650 research outputs found
Topological Characterization of Task Solvability in General Models of Computation
The famous asynchronous computability theorem (ACT) relates the existence of
an asynchronous wait-free shared memory protocol for solving a task with the
existence of a simplicial map from a subdivision of the simplicial complex
representing the inputs to the simplicial complex representing the allowable
outputs. The original theorem relies on a correspondence between protocols and
simplicial maps in round-structured models of computation that induce a compact
topology. This correspondence, however, is far from obvious for computation
models that induce a non-compact topology, and indeed previous attempts to
extend the ACT have failed.
This paper shows that in every non-compact model, protocols solving tasks
correspond to simplicial maps that need to be continuous. It first proves a
generalized ACT for sub-IIS models, some of which are non-compact, and applies
it to the set agreement task. Then it proves that in general models too,
protocols are simplicial maps that need to be continuous, hence showing that
the topological approach is universal. Finally, it shows that the approach used
in ACT that equates protocols and simplicial complexes actually works for every
compact model.
Our study combines, for the first time, combinatorial and point-set
topological aspects of the executions admitted by the computation model
Design of decorative 3D models: from geodesic ornaments to tangible assemblies
L'obiettivo di questa tesi è sviluppare strumenti utili per creare opere d'arte decorative digitali in 3D. Uno dei processi decorativi più comunemente usati prevede la creazione di pattern decorativi, al fine di abbellire gli oggetti. Questi pattern possono essere dipinti sull'oggetto di base o realizzati con l'applicazione di piccoli elementi decorativi. Tuttavia, la loro realizzazione nei media digitali non è banale. Da un lato, gli utenti esperti possono eseguire manualmente la pittura delle texture o scolpire ogni decorazione, ma questo processo può richiedere ore per produrre un singolo pezzo e deve essere ripetuto da zero per ogni modello da decorare. D'altra parte, gli approcci automatici allo stato dell'arte si basano sull'approssimazione di questi processi con texturing basato su esempi o texturing procedurale, o con sistemi di riproiezione 3D. Tuttavia, questi approcci possono introdurre importanti limiti nei modelli utilizzabili e nella qualità dei risultati. Il nostro lavoro sfrutta invece i recenti progressi e miglioramenti delle prestazioni nel campo dell'elaborazione geometrica per creare modelli decorativi direttamente sulle superfici. Presentiamo una pipeline per i pattern 2D e una per quelli 3D, e dimostriamo come ognuna di esse possa ricreare una vasta gamma di risultati con minime modifiche dei parametri. Inoltre, studiamo la possibilità di creare modelli decorativi tangibili. I pattern 3D generati possono essere stampati in 3D e applicati a oggetti realmente esistenti precedentemente scansionati. Discutiamo anche la creazione di modelli con mattoncini da costruzione, e la possibilità di mescolare mattoncini standard e mattoncini custom stampati in 3D. Ciò consente una rappresentazione precisa indipendentemente da quanto la voxelizzazione sia approssimativa. I principali contributi di questa tesi sono l'implementazione di due diverse pipeline decorative, un approccio euristico alla costruzione con mattoncini e un dataset per testare quest'ultimo.The aim of this thesis is to develop effective tools to create digital decorative 3D artworks. Real-world art often involves the use of decorative patterns to enrich objects. These patterns can be painted on the base or might be realized with the application of small decorative elements. However, their creation in digital media is not trivial. On the one hand, users can manually perform texture paint or sculpt each decoration, in a process that can take hours to produce a single piece and needs to be repeated from the ground up for every model that needs to be decorated. On the other hand, automatic approaches in state of the art rely on approximating these processes with procedural or by-example texturing or with 3D reprojection. However, these approaches can introduce significant limitations in the models that can be used and in the quality of the results. Instead, our work exploits the recent advances and performance improvements in the geometry processing field to create decorative patterns directly on surfaces. We present a pipeline for 2D and one for 3D patterns and demonstrate how each of them can recreate a variety of results with minimal tweaking of the parameters. Furthermore, we investigate the possibility of creating decorative tangible models. The 3D patterns we generate can be 3D printed and applied to previously scanned real-world objects. We also discuss the creation of models with standard building bricks and the possibility of mixing standard and custom 3D-printed bricks. This allows for a precise representation regardless of the coarseness of the voxelization. The main contributions of this thesis are the implementation of two different decorative pipelines, a heuristic approach to brick construction, and a dataset to test the latter
Leveraging Manifold Theory for Trajectory Design - A Focus on Futuristic Cislunar Missions
Optimal control methods for designing trajectories have been studied extensively by astro-dynamicists. Direct and indirect methods provide separate approaches to arrive at the optimal solution, each having their associated advantages and challenges. Among the realm of optimized transfer trajectories, fuel-optimal trajectories are typically most sought and characterized by se-quential thrust and coast arcs.
On the other hand, it is well known that a simplified dynamical model like the CR3BP analyzed in a rotating coordinate system, reveal fixed points known as Lagrange points. These spatial points can be orbited, with researchers categorizing periodic orbits around them starting from the simple planar Lyapunov orbits and continuing to the more enigmatic butterfly orbits. Studying linearized dynamics using eigenanalysis in the vicinity of a point on these periodic orbits lead to interesting departures spatially manifesting into the invariant manifolds.
This thesis delves into the novel idea of merging aspects of invariant manifold theory and indirect optimal control methods to provide efficient computation of feasible transfer trajectories. The marriage of these ideas provide the possibility of alleviating the challenges of an end-to end optimization using indirect methods for a long mission by utilizing the pre-computed and analyzed manifolds for insertion points of a long terminal coast arc. In addition to this, realistic and accurate mission scenarios require consideration of a high-fidelity dynamical model as well as shadow constraints. A methodology to use the “manifold analogues” in such cases has been discussed and utilized in this thesis along with modelling of eclipses during optimization, providing mission designers a basis for efficient and accurate/mission-ready trajectory design. This overcomes the shortcomings in state of the art software packages such as MYSTIC and COPERNICUS
Generalised subbundles and distributions: A comprehensive review
Distributions, i.e., subsets of tangent bundles formed by piecing together
subspaces of tangent spaces, are commonly encountered in the theory and
application of differential geometry. Indeed, the theory of distributions is a
fundamental part of mechanics and control theory.
The theory of distributions is presented in a systematic way, and
self-contained proofs are given of some of the major results. Parts of the
theory are presented in the context of generalised subbundles of vector
bundles. Special emphasis is placed on understanding the r\^ole of sheaves and
understanding the distinctions between the smooth or finitely differentiable
cases and the real analytic case. The Orbit Theorem and applications, including
Frobenius's Theorem and theorems on the equivalence of families of vector
fields, are considered in detail. Examples illustrate the phenomenon that can
occur with generalised subbundles and distributions
Properties of Discrete Sliced Wasserstein Losses
The Sliced Wasserstein (SW) distance has become a popular alternative to the
Wasserstein distance for comparing probability measures. Widespread
applications include image processing, domain adaptation and generative
modelling, where it is common to optimise some parameters in order to minimise
SW, which serves as a loss function between discrete probability measures
(since measures admitting densities are numerically unattainable). All these
optimisation problems bear the same sub-problem, which is minimising the Sliced
Wasserstein energy. In this paper we study the properties of , i.e. the SW distance between
two uniform discrete measures with the same amount of points as a function of
the support of one of the measures. We
investigate the regularity and optimisation properties of this energy, as well
as its Monte-Carlo approximation (estimating the expectation in
SW using only samples) and show convergence results on the critical points
of to those of , as well as an almost-sure uniform
convergence. Finally, we show that in a certain sense, Stochastic Gradient
Descent methods minimising and converge towards
(Clarke) critical points of these energies
Data analysis with merge trees
Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram
Electromagnetic scattering from thin tubular objects and an application in electromagnetic chirality
Asymptotic perturbation formulas characterize the effective behavior of waves as the volume of the scattering object tends to zero.
In this work, wave propagation is described by time-harmonic Maxwell\u27s equations in free space and the corresponding scattering objects are thin tubular objects that feature a different electric permittivity and a different magnetic permeability than their surrounding medium.
For this setting, we derive an asymptotic representation of the scattered electric field away from the thin tubular object and use the corresponding leading order term in a shape identification problem and for designing highly electromagnetically chiral objects.
In inverse problems, the leading order term may be used to find the center curve of a thin wire that is supposed to emit a scattered field, which is reasonably close to a given measured field.
For the optimal design of electromagnetically chiral structures,
the representation formula provides an explicit formula for the leading order term of an asymptotic far field operator expansion.
A chirality measure, usually requiring the far field operator, will now map aforementioned leading order term to a value between and dependent on the level of electromagnetic chirality of the thin tubular scatterer.
This approximation greatly simplifies the challenge to maximize the chirality measure with respect to thin tubular objects.
The fact that neither the evaluation of the leading order term nor the calculation of corresponding derivatives require a Maxwell system to be solved implies that the shape optimization scheme is highly efficient compared to shape optimization algorithms that use e.g. domain derivatives.
In the visible range, the metallic nanowires obtained by our optimization scheme
attain high values of electromagnetic chirality and even exceed those attained by traditional metallic helices
Ghost polygons, Poisson bracket and convexity
The moduli space of Anosov representations of a surface group in a semisimple
group, which is an open set in the character variety, admits many more natural
functions than the regular functions. We will study in particular length
functions and, correlation functions. Our main result is a formula that
computes the Poisson bracket of those functions using some combinatorial
devices called {\em ghost polygons} and {\em ghost bracket} encoded in a formal
algebra called {\em ghost algebra} related in some cases to the swapping
algebra introduced by the second author. As a consequence of our main theorem,
we show that the set of those functions -- length and correlation -- is stable
under the Poisson bracket. We give two applications: firstly in the presence of
positivity we prove the convexity of length functions, generalising a result of
Kerckhoff in Teichm\"uller space, secondly we exhibit subalgebras of commuting
functions. An important tool is the study of {\em uniformly hyperbolic bundles}
which is a generalisation of Anosov representations beyond periodicity.Comment: 65 pages, 7 figure
Electrical and Optical Modeling of Thin-Film Photovoltaic Modules
Heutzutage ist durch viele wissenschaftliche Studien nachgewiesen, dass die Erde längst dem Klimawandel unterworfen ist. Daher muss die gesamte Menschheit vereint handeln, um die schlimmsten Katastrophenszenarien zu verhindern. Ein vielversprechender Ansatz - wenn nicht sogar der vielversprechendste überhaupt - um diese angesprochene, größte Herausforderung in der Geschichte der Menschheit zu bewältigen, ist es, den Energiehunger der Menschheit durch die Erzeugung erneuerbarer und unerschöpflicher Energie zu sättigen. Die Photovoltaik (PV)-Technologie ist ein vielversprechender Anwärter, die leistungsstärkste erneuerbare Energiequelle zu stellen, und spielt aufgrund ihrer direkten Umwandlung des Sonnenlichtes und ihrer skalierbaren Anwendbarkeit in Form von großflächigen Solarmodulen bereits jetzt eine große Rolle bei der Erzeugung erneuerbarer Energie. Im PV-Sektor sind Solarmodule aus Siliziumwafern die derzeit vorherrschende Technologie. Neu aufkommende PV-Technologien wie die Dünnschichttechnologie haben jedoch vorteilhafte Eigenschaften wie einen sehr geringen Kohlenstoffdioxid (CO2)-Fußabdruck, eine kurze energetische Amortisierungszeit und das Potenzial für eine kostengünstige monolithische Massenproduktion, obwohl diese derzeit noch nicht final ausgereift ist. Um die Dünnschichttechnologie jedoch gezielt in Richtung einer breiten Marktreife zu entwickeln, sind numerische Simulationen eine wichtige Säule für das wissenschaftliche Verständnis und die technologische Optimierung. Während sich traditionelle Simulationsliteratur häufig mit materialspezifischen Herausforderungen befasst, konzentriert sich diese Arbeit auf industrieorientierte Herausforderungen auf Modulebene, ohne die zugrundeliegenden Materialparameter zu verändern.
Um ein allumfassendes, digitales Modell eines Solarmoduls zu erstellen, werden in dieser Arbeit mehrere Simulationsansätze aus verschiedenen physikalischen Bereichen kombiniert. Zur Abbildung elektrischer Effekte, einschließlich der räumlichen Spannungsvariation innerhalb des Moduls, wird eine Finite Elemente Methode (FEM) zur Lösung der räumlich quantisierten Poisson-Gleichung verwendet. Um optische Effekte zu berücksichtigen, wird eine generalisierte Transfermatrix-Methode (TMM) verwendet. Alle Simulationsmethoden sind in dieser Arbeit von Grund auf neu programmiert worden, um eine Verknüpfung aller Simulationsebenen mit dem höchstmöglichen Grad an Anpassung und Verknüpfung zu ermöglichen. Die Simulation und die Korrektheit der Parameter wird durch externe Quanteneffizienz (EQE)-Messungen, experimentelle Reflexionsdaten und gemessene Strom-Spannungs (I-U)-Kennlinien verifiziert. Der Kernpunkt der Vorgehensweise dieser Arbeit ist eine ganzheitliche Simulationsmethodik auf Modulebene. Dies ermöglicht es, die Lücke zwischen der Simulation auf Materialebene über die Berechnung von Laborwirkungsgraden bis hin zur Bestimmung der von zahlreichen Umweltfaktoren beeinflusste Leistung der Module im Freifeld zu überbrücken. Durch diese Verknüpfung von Zellsimulation und Systemdesign ist es lediglich aus Laboreigenschaften möglich, das Freifeldverhalten von Solarmodulen zu prognostizieren. Sogar das Zurückrechnen von experimentellen Messungen zu Materialparameter ist mittels des in dieser Arbeit entwickelten Verfahrens des Reverse Engineering Fittings (REF) möglich.
Das in dieser Arbeit entwickelte numerische Verfahren kann für mehrere Anwendungen genutzt werden. Zunächst können durch die Kombination von elektrischen und optischen Simulationen ganzheitliche Top-Down-Verlustanalysen durchgeführt werden. Dies ermöglicht eine wissenschaftliche Einordnung und einen quantitativen Vergleich aller Verlustleistungsmechanismen auf einen Blick, was die zukünftige Forschung und Entwicklung in Richtung von technologischen Schwachstellen von Solarmodulen lenkt. Darüber hinaus ermöglicht die Kombination von Elektrik und Optik die Detektion von Verlusten, die auf dem nichtlinearen Zusammenspiel dieser beiden Ebenen beruhen und auf eine räumliche Spannungsverteilung im Solarmodul zurückzuführen sind.
Diese Arbeit verwendet die entwickelten numerischen Modelle ebenfalls für Optimierungsprobleme, die an digitalen Modellen realer Solarmodule durchgeführt werden. Häufig auftretende Fragestellungen bei der Entwicklung von Solarmodulen sind beispielsweise die Schichtdicke des vorderen optisch transparenten, elektrisch leitfähigen Oxids (TCO) oder die Breite von monolithisch verschalteten Zellen. Die Bestimmung des Optimums dieser mehrdimensionalen Abwägungen zwischen optischer Transparenz, elektrischer Leitfähigkeit und geometrisch inaktiver Fläche zwischen den einzelnen Zellen ist ein Hauptmerkmal der Methodik dieser Arbeit. Mittels des FEM-Ansatzes dieser Arbeit ist es möglich, alle gegenseitigen Wechselwirkungen über verschiedene physikalische Ebenen hinweg zu berücksichtigen und ein ganzheitlich optimiertes Moduldesign zu finden. Auch topologisch komplexere Probleme, wie das Finden eines geeigneten Designs für das Metallisierungsgitter, können auf Grundlage der Simulation mittels der Methode der Topologie-Optimierung (TO) gelöst werden. In dieser Arbeit wurde das TO-Verfahren zum ersten Mal für monolithisch integrierte Zellen eingesetzt. Darüber hinaus wurde gezeigt, dass sowohl einfache Optimierungen der TCO-Schichtdicken als auch Topologie-Optimierungen stark von den vorherrschenden Beleuchtungsverhältnissen abhängen. Daher ist eine Optimierung auf den Jahresertrag anstelle des Laborwirkungsgrades für industrienahe Anwendungen wesentlich sinnvoller, da die mittleren Jahreseinstrahlungen deutlich von den Laborbedingungen abweichen. Mit Hilfe dieser Ertragsoptimierung wurde in dieser Arbeit für die Kupfer-Indium-Gallium-Diselenid CuInGaSe (CIGS)-Technologie ein Leistungsgewinn von über 1 % im Ertrag für einige geografische Standorte und gleichzeitig eine Materialeinsparung für die Metallisierungs- und TCO-Schicht von bis zu 50 % errechnet.
Mit Hilfe der numerischen Simulationen dieser Arbeit können alle denkbaren technologischen Verbesserungen auf Modulebene in das Modell eingebracht werden. Auf diese Weise wurde das aktuelle technologische Limit für CIGS-Dünnschicht-Solarmodule berechnet. Unter Verwendung der Randbedingungen der derzeit verfügbaren Materialien, Technologie- und Fertigungstoleranzen und des derzeit besten in der Literatur veröffentlichten CIGS-Materials ergibt sich ein theoretisches Wirkungsgradmaximum von 24 % auf Modulebene. Das derzeit beste veröffentlichte Modul mit den gegebenen Restriktionen weist einen Wirkungsgrad von 19,2 % auf [1]. Verbessert sich der CIGS-Absorber vergleichbar mit jenem von Galliumarsenid (GaAs) im Hinblick auf dessen Rekombinationsrate, ergibt sich ein erhöhtes Wirkungsgradlimit von etwa 28 %. Im Falle eines idealen CIGS-Absorbers ohne intrinsische Rekombinationsverluste wird in dieser Arbeit eine maximale Effizienzobergrenze von 29 % berechnet
The Geometry of Monotone Operator Splitting Methods
We propose a geometric framework to describe and analyze a wide array of
operator splitting methods for solving monotone inclusion problems. The initial
inclusion problem, which typically involves several operators combined through
monotonicity-preserving operations, is seldom solvable in its original form. We
embed it in an auxiliary space, where it is associated with a surrogate
monotone inclusion problem with a more tractable structure and which allows for
easy recovery of solutions to the initial problem. The surrogate problem is
solved by successive projections onto half-spaces containing its solution set.
The outer approximation half-spaces are constructed by using the individual
operators present in the model separately. This geometric framework is shown to
encompass traditional methods as well as state-of-the-art asynchronous
block-iterative algorithms, and its flexible structure provides a pattern to
design new ones
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