Optimization of mesh hierarchies in Multilevel Monte Carlo samplers

We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and show that geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity as non-geometric optimal hierarchies. We discuss how enforcing constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These constraints include an upper and a lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm. The first example considers a three-dimensional elliptic partial differential equation with random inputs. Its space discretization is based on continuous piecewise trilinear finite elements and the corresponding linear system is solved by either a direct or an iterative solver. The second example considers a one-dimensional It\^o stochastic differential equation discretized by a Milstein scheme

Geometric Separation and Packing Problems

The first part of this thesis investigates combinatorial and algorithmic aspects of geometric separation problems in the plane. In such a setting one is given a set of points and a set of separators such as lines, line segments or disks. The goal is to select a small subset of those separators such that every path between any two points is intersected by at least one separator. We first look at several problems which arise when one is given a set of points and a set of unit disks embedded in the plane and the goal is to separate the points using a small subset of the given disks. Next, we focus on a separation problem involving only one region: Given a region in the plane, bounded by a piecewise linear closed curve, such as a fence, place few guards inside the fenced region such that wherever an intruder cuts through the fence, the closest guard is at most a distance one away. Restricting the separating objects to be lines, we investigate combinatorial aspects which arise when we use them to pairwise separate a set of points in the plane; hereafter we generalize the notion of separability to arbitrary sets and present several enumeration results. Lastly, we investigate a packing problem with a non-convex shape in ℝ3. We show that ℝ3 can be packed at a density of 0.222 with tori of major radius one and minor radius going to zero. Furthermore, we show that the same torus arrangement yields the asymptotically optimal number of pairwise linked tori

Scale-based surface understanding using diffusion smoothing

The research discussed in this thesis is concerned with surface understanding from the viewpoint of recognition-oriented, scale-related processing based on surface curvatures and diffusion smoothing. Four problems below high level visual processing are investigated: 1) 3-dimensional data smoothing using a diffusion process; 2) Behaviour of shape features across multiple scales, 3) Surface segmentation over multiple scales; and 4) Symbolic description of surface features at multiple scales. In this thesis, the noisy data smoothing problem is treated mathematically as a boundary value problem of the diffusion equation instead of the well-known Gaussian convolution, In such a way, it provides a theoretical basis to uniformly interpret the interrelationships amongst diffusion smoothing, Gaussian smoothing, repeated averaging and spline smoothing. It also leads to solving the problem with a numerical scheme of unconditional stability, which efficiently reduces the computational complexity and preserves the signs of curvatures along the surface boundaries. Surface shapes are classified into eight types using the combinations of the signs of the Gaussian curvature K and mean curvature H, both of which change at different scale levels. Behaviour of surface shape features over multiple scale levels is discussed in terms of the stability of large shape features, the creation, remaining and fading of small shape features, the interaction between large and small features and the structure of behaviour of the nested shape features in the KH sign image. It provides a guidance for tracking the movement of shape features from fine to large scales and for setting up a surface shape description accordingly. A smoothed surface is partitioned into a set of regions based on curvature sign homogeneity. Surface segmentation is posed as a problem of approximating a surface up to the degree of Gaussian and mean curvature signs using the depth data alone How to obtain feasible solutions of this under-determined problem is discussed, which includes the surface curvature sign preservation, the reason that a sculptured surface can be segmented with the KH sign image alone and the selection of basis functions of surface fitting for obtaining the KH sign image or for region growing. A symbolic description of the segmented surface is set up at each scale level. It is composed of a dual graph and a geometrical property list for the segmented surface. The graph describes the adjacency and connectivity among different patches as the topological-invariant properties that allow some object's flexibility, whilst the geometrical property list is added to the graph as constraints that reduce uncertainty. With this organisation, a tower-like surface representation is obtained by tracking the movement of significant features of the segmented surface through different scale levels, from which a stable description can be extracted for inexact matching during object recognition

Geometry–aware finite element framework for multi–physics simulations: an algorithmic and software-centric perspective

In finite element simulations, the handling of geometrical objects and their discrete representation is a critical aspect in both serial and parallel scientific software environments. The development of codes targeting such envinronments is subject to great development effort and man-hours invested. In this thesis we approach these issues from three fronts. First, stable and efficient techniques for the transfer of discrete fields between non matching volume or surface meshes are an essential ingredient for the discretization and numerical solution of coupled multi-physics and multi-scale problems. In particular L2-projections allows for the transfer of discrete fields between unstructured meshes, both in the volume and on the surface. We present an algorithm for parallelizing the assembly of the L2-transfer operator for unstructured meshes which are arbitrarily distributed among different processes. The algorithm requires no a priori information on the geometrical relationship between the different meshes. Second, the geometric representation is often a limiting factor which imposes a trade-off between how accurately the shape is described, and what methods can be employed for solving a system of differential equations. Parametric finite-elements and bijective mappings between polygons or polyhedra allow us to flexibly construct finite element discretizations with arbitrary resolutions without sacrificing the accuracy of the shape description. Such flexibility allows employing state-of-the-art techniques, such as geometric multigrid methods, on meshes with almost any shape.t, the way numerical techniques are represented in software libraries and approached from a development perspective, affect both usability and maintainability of such libraries. Completely separating the intent of high-level routines from the actual implementation and technologies allows for portable and maintainable performance. We provide an overview on current trends in the development of scientific software and showcase our open-source library utopia

Mathematical Imaging and Surface Processing

Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains

Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005

This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop

Algorithms for Geometric Covering and Piercing Problems

This thesis involves the study of a range of geometric covering and piercing problems, where the unifying thread is approximation using disks. While some of the problems addressed in this work are solved exactly with polynomial time algorithms, many problems are shown to be at least NP-hard. For the latter, approximation algorithms are the best that we can do in polynomial time assuming that P is not equal to NP. One of the best known problems involving unit disks is the Discrete Unit Disk Cover (DUDC) problem, in which the input consists of a set of points P and a set of unit disks in the plane D, and the objective is to compute a subset of the disks of minimum cardinality which covers all of the points. Another perspective on the problem is to consider the centre points (denoted Q) of the disks D as an approximating set of points for P. An optimal solution to DUDC provides a minimal cardinality subset Q*, a subset of Q, so that each point in P is within unit distance of a point in Q*. In order to approximate the general DUDC problem, we also examine several restricted variants. In the Line-Separable Discrete Unit Disk Cover (LSDUDC) problem, P and Q are separated by a line in the plane. We write that l^- is the half-plane defined by l containing P, and l^+ is the half-plane containing Q. LSDUDC may be solved exactly in O(m^2n) time using a greedy algorithm. We augment this result by describing a 2-approximate solution for the Assisted LSDUDC problem, where the union of all disks centred in l^+ covers all points in P, but we consider using disks centred in l^- as well to try to improve the solution. Next, we describe the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem, where P and Q are confined to a strip of the plane of height h. We show that this problem is NP-complete, and we provide a range of approximation algorithms for the problem with trade-offs between the approximation factor and running time. We outline approximation algorithms for the general DUDC problem which make use of the algorithms for LSDUDC and WSDUDC. These results provide the fastest known approximation algorithms for DUDC. As with the WSDUDC results, we present a set of algorithms in which better approximation factors may be had at the expense of greater running time, ranging from a 15-approximate algorithm which runs in O(mn + m log m + n log n) time to a 18-approximate algorithm which runs in O(m^6n+n log n) time. The next problems that we study are Hausdorff Core problems. These problems accept an input polygon P, and we seek a convex polygon Q which is fully contained in P and minimizes the Hausdorff distance between P and Q. Interestingly, we show that this problem may be reduced to that of computing the minimum radius of disk, call it k_opt, so that a convex polygon Q contained in P intersects all disks of radius k_opt centred on the vertices of P. We begin by describing a polynomial time algorithm for the simple case where P has only a single reflex vertex. On general polygons, we provide a parameterized algorithm which performs a parametric search on the possible values of k_opt. The solution to the decision version of the problem, i.e. determining whether there exists a Hausdorff Core for P given k_opt, requires some novel insights. We also describe an FPTAS for the decision version of the Hausdorff Core problem. Finally, we study Generalized Minimum Spanning Tree (GMST) problems, where the input consists of imprecise vertices, and the objective is to select a single point from each imprecise vertex in order to optimize the weight of the MST over the points. In keeping with one of the themes of the thesis, we begin by using disks as the imprecise vertices. We show that the minimization and maximization versions of this problem are NP-hard, and we describe some parameterized and approximation algorithms. Finally, we look at the case where the imprecise vertices consist of just two vertices each, and we show that the minimization version of the problem (which we call 2-GMST) remains NP-hard, even in the plane. We also provide an algorithm to solve the 2-GMST problem exactly if the combinatorial structure of the optimal solution is known. We identify a number of open problems in this thesis that are worthy of further study. Among them: Is the Assisted LSDUDC problem NP-complete? Can the WSDUDC results be used to obtain an improved PTAS for DUDC? Are there classes of polygons for which the determination of the Hausdorff Core is easy? Is there a PTAS for the maximum weight GMST problem on (unit) disks? Is there a combinatorial approximation algorithm for the 2-GMST problem (particularly with an approximation factor under 4)