A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing

The motivation for this thesis was to recast quantum self-testing [MY98,MY04] in operational terms. The result is a category-theoretic framework for discussing the following general question: How do different implementations of the same input-output process compare to each other? In the proposed framework, an input-output process is modelled by a causally structured channel in some fixed theory, and its implementations are modelled by causally structured dilations formalising hidden side-computations. These dilations compare through a pre-order formalising relative strength of side-computations. Chapter 1 reviews a mathematical model for physical theories as semicartesian symmetric monoidal categories. Many concrete examples are discussed, in particular quantum and classical information theory. The key feature is that the model facilitates the notion of dilations. Chapter 2 is devoted to the study of dilations. It introduces a handful of simple yet potent axioms about dilations, one of which (resembling the Purification Postulate [CDP10]) entails a duality theorem encompassing a large number of classic no-go results for quantum theory. Chapter 3 considers metric structure on physical theories, introducing in particular a new metric for quantum channels, the purified diamond distance, which generalises the purified distance [TCR10,Tom12] and relates to the Bures distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for causality in terms of '(constructible) causal channels' and 'contractions'. It simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in monoidal categories [JSV96]. The formalism allows for the definition of 'causal dilations' and the establishment of a non-trivial theory of such dilations. Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus pointing towards the first known operational foundation for self-testing.Comment: PhD thesis submitted to the University of Copenhagen (ISBN 978-87-7125-039-8). Advised by prof. Matthias Christandl, submitted 1st of December 2020, defended 11th of February 2021. Keywords: dilations, applied category theory, quantum foundations, causal structure, quantum self-testing. 242 pages, 1 figure. Comments are welcom

Variational Methods in Shape Space

This dissertation deals with the application of variational methods in spaces of geometric shapes. In particular, the treated topics include shape averaging, principal component analysis in shape space, computation of geodesic paths in shape space, as well as shape optimisation. Chapter 1 provides a brief overview over the employed models of shape space. Geometric shapes are identified with two- or three-dimensional, deformable objects. Deformations will be described via physical models; in particular, the objects will be interpreted as consisting of either a hyperelastic solid or a viscous liquid material. Furthermore, the description of shapes via phase fields or level sets is briefly introduced. Chapter 2 reviews different and related approaches to shape space modelling. References to related topics in image segmentation and registration are also provided. Finally, the relevant shape optimisation literature is introduced. Chapter 3 recapitulates the employed concepts from continuum mechanics and phase field modelling and states basic theoretical results needed for the later analysis. Chapter 4 addresses the computation of shape averages, based on a hyperelastic notion of shape dissimilarity: The dissimilarity between two shapes is measured as the minimum deformation energy required to deform the first into the second shape. A corresponding phase-field model is introduced, analysed, and finally implemented numerically via finite elements. A principal component analysis of shapes, which is consistent with the previously introduced average, is considered in Chapter 5. Elastic boundary stresses on the average shape are used as representatives of the input shapes in a linear vector space. On these linear representatives, a standard principal component analysis can be performed, where the employed covariance metric should be properly chosen to depend on the input shapes. Chapter 6 interprets shapes as belonging to objects made of a viscous liquid and correspondingly defines geodesic paths between shapes. The energy of a path is given as the total physical dissipation during the deformation of an object along the path. A rigid body motion invariant time discretisation is achieved by approximating the dissipation along a path segment by the deformation energy of a small solid deformation. The numerical implementation is based on level sets. Chapter 7 is concerned with the optimisation of the geometry and topology of solid structures that are subject to a mechanical load. Given the load configuration, the structure rigidity, its volume, and its surface area shall be optimally balanced. A phase field model is devised and analysed for this purpose. In this context, the use of nonlinear elasticity allows to detect buckling phenomena which would be ignored in linearised elasticity

Problèmes inverses en mécanique des fluides résolus par des stratégies de jeux

This thesis aims to study the ability of theoretic game approaches to deal with ill-posed problems. The first part of the thesis is dedicated to the Stokes system’s linear problem, with the goal of detecting unknown geometric inclusions or pointwise sources in a stationary viscous fluid, using a single compatible pair of Dirichlet and Neumann data, available only on a partially accessible part of the boundary. Inverse geometric-or-source identification for the Cauchy-Stokes problem is severely ill-posed (in the sense of Hadamard) for both the inclusions or sources and the missing data reconstructions, and designing stable and efficient algorithms is challenging. To solve the joint completion/detection problem, we reformulate it as a three players Nash game. The two first players aim at recovering the missing data (Dirichlet and Neumann conditions prescribed over the inaccessible boundary), while the third player seeks to identify the shape and locations of the inclusions (in Chapter 2) or determine the source term (in Chapter 3). We then introduce new algorithms dedicated to the Nash equilibria, which is expected to approximate the original coupled problems’ solutions. We present different numerical experiments to illustrate the efficiency and robustness of our 3- player Nash game strategy. The extension of this work to another situation, such as identifying small objects, has been carried out (in Chapter 4). The second purpose of this thesis is to extend those results to the case of quasi-Newtonian fluid flow whose viscosity is assumed to be a nonlinear function that varies upon the imposed rate of deformation. The considered problem then is a nonlinear Cauchy type because of the non-linearity of the viscosity function. Two different iterative procedures, control-type and Nash game algorithms, are considered to solve it. From a computational point of view, the non-linearity needs some particular algorithms. We propose a novel one-shot algorithm to solve the nonlinear state equations during a recovery process, representing a different idea to treat the nonlinear Cauchy problems. Some numerical experiments are provided to demonstrate our algorithm’s efficiency in the noise-free and noisy data cases. A comparison between the one-shot scheme and the fixed-point method was performed. Finally, we introduce an algorithm to jointly recover the missing boundary data and the location and shape of the inclusions for nonlinear Stokes models based on the Game-Theoretic approach.Dans cette thèse, on s’intéresse à étudier la capacité de l’approche de la théorie des jeux à traiter certains problèmes inverses ‘mal posé’, gouvernés par les équations de Stokes ou quasi-Stokes. La première partie concerne la détection d’un ou plusieurs objets (Chapitre 2), et l’identification de sources ponctuelles dans un écoulement (Chapitre 3), en utilisant des données du type Cauchy qui seront ainsi fournies seulement sur une partie frontière de l’écoulement. Ce type de problème est mal posé au sens d’Hadamard du fait de l’absence de solution si les données ne sont pas compatibles mais surtout du fait de son extrême sensibilité aux données bruitées, dans le sens où une légère perturbation des données entraine une grande perturbation de la solution. Cette difficulté de stabilité fournit aux chercheurs un défi intéressant pour la mise au point de méthodes numériques permettant d’approcher de la solution du problème inverse original. L’approche développée ici est différente de celles existantes, elle a traité simultanément la question de la reconstruction des données manquantes avec celle de l’identification des inclusions ou de sources ponctuelles dans un fluide visqueux, incompressible et stationnaire. En considérant une méthode de type minimisation de critères, la solution est réinterprétée en termes d’équilibre de Nash entre les deux problèmes complétion/identification. Des nouveaux algorithmes originaux dédiés au calcul d’équilibre de Nash sont présenté et implémenté avec FreeFem ++. Une extension pour le problème d’identification de petits objets de l’approche proposée de jeu de Nash a été réalisé (Chapitre 4). La deuxième partie est consacrée à la résolution des problèmes inverses non linéaires dans le cadre des écoulements de fluide quasi-newtonien (Chapitre 5). La viscosité est supposée une fonction non linéaire, varie en fonction du tenseur des déformations. Un problème inverse non linéaire du type Cauchy est reformulé comme un problème du contrôle optimal, puis comme un jeu de Nash à deux joueurs. Deux algorithmes ont été utilisés et comparés afin de résoudre les problèmes aux limites non linéaires : un algorithme classique de point fixe et un nouveau schéma proposé ‘one-shots’. Enfin, on applique la théorie des jeux pour la résolution du problème de couplage de complétion des données etidentification des inclusions pour le modèle de quasi-Stokes

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Proceedings of the 3rd Annual Conference on Aerospace Computational Control, volume 1

Conference topics included definition of tool requirements, advanced multibody component representation descriptions, model reduction, parallel computation, real time simulation, control design and analysis software, user interface issues, testing and verification, and applications to spacecraft, robotics, and aircraft