A method for dense packing discovery

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density $\phi=128/219\approx0.5845$ and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure

On-line stable state determination in decentralized power grid management

Both the coordination of international energy transfer and the integration of a rapidly growing number of decentralized energy resources (DER) throughout most countries causes novel problems for avoiding voltage band violations and line overloads. Traditional approaches are typically based on global off-line scheduling under globally available information and rely on iterative procedures that can guarantee neither convergence nor execution time. In this paper we focus on stability problems in power grids based on widely dispersed (renewable) energy sources. In this paper we will introduce an extension of the DEZENT algorithm, a multi-agent based coordination system for DER, that allows for the feasibility verification in constant and predetermined time. We give a numerical example showing the legitimacy of our approach and mention ongoing and future work regarding the implementation and utilization

The Convex Hull Problem in Practice : Improving the Running Time of the Double Description Method

The double description method is a simple but widely used algorithm for computation of extreme points in polyhedral sets. One key aspect of its implementation is the question of how to efficiently test extreme points for adjacency. In this dissertation, two significant contributions related to adjacency testing are presented. First, the currently used data structures are revisited and various optimizations are proposed. Empirical evidence is provided to demonstrate their competitiveness. Second, a new adjacency test is introduced. It is a refinement of the well known algebraic test featuring a technique for avoiding redundant computations. Its correctness is formally proven. Its superiority in multiple degenerate scenarios is demonstrated through experimental results. Parallel computation is one further aspect of the double description method covered in this work. A recently introduced divide-and-conquer technique is revisited and considerable practical limitations are demonstrated

Lineup polytopes of product of simplices

Consider a real point configuration $\mathbf{A}$ of size $n$ and an integer $r \leq n$. The vertices of the $r$-lineup polytope of $\mathbf{A}$ correspond to the possible orderings of the top $r$ points of the configuration obtained by maximizing a linear functional. The motivation behind the study of lineup polytopes comes from the representability problem in quantum chemistry. In that context, the relevant point configurations are the vertices of hypersimplices and the integer points contained in an inflated regular simplex. The central problem consists in providing an inequality representation of lineup polytopes as efficiently as possible. In this article, we adapt the developed techniques to the quantum information theory setup. The appropriate point configurations become the vertices of products of simplices. A particular case is that of lineup polytopes of cubes, which form a type $B$ analog of hypersimplices, where the symmetric group of type~$A$ naturally acts. To obtain the inequalities, we center our attention on the combinatorics and the symmetry of products of simplices to obtain an algorithmic solution. Along the way, we establish relationships between lineup polytopes of products of simplices with the Gale order, standard Young tableaux, and the Resonance arrangement.Comment: 19 pages, 8 figure

A decomposition theory for vertex enumeration of convex polyhedra

In the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorit

Computing Volumes and Convex Hulls: Variations and Extensions

Geometric techniques are frequently utilized to analyze and reason about multi-dimensional data. When confronted with large quantities of such data, simplifying geometric statistics or summaries are often a necessary first step. In this thesis, we make contributions to two such fundamental concepts of computational geometry: Klee's Measure and Convex Hulls. The former is concerned with computing the total volume occupied by a set of overlapping rectangular boxes in d-dimensional space, while the latter is concerned with identifying extreme vertices in a multi-dimensional set of points. Both problems are frequently used to analyze optimal solutions to multi-objective optimization problems: a variant of Klee's problem called the Hypervolume Indicator gives a quantitative measure for the quality of a discrete Pareto Optimal set, while the Convex Hull represents the subset of solutions that are optimal with respect to at least one linear optimization function.In the first part of the thesis, we investigate several practical and natural variations of Klee's Measure Problem. We develop a specialized algorithm for a specific case of Klee's problem called the “grounded” case, which also solves the Hypervolume Indicator problem faster than any earlier solution for certain dimensions. Next, we extend Klee's problem to an uncertainty setting where the existence of the input boxes are defined probabilistically, and study computing the expectation of the volume. Additionally, we develop efficient algorithms for a discrete version of the problem, where the volume of a box is redefined to be the cardinality of its overlap with a given point set.The second part of the thesis investigates the convex hull problem on uncertain input. To this extent, we examine two probabilistic uncertainty models for point sets. The first model incorporates uncertainty in the existence of the input points. The second model extends the first one by incorporating locational uncertainty. For both models, we study the problem of computing the probability that a given point is contained in the convex hull of the uncertain points. We also consider the problem of finding the most likely convex hull, i.e., the mode of the convex hull random variable

Polyhedral Tools for Control

Polyhedral operations play a central role in constrained control. One of the most fundamental operations is that of projection, required both by addition and multiplication. This thesis investigates projection and its relation to multi-parametric linear optimisation for the types of problems that are of particular interest to the control community. The first part of the thesis introduces an algorithm for the projection of polytopes in halfspace form, called Equality Set Projection (ESP). ESP has the desirable property of output sensitivity for non-degenerate polytopes. That is, a linear number of linear programs are needed per output facet of the projection. It is demonstrated that ESP is particularly well suited to control problems and comparative simulations are given, which greatly favour ESP. Part two is an investigation into the multi-parametric linear program (mpLP). The mpLP has received a lot of attention in the control literature as certain model predictive control problems can be posed as mpLPs and thereby pre-solved, eliminating the need for online optimisation. The structure of the solution to the mpLP is studied and an approach is pre- sented that eliminates degeneracy. This approach causes the control input to be continuous, preventing chattering, which is a significant problem in control with a linear cost. Four new enumeration methods are presented that have benefits for various control problems and comparative simulations demonstrate that they outperform existing codes. The third part studies the relationship between projection and multi-parametric linear programs. It is shown that projections can be posed as mpLPs and mpLPs as projections, demonstrating the fundamental nature of both of these problems. The output of a multi-parametric linear program that has been solved for the MPC control inputs offline is a piecewise linear controller defined over a union of polyhedra. The online work is then to determine which region the current measured state is in and apply the appropriate linear control law. This final part introduces a new method of searching for the appropriate region by posing the problem as a nearest neighbour search. This search can be done in logarithmic time and we demonstrate speed increases from 20Hz to 20kHz for a large example system

Molecular movies and geometry reconstruction using Coulomb explosion imaging

Coulomb explosion imaging is a technique of imaging the structure of small molecules in the gas phase and their ultrafast dynamics by inducing the rapid ionization and dissociation of the molecule into its constituent atomic fragments. The momentum vectors of the atomic fragments facilitate the retrieval of the molecule's structure, however, few attempts at geometry reconstruction appear in the published literature, whose vague methodology casts serious doubts on the geometry reconstructions that have been performed, and motivating the need for an investigation into the feasibility of geometry reconstruction. We develop a method for the fast and precise reconstruction of triatomic molecular geometries by casting the task as a nonlinear constrained optimization problem. We use this method to investigate the uncertainty in geometry reconstructions as a function of measurement uncertainty as well as the existence and nature of multiple solutions to the geometry reconstruction problem. We map out the conditions under which molecular geometries may be accurately reconstructed and propose a framework for reconstructing geometries, and therefore producing molecular movies using Coulomb explosion imaging

Ship Hull Representation by Non-Uniform Rational B-Spline Surface Patches

The purpose of this work is to propose a new method for representing the ship hull shape with mathematic surfaces so that geometric data can be generated for any point on the hull where required to assist the production process. An extensive survey of previous work is presented covering both the use of parametric curves and surfaces to model the ship hull and also the most relevant software systems developed for that purpose. The main methods and algorithms available for the generation and edition of curves and surfaces are presented and compared taking into consideration the intended application. From the analysis of the formulations available it was concluded that the most adequate one, which however had not yet been extensively used to model ship hulls was the Non-Uniform Rational B-Splines (NURBS), due to the potential of their capability to represent exactly conic curves and surfaces. Therefore these surfaces were selected as the basis of the method developed in this thesis. A procedure is proposed for the representation of a given hull form in a two step approach, creating first a wireframe model over which the surface patches are generated. Both curves and surfaces are based on the NURBS formulation. To create the wireframe model, first a set of longitudinal boundary lines is selected, dividing the surface into areas of similar shape. Then, these lines are fitted by curves and faired to some extent. Next, transverse sections are defined and split by the boundary lines. Surface patches are then generated over the transverse section curves within the limits of each patch. Finally, to obtain the traditional representation of the ship surface by transverse sections, buttocks and waterlines, contour lines are generated for constant values of x, y and z coordinates. A computer system has been developed incorporating an interface that allows the visualization of the curves and surfaces being modeled. The system incorporates several algorithms for generation and edition of curves and surfaces, in addition to the main contribution of this thesis which is the use of NURBS to represent the ship hull surface. The system also incorporates curve and surface analysis tools and some basic fairing algorithms so that during the several steps of the creation of the model, the fairness of the curves and surfaces can be evaluated and improved to some extent. The procedure is tested and compared with an existing commercial system through some application examples, of a complete hull and in more detail in the bow region, showing that good results can be obtained with the system presented here