose three methods to generate sets of dummy points for each surface and present the advantages and shortcomings of each method. We identify a suitable subset of the hyperbolic plane that contains at least one representative for each face of a Delaunay triangulation of the surface, and we define a canonical representative in the hyperbolic plane for each face on the surface. We describe a data structure to represent such a triangulation via the canonical representatives of its faces, and give algorithms for the initialization of the triangulation with dummy points. Finally, we discuss a preliminary implementation in which we examine the difficulties of having efficient exact predicates for the construction of Delaunay triangulations of symmetric hyperbolic surfaces

This thesis is a contribution to a classical problem in computational and combinatorial geometry: the study of the k-sets of a set V of n points in the plane. First we introduce the notion of convex inclusion chain that is an ordering of the points of V such that no point is inside the convex hull of the points that precede it. Every k-set of an initial sub-sequence of the chain is called a k-set of the chain. We prove that the number of these k-sets is an invariant of V and is equal to the number of regions in the order-k Voronoi diagram of V. We then deduce an online algorithm for the construction of the k-sets of the vertices of a simple polygonal line such that every vertex of this line is outside the convex hull of all its preceding vertices on the line. If c is the total number of k-sets built with this algorithm, the complexity of our algorithm is in O(n log n + c log^2k) and is equal, per constructed k-set, to the complexity of the best algorithm known. Afterward, we prove that the classical divide and conquer algorithmic method can be adapted to the construction of the k-sets of V. The algorithm has a complexity of O(n log n + c log^2k log(n/k)), where c is the maximum number of k-sets of a set of n points. We finally prove that the centers of gravity of the k-sets of a convex inclusion chain are the vertices of a triangulation belonging to the family of so-called centroid triangulations. This family notably contains the dual of the order-k Voronoi diagram. We give an algorithm that builds particular centroid triangulations in O(n log n + k(n- k) log^2 k) time, which is more efficient than all the currently known algorithms.

This thesis is devoted to the problem of finding a suitable geometrical de- scription of the domain for the Finite Element Method (FEM). We present the most important methods used in generation and improvement of unstructured triangular meshes (grids) for two dimensional FEM. Possible measures of mesh quality are discussed with respect to their usage in linear Lagrange FEM. The relationship between mesh geometry (especially angles of particular triangles), discretization error and stiffness matrix condition number is examined. Two methods of mesh improvement, based on Centroidal Voronoi Tessellations (CVT) and Optimal Delaunay Triangulations (ODT), are discussed in detail and some results on convergence of CVT based methods are reviewed. Some aspects of these methods, e.g. the relation between density of boundary points and interior mesh vertices and the treatment of the boundary triangles is reconsidered in a new way. We have implemented these two methods and we discuss possible im- provements and new algorithms. A geometrically very interesting idea of recent alternative to FEM, Isogeometric Analysis (IGA), is outlined and demonstrated on a simple example. Several numerical tests are made in order to the compare the accuracy of solutions of isotropic PDEs obtained by FEM on bad mesh, mesh improved...

Cette thèse explore divers aspects des cartes aléatoires par l'étude de trois modèles. Dans un premier temps, nous examinons les propriétés d’une mesure définie sur l’ensemble des triangulations de Delaunay planaires comportant n sommets, qui est un modèle de cartes où les arêtes sont décorées par des angles. Nous montrons ainsi que la mesure est égale à la mesure de Weil-Petersson sur l’espace des modules des surfaces de Riemann planaires marquées. Sont aussi montrées deux propriétés de la mesures, premiers pas d'une étude de la limite continue de ce modèle. Dans un deuxième temps, nous définissons des fonctions de corrélations sur les graphes de Strebel planaires isopérimétriques à n faces, qui sont des cartes métriques trivalentes. Les périmètres des faces sont fixés. Nous recourons au théorème de Kontsevich pour calculer les fonctions de corrélations en termes de nombres d’intersection de classes de Chern sur l’espace des modules des surfaces de Riemann. Pour la fonction à une face marquée, la limite des grandes cartes est examinée via l’approximation du point-selle, pour différents régimes du périmètre de la face marquée, et nous déduisons le régime où le comportement de la fonction de corrélation n’est pas trivial. Les fonctions de corrélations peuvent être calculées de manière systématique par la récurrence topologique. Partant, nous calculons la courbe spectrale de notre modèle, ce qui nous permet de montrer qu’il existe une courbe spectrale critique. Nous déduisons de cette courbe critique que la limite continue des graphes de Strebel isopérimétriques est un modèle minimal de type (3,2), habillé par la théorie de Liouville. Cela correspond bien à la gravité pure. Enfin, nous abordons la question des symétries dans le modèle d’Ising sur cartes aléatoires. Certaines fonctions de corrélations de ce modèle comptent le nombre de cartes bicolores avec des faces marquées, les bords, ayant des conditions aux bords mixtes, calculées par récurrence à partir de la courbe spectrale du modèle. Nous prouvons ici que, pour des courbes spectrales génériques, les fonctions de corrélations des cartes à un bord mixte sont symétriques par rotation et par inversion du bord mixte. Nous décrivons ensuite les conséquences de telles symétries, suggérant une possible reformulation du modèle en termes de chaînes de spins. / This thesis explore several aspects of random maps through the study of three models. First, we examine the properties of a measure defined on the set of planar Delaunay triangulations with n vertices, a model in which the edges of the maps are decorated with angles. We show that the measure is the Weil-Petersson volume form on the moduli space of planar Riemann surfaces having n marked points. Two other properties, first steps toward the continuous limit study of the model, are also shown. Second, we define correlation functions on isoperimetric planar Strebel graphs with n faces, which are trivalent maps whose edges are decorated by positive lengths, and whose faces have a fixed perimeter. Kontsevich's theorem allows us to compute the correlation functions in terms of the intersection numbers of Chern classes of moduli space of Riemann surfaces. The continuous limit of the one-point function is computed in different regimes for the perimeter of the marked face via the saddle-point approximation. We identify the regime in which the behaviour of the one-point function is not trivial. The correlation functions can be computed in a systematic way by the Topological Recursion. To do so, we compute the spectral curve of the model, and show that there exists a critical spectral curve. We deduce from the latter that the continuous limit of isoperimetric Strebel graphs is a (3,2) minimal model dressed by Liouville theory: it corresponds to pure gravity. Last, we address the problem of symmetries in the Ising model on random maps. Some correlation functions of this model count the bi-colored maps with marked faces having mixed boundary conditions. They are computed via a recursive formula and the spectral curve of the model. We prove here that the correlation functions of maps with one mixed boundary, computed from the recursive relation with generic spectral curve, are invariant under rotation and inversion of the mixed boundary. We describe the consequences of such symmetries, suggesting a possible reformulation of the model in terms of spin chains.

Cette thèse est une contribution à un problème classique de la géométrie algorithmique et combinatoire : l’étude des k-sets d’un ensemble V de n points du plan. Nous introduisons d’abord la notion de chaîne d’inclusion de convexes qui est un ordonnancement des points de V tel qu’aucun point n’appartient à l’enveloppe convexe de ceux qui le précèdent. Tout k-set d’une sous-suite commençante de la chaîne est appelé un k-set de la chaîne. Nous montrons que le nombre de ces k-sets est un invariant de V et qu’il est égal au nombre de régions du diagramme de Voronoï d’ordre k de V. Nous en déduisons un algorithme en ligne pour construire les k-sets des sommets d’une ligne polygonale simple dont chaque sommet est à l’extérieur de l’enveloppe convexe des sommets qui le précèdent sur la ligne. Si c est le nombre total de k-sets construits, la complexité de notrealgorithme est en O(n log n+c log^2 k) et est équivalente, par k-set construit, à celle du meilleur algorithme connu. Nous montrons ensuite que la méthode algorithmique classique de division-fusion peut être adaptée à la construction des k-sets de V. L’algorithme qui en résulte a une complexité enO(n log n+c log^2 k log(n/k)), où c est le nombre maximum de k-sets d’un ensemble de n points.Nous prouvons enfin que les centres de gravité des k-sets d’une chaîne d’inclusion de convexes sont les sommets d’une triangulation qui appartient à la même famille de triangulations, dites centroïdes, que le dual du diagramme de Voronoï d’ordre k. Nous en d´déduisons un algorithme qui construit des triangulations centroïdes particulières en temps O(n log n+k(n-k) log^2 k), ce qui est plus efficace que les algorithmes connus jusque là. / This thesis is a contribution to a classical problem in computational and combinatorial geometry: the study of the k-sets of a set V of n points in the plane. First we introduce the notion of convex inclusion chain that is an ordering of the points of V such that no point is inside the convex hull of the points that precede it. Every k-set of an initial sub-sequence of the chain is called a k-set of the chain. We prove that the number of these k-sets is an invariant of V and is equal to the number of regions in the order-k Voronoi diagram of V. We then deduce an online algorithm for the construction of the k-sets of the vertices of a simple polygonal line such that every vertex of this line is outside the convex hull of all its preceding vertices on the line. If c is the total number of k-sets built with this algorithm, the complexity of our algorithm is in O(n log n + c log^2k) and is equal, per constructed k-set, to the complexity of the best algorithm known. Afterward, we prove that the classical divide and conquer algorithmic method can be adapted to the construction of the k-sets of V. The algorithm has a complexity of O(n log n + c log^2k log(n/k)), where c is the maximum number of k-sets of a set of n points. We finally prove that the centers of gravity of the k-sets of a convex inclusion chain are the vertices of a triangulation belonging to the family of so-called centroid triangulations. This family notably contains the dual of the order-k Voronoi diagram. We give an algorithm that builds particular centroid triangulations in O(n log n + k(n- k) log^2 k) time, which is more efficient than all the currently known algorithms.

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