Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Algorithm engineering in geometric network planning and data mining

The geometric nature of computational problems provides a rich source of solution strategies as well as complicating obstacles. This thesis considers three problems in the context of geometric network planning, data mining and spherical geometry. Geometric Network Planning: In the d-dimensional Generalized Minimum Manhattan Network problem (d-GMMN) one is interested in finding a minimum cost rectilinear network N connecting a given set of n pairs of points in ℝ^d such that each pair is connected in N via a shortest Manhattan path. The decision version of this optimization problem is known to be NP-hard. The best known upper bound is an O(log^{d+1} n) approximation for d>2 and an O(log n) approximation for 2-GMMN. In this work we provide some more insight in, whether the problem admits constant factor approximations in polynomial time. We develop two new algorithms, a `scale-diversity aware' algorithm with an O(D) approximation guarantee for 2-GMMN. Here D is a measure for the different `scales' that appear in the input, D ∈ O(log n) but potentially much smaller, depending on the problem instance. The other algorithm is based on a primal-dual scheme solving a more general, combinatorial problem - which we call Path Cover. On 2-GMMN it performs well in practice with good a posteriori, instance-based approximation guarantees. Furthermore, it can be extended to deal with obstacle avoiding requirements. We show that the Path Cover problem is at least as hard to approximate as the Hitting Set problem. Moreover, we show that solutions of the primal-dual algorithm are 4ω^2 approximations, where ω ≤ n denotes the maximum overlap of a problem instance. This implies that a potential proof of O(1)-inapproximability for 2-GMMN requires gadgets of many different scales and non-constant overlap in the construction. Geometric Map Matching for Heterogeneous Data: For a given sequence of location measurements, the goal of the geometric map matching is to compute a sequence of movements along edges of a spatially embedded graph which provides a `good explanation' for the measurements. The problem gets challenging as real world data, like traces or graphs from the OpenStreetMap project, does not exhibit homogeneous data quality. Graph details and errors vary in areas and each trace has changing noise and precision. Hence, formalizing what a `good explanation' is becomes quite difficult. We propose a novel map matching approach, which locally adapts to the data quality by constructing what we call dominance decompositions. While our approach is computationally more expensive than previous approaches, our experiments show that it allows for high quality map matching, even in presence of highly variable data quality without parameter tuning. Rational Points on the Unit Spheres: Each non-zero point in ℝ^d identifies a closest point x on the unit sphere S^{d-1}. We are interested in computing an ε-approximation y ∈ ℚ^d for x, that is exactly on S^{d-1} and has low bit-size. We revise lower bounds on rational approximations and provide explicit spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in ℝ^2 and ℝ^3. However, we show how to construct a rational point with denominators of at most 10(d-1)/ε^2 for any given ε ∈ (0, 1/8], improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation. Our open-source implementation and experiments demonstrate the practicality of our approach in the context of massive data sets, geo-referenced by latitude and longitude values.Die geometrische Gestalt von Berechnungsproblemen liefert vielfältige Lösungsstrategieen aber auch Hindernisse. Diese Arbeit betrachtet drei Probleme im Gebiet der geometrischen Netzwerk Planung, des geometrischen Data Minings und der sphärischen Geometrie. Geometrische Netzwerk Planung: Im d-dimensionalen Generalized Minimum Manhattan Network Problem (d-GMMN) möchte man ein günstigstes geradliniges Netzwerk finden, welches jedes der gegebenen n Punktepaare aus ℝ^d mit einem kürzesten Manhattan Pfad verbindet. Es ist bekannt, dass die Entscheidungsvariante dieses Optimierungsproblems NP-hart ist. Die beste bekannte obere Schranke ist eine O(log^{d+1} n) Approximation für d>2 und eine O(log n) Approximation für 2-GMMN. Durch diese Arbeit geben wir etwas mehr Einblick, ob das Problem eine Approximation mit konstantem Faktor in polynomieller Zeit zulässt. Wir entwickeln zwei neue Algorithmen. Ersterer nutzt die `Skalendiversität' und hat eine O(D) Approximationsgüte für 2-GMMN. Hierbei ist D ein Maß für die in Eingaben auftretende `Skalen'. D ∈ O(log n), aber potentiell deutlichen kleiner für manche Problem Instanzen. Der andere Algorithmus basiert auf einem Primal-Dual Schema zur Lösung eines allgemeineren, kombinatorischen Problems, welches wir Path Cover nennen. Die praktisch erzielten a posteriori Approximationsgüten auf Instanzen von 2-GMMN verhalten sich gut. Dieser Algorithmus kann für Netzwerk Planungsprobleme mit Hindernis-Anforderungen angepasst werden. Wir zeigen, dass das Path Cover Problem mindestens so schwierig zu approximieren ist wie das Hitting Set Problem. Darüber hinaus zeigen wir, dass Lösungen des Primal-Dual Algorithmus 4ω^2 Approximationen sind, wobei ω ≤ n die maximale Überlappung einer Probleminstanz bezeichnet. Daher müssen potentielle Beweise, die konstante Approximationen für 2-GMMN ausschließen möchten, Instanzen mit vielen unterschiedlichen Skalen und nicht konstanter Überlappung konstruieren. Geometrisches Map Matching für heterogene Daten: Für eine gegebene Sequenz von Positionsmessungen ist das Ziel des geometrischen Map Matchings eine Sequenz von Bewegungen entlang Kanten eines räumlich eingebetteten Graphen zu finden, welche eine `gute Erklärung' für die Messungen ist. Das Problem wird anspruchsvoll da reale Messungen, wie beispielsweise Traces oder Graphen des OpenStreetMap Projekts, keine homogene Datenqualität aufweisen. Graphdetails und -fehler variieren in Gebieten und jeder Trace hat wechselndes Rauschen und Messgenauigkeiten. Zu formalisieren, was eine `gute Erklärung' ist, wird dadurch schwer. Wir stellen einen neuen Map Matching Ansatz vor, welcher sich lokal der Datenqualität anpasst indem er sogenannte Dominance Decompositions berechnet. Obwohl unser Ansatz teurer im Rechenaufwand ist, zeigen unsere Experimente, dass qualitativ hochwertige Map Matching Ergebnisse auf hoch variabler Datenqualität erzielbar sind ohne vorher Parameter kalibrieren zu müssen. Rationale Punkte auf Einheitssphären: Jeder, von Null verschiedene, Punkt in ℝ^d identifiziert einen nächsten Punkt x auf der Einheitssphäre S^{d-1}. Wir suchen eine ε-Approximation y ∈ ℚ^d für x zu berechnen, welche exakt auf S^{d-1} ist und niedrige Bit-Größe hat. Wir wiederholen untere Schranken an rationale Approximationen und liefern explizite, sphärische Instanzen. Wir beweisen, dass Floating-Point Zahlen nur triviale Lösungen zur Sphären-Gleichung in ℝ^2 und ℝ^3 liefern können. Jedoch zeigen wir die Konstruktion eines rationalen Punktes mit Nennern die maximal 10(d-1)/ε^2 sind für gegebene ε ∈ (0, 1/8], was ein bekanntes Resultat verbessert. Darüber hinaus profitiert die Methode von Algorithmen für simultane Diophantische Approximationen. Unsere quell-offene Implementierung und die Experimente demonstrieren die Praktikabilität unseres Ansatzes für sehr große, durch geometrische Längen- und Breitengrade referenzierte, Datensätze

Interpretable Network Representations

Networks (or interchangeably graphs) have been ubiquitous across the globe and within science and engineering: social networks, collaboration networks, protein-protein interaction networks, infrastructure networks, among many others. Machine learning on graphs, especially network representation learning, has shown remarkable performance in network-based applications, such as node/graph classification, graph clustering, and link prediction. Like performance, it is equally crucial for individuals to understand the behavior of machine learning models and be able to explain how these models arrive at a certain decision. Such needs have motivated many studies on interpretability in machine learning. For example, for social network analysis, we may need to know the reasons why certain users (or groups) are classified or clustered together by the machine learning models, or why a friend recommendation system considers some users similar so that they are recommended to connect with each other. Therefore, an interpretable network representation is necessary and it should carry the graph information to a level understandable by humans. Here, we first introduce our method on interpretable network representations: the network shape. It provides a framework to represent a network with a 3-dimensional shape, and one can customize network shapes for their need, by choosing various graph sampling methods, 3D network embedding methods and shape-fitting methods. In this thesis, we introduce the two types of network shape: a Kronecker hull which represents a network as a 3D convex polyhedron using stochastic Kronecker graphs as the network embedding method, and a Spectral Path which represents a network as a 3D path connecting the spectral moments of the network and its subgraphs. We demonstrate that network shapes can capture various properties of not only the network, but also its subgraphs. For instance, they can provide the distribution of subgraphs within a network, e.g., what proportion of subgraphs are structurally similar to the whole network? Network shapes are interpretable on different levels, so one can quickly understand the structural properties of a network and its subgraphs by its network shape. Using experiments on real-world networks, we demonstrate that network shapes can be used in various applications, including (1) network visualization, the most intuitive way for users to understand a graph; (2) network categorization (e.g., is this a social or a biological network?); (3) computing similarity between two graphs. Moreover, we utilize network shapes to extend biometrics studies to network data, by solving two problems: network identification (Given an anonymized graph, can we identify the network from which it is collected? i.e., answering questions such as ``where is this anonymized graph sampled from, Twitter or Facebook? ) and network authentication (If one claims the graph is sampled from a certain network, can we verify this claim?). The overall objective of the thesis is to provide a compact, interpretable, visualizable, comparable and efficient representation of networks

Hydrodynamics

The phenomena related to the flow of fluids are generally complex, and difficult to quantify. New approaches - considering points of view still not explored - may introduce useful tools in the study of Hydrodynamics and the related transport phenomena. The details of the flows and the properties of the fluids must be considered on a very small scale perspective. Consequently, new concepts and tools are generated to better describe the fluids and their properties. This volume presents conclusions about advanced topics of calculated and observed flows. It contains eighteen chapters, organized in five sections: 1) Mathematical Models in Fluid Mechanics, 2) Biological Applications and Biohydrodynamics, 3) Detailed Experimental Analyses of Fluids and Flows, 4) Radiation-, Electro-, Magnetohydrodynamics, and Magnetorheology, 5) Special Topics on Simulations and Experimental Data. These chapters present new points of view about methods and tools used in Hydrodynamics

Computational Modelling of Concrete and Concrete Structures

Computational Modelling of Concrete and Concrete Structures contains the contributions to the EURO-C 2022 conference (Vienna, Austria, 23-26 May 2022). The papers review and discuss research advancements and assess the applicability and robustness of methods and models for the analysis and design of concrete, fibre-reinforced and prestressed concrete structures, as well as masonry structures. Recent developments include methods of machine learning, novel discretisation methods, probabilistic models, and consideration of a growing number of micro-structural aspects in multi-scale and multi-physics settings. In addition, trends towards the material scale with new fibres and 3D printable concretes, and life-cycle oriented models for ageing and durability of existing and new concrete infrastructure are clearly visible. Overall computational robustness of numerical predictions and mathematical rigour have further increased, accompanied by careful model validation based on respective experimental programmes. The book will serve as an important reference for both academics and professionals, stimulating new research directions in the field of computational modelling of concrete and its application to the analysis of concrete structures. EURO-C 2022 is the eighth edition of the EURO-C conference series after Innsbruck 1994, Bad Gastein 1998, St. Johann im Pongau 2003, Mayrhofen 2006, Schladming 2010, St. Anton am Arlberg 2014, and Bad Hofgastein 2018. The overarching focus of the conferences is on computational methods and numerical models for the analysis of concrete and concrete structures

Sobre o problema Euclidiano de Steiner no Rn

The Euclidean Steiner Problem (ESP) asks for a network of minimum length interconnecting a finite set of given points in Rn. The distances considered are Euclidean and it’s allowed to add additional points to decrease the overall length of the network. Problems of this nature are often found in several areas of mathematics, engineering, etc. In this work, we study the origins of ESP, their properties, complexity, and resolution methods. We conclude by analyzing a conjecture proposed in 1992 by Warren Smith on the application of this problem to the vertices of an n-dimensional hypercube, which has remained open since its publication.O Problema Euclidiano de Steiner (PES) tem como objetivo determinar uma rede de comprimento mínimo que conecte um conjunto finito de pontos do Rn previamente escolhidos. A norma utilizada é a euclidiana e é permitido o uso de pontos extras que possam contribuir para a redução do comprimento final da rede. Problemas desta natureza são frequentemente encontrados em diversas áreas da matemática, engenharia, etc. Neste trabalho, estudamos as origens do PES, suas propriedades, complexidade e métodos de resolução. Encerramos analisando uma conjectura proposta em 1992 por Warren Smith sobre a aplicação desse problema aos vértices de um hipercubo n-dimensional, a qual está em aberto desde sua publicação