One machine, one minute, three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh

Non-Equilibrium Electron Transport in Two-Dimensional Nano-Structures Modeled by Green's Functions and the Finite-Element Method

We use the effective-mass approximation and the density-functional theory with the local-density approximation for modeling two-dimensional nano-structures connected phase-coherently to two infinite leads. Using the non-equilibrium Green's function method the electron density and the current are calculated under a bias voltage. The problem of solving for the Green's functions numerically is formulated using the finite-element method (FEM). The Green's functions have non-reflecting open boundary conditions to take care of the infinite size of the system. We show how these boundary conditions are formulated in the FEM. The scheme is tested by calculating transmission probabilities for simple model potentials. The potential of the scheme is demonstrated by determining non-linear current-voltage behaviors of resonant tunneling structures.Comment: 13 pages,15 figure

Contours in Visualization

This thesis studies the visualization of set collections either via or defines as the relations among contours. In the first part, dynamic Euler diagrams are used to communicate and improve semimanually the result of clustering methods which allow clusters to overlap arbitrarily. The contours of the Euler diagram are rendered as implicit surfaces called blobs in computer graphics. The interaction metaphor is the moving of items into or out of these blobs. The utility of the method is demonstrated on data arising from the analysis of gene expressions. The method works well for small datasets of up to one hundred items and few clusters. In the second part, these limitations are mitigated employing a GPU-based rendering of Euler diagrams and mixing textures and colors to resolve overlapping regions better. The GPU-based approach subdivides the screen into triangles on which it performs a contour interpolation, i.e. a fragment shader determines for each pixel which zones of an Euler diagram it belongs to. The rendering speed is thus increased to allow multiple hundred items. The method is applied to an example comparing different document clustering results. The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Adapting popular graph drawing approaches to the problem of contour tree drawing it is found that they are unable to convey this information. Five aesthetic criteria for drawing contour trees are proposed and a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria is presented. The implementation is fast and effective for contour tree sizes usually used in interactive systems and also produces readable pictures for larger trees. Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods to analyze these model\''s validity. The fourth part of the thesis focuses on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; they are huge and high dimensional. Their most salient features are described by their so-called barrier tree -- a contour tree for discrete observation spaces. The changing folding landscapes of a growing RNA chain are visualized as an animation of the corresponding barrier tree sequence. The animation is created as an adaption of the foresight layout with tolerance algorithm for dynamic graph layout. The adaptation requires changes to the concept of supergraph and it layout. The thesis finishes with some thoughts on how these approaches can be combined and how the task the application should support can help inform the choice of visualization modality

Family names as indicators of Britain’s changing regional geography

In recent years the geography of surnames has become increasingly researched in genetics, epidemiology, linguistics and geography. Surnames provide a useful data source for the analysis of population structure, migrations, genetic relationships and levels of cultural diffusion and interaction between communities. The Worldnames database (www.publicprofiler.org/worldnames) of 300 million people from 26 countries georeferenced in many cases to the equivalent of UK Postcode level provides a rich source of surname data. This work has focused on the UK component of this dataset, that is the 2001 Enhanced Electoral Role, georeferenced to Output Area level. Exploratory analysis of the distribution of surnames across the UK shows that clear regions exist, such as Cornwall, Central Wales and Scotland, in agreement with anecdotal evidence. This study is concerned with applying a wide range of methods to the UK dataset to test their sensitivity and consistency to surname regions. Methods used thus far are hierarchical and non-hierarchical clustering, barrier algorithms, such as the Monmonier Algorithm, and Multidimensional Scaling. These, to varying degrees, have highlighted the regionality of UK surnames and provide strong foundations to future work and refinement in the UK context. Establishing a firm methodology has enabled comparisons to be made with data from the Great British 1881 census, developing insights into population movements from within and outside Great Britain

Topological Segmentation of 2D Vector Fields

Vector field topology has a long tradition as a visualization tool. The separatrices segment the domain visually into canonical regions in which all streamlines behave qualitatively the same. But application scientists often need more than just a nice image for their data analysis, and, to best of our knowledge, so far no workflow has been proposed to extract the critical points, the associated separatrices, and then provide the induced segmentation on the data level. We present a workflow that computes the segmentation of the domain of a 2D vector field based on its separatrices. We show how it can be used for the extraction of quantitative information about each segment in two applications: groundwater flow and heat exchange

A computational geometric approach for an ensemble-based topological entropy calculation in two and three dimensions

From the stirring of dye in viscous fluids to the availability of essential nutrients spreading over the surface of a pond, nature is rife with examples of mixing in two-dimensional fluids. The long-time exponential growth rate of a thin filament of dye stretched by the fluid is a well-known proxy for the quality of mixing in two dimensions. This growth rate in turn gives a lower bound on the flow's topological entropy, a measure quantifying the complexity of chaotic dynamics. In the real-world study of mixing, topological entropy may be hard to compute; the velocity field may not be known or may be expensive to recover or approximate, thus limiting our knowledge of the governing system and underlying mechanics driving the mixing. Central to this study are two questions: \emph{How can stretching rates in two-dimensional planar flows best be computed using only trajectory data?}, and \emph{Can a method for computing stretching rates in higher dimensions from only trajectory data be developed?}. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation (E-tec), a method to derive a lower-bound on topological entropy that requires only finite number of system trajectories, like those obtained from ocean drifters, and no detailed knowledge of the velocity field. E-tec is demonstrated to be computationally more efficient than other competing methods in two dimensions that accommodate trajectory data. This is accomplished by considering the evolution of a ``rubber band" wrapped around the data points and evolving with their trajectories. E-tec records the growth of this band as the collective motion of trajectories strike, deform, and stretch it. This exponential growth rate acts as a lower bound on the topological entropy. In this manuscript, I demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time. Driving the efficiency of E-tec in two dimensions is the use of computational geometry tools. Not only this, by computing stretching rates in this new computational geometry framework, I extend E-tec to three dimensions using two methods. First, I consider a two-dimensional rubber sheet stretched around a collection of points in a three-dimensional flow. Similar to the band-stretching component of two-dimensional E-tec, a three-dimensional triangulation is used to record the growth of the sheet as it is stretched and deformed by points evolving in time. Second, I calculate the growth rates of one-dimensional rubber strings as they are stretched by the edges of this dynamic, moving triangulation

Approximation algorithms for multi-facility location

This thesis deals with the development and implementation of efficient algorithms to obtain acceptable solutions for the location of several facilities to serve customer sites. The general version of facility location problem is known to be NP-hard; For locating multiple facilities we use Voronoi diagram of initial facility locations to partition the customer sites into k clusters. On each Voronoi region, solutions for single facility problem is obtained by using both Weizfield\u27s algorithm and Center of Gravity. The customer space is again partitioned by using the newly computed locations. This iteration is continued to obtain a better solution for multi-facility location problem. We call the resulting algorithm: Voronoi driven k-median algorithm ; We report experimental results on several test data that include randomly distributed customers and distinctly clustered customers. The observed results show that the proposed approximation algorithm produces good results

Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Grid generation for reservoir simulation must honor classical key constraints and ensure boundary alignment such that control-volume boundaries are aligned with geological features including layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. Novel unstructured grid generation methods are proposed that automate control-volume and/or control point boundary alignment and yield perpendicular-bisector (PEBI) meshes both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles that contain segments of key geological boundaries, while in order to generate a dual-cell feature aligned grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex flux-continuous control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom and thus ensure application of the most efficient methods. The CVD-MPFA formulation (c.f. Edwards et al.) in cell-centred and cell-vertex modes is somewhat analogous and requires switching control-volume from primal to dual or vice versa, together with appropriate data structures and boundary conditions, however dual-cells are generated after primal grid generation. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are contrasted in terms of flow resolution and degrees of freedom required