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Chamber Graphs of Minimal Parabolic Geometries of Type M_24
No full textThis paper investigates the structure of the chamber graph associated with
the minimal parabolic geometries of rank 3 for the groups M24, the Mathieu
group of degree 24, He, Held's simple group, and 37_Sp6(2), a non-split
extension of an elementary abelian 3-group with Sp6(2). These three minmal
parabolic geometries all have the same diagram. Executable �les containing
data describing the disc structure of a �xed chamber of these graphs
accompany this paper. Those chambers at maximal distance from a given
chamber are studied, as are the maximal opposite sets. Also the geodesic
closure of opposite chambers are described for each of these three geometries
Evolving Graphs and Similarity-based Graphs with Applications
Full text linkA graph is a mathematical structure for modelling the pairwise relations between objects. This thesis studies two types of graphs, namely, similarity-based graphs and evolving graphs.
We look at ways to traverse an evolving graph. In particular, we examine the influence of temporal information on node centrality. In the process, we develop EvolvingGraphs.jl, a software package for analyzing time-dependent networks.
We develop Etymo, a search system for discovering interesting research papers. Etymo utilizes both similarity-based graphs and evolving graphs to build a knowledge graph of research articles in order to help users to track the development of ideas. We construct content similarity-based graphs using the full text of research papers. And we extract key concepts from research papers and exploit the temporal information in research papers to construct a concepts evolving graph
Optimality of the Paterson-Stockmeyer method for evaluating matrix polynomials and rational matrix functions
Full text linkMany state-of-the-art algorithms reduce the computation of transcendental matrix functions to the evaluation of polynomial or rational approximants at a matrix argument. This task can be accomplished efficiently by resorting to the Paterson–Stockmeyer method, an evaluation scheme originally developed for matrix polynomials that extends quite naturally to rational functions. An important feature of this technique is that the number of matrix multiplications required to evaluate an approximant of order n grows slower than n itself, with the result that different approximants yield the same asymptotic computational cost. We analyze the number of matrix multiplications required by the Paterson–Stockmeyer method and by two widely used generalizations, one for evaluating diagonal Padé approximants of general functions and one specifically tailored to those of the exponential. In all three cases, we identify the approximants of maximum order for any given computational cost
The classification and dynamics of the momentum polytopes of the SU(3) action on points in the complex projective plane with an application to point vortices
Full text linkWe have fully classified the momentum polytopes of the SU(3) action on CP2 x CP2 and CP2 x CP2 x CP2, both actions with weighted symplectic forms, and their corresponding transition momentum polytopes. For CP2 x CP2 the
momentum polytopes are distinct line segments. The action on CP2 x CP2 x CP2 has 8 different momentum polytopes. The vertices of the momentum polytopes of the SU(3) action on CP2 x CP2 x CP2 fall into two categories: definite and
indefinite vertices. The reduced space corresponding to momentum map image values at definite vertices is isomorphic to the 2-sphere. We show that these results can be applied to assess the dynamics by introducing and computing the
space of allowed velocity vectors for the different configurations of two-vortex systems
A New Preconditioner that Exploits Low-Rank Approximations to Factorization Error
No full textWe consider ill-conditioned linear systems b that are to be solved
iteratively, and assume that a low accuracy LU factorization
is available for use in a
preconditioner. We have observed that for ill-conditioned matrices
arising in practice, tends to be numerically low rank, that is, it
has a small number of large singular values. Importantly, the error matrix
tends to have the same
property. To understand this phenomenon we give bounds for the distance
from to a low-rank matrix in terms of the corresponding distance for
. We then design a novel preconditioner that exploits the low-rank
property of the error to accelerate the convergence of iterative
methods. We apply this new preconditioner in three different contexts
fitting our general framework: low floating-point precision (e.g., half
precision) LU factorization, incomplete LU factorization, and block
low-rank LU factorization.
In numerical experiments with GMRES-based iterative refinement we show
that our preconditioner can achieve a significant reduction in the number
of iterations required to solve a variety of real-life problems
Generalized point vortex dynamics on CP^2
No full textThis is the second of two companion papers.
We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space CP^2 interacting via a simple Hamiltonian function. The system has symmetry group SU(3). The first paper describes all possible momentum polytopes for this system, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of interacting generalized vortices.
The different types of polytope depend on the values of the `vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of CP^2. We show that the reduced spaces for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction.
For 2 vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian
Face recognition based on texture information and geodesic distance approximations between multivariate normal distributions
No full textGeodesic distance is a natural dissimilarity measure between
probability distributions of a specific type, and can be used to
discriminate texture in image-based measurements. Furthermore, since
there is no known closed-form solution for the geodesic distance
between general multivariate normal distributions, we propose two
efficient approximations to be used as texture dissimilarity metrics
in the context of face recognition. A novel face recognition
approach based on texture discrimination in high-resolution color
face images is proposed, unlike the typical appearance-based
approach that relies on low-resolution grayscale face images. In our
face recognition approach, sparse facial features are extracted
using predefined landmark topologies, that identify discriminative
image locations on the face images. Given this landmark topology,
the dissimilarity between distinct face images are scored in terms
of the dissimilarities between their corresponding face landmarks,
and the texture in each one of these landmarks is represented by
multivariate normal distributions, expressing the color distribution
in the vicinity of each landmark location. The classification of new
face image samples occurs by determining the face image sample in
the training set which minimizes the dissimilarity score, using the
nearest neighbor rule. The proposed face recognition method was
compared to methods representative of the state-of-the-art, using
color or grayscale face images, and presented higher recognition
rates. Moreover, the proposed texture dissimilarity metric also is
efficient in general texture discrimination (e.g., texture
recognition of material images), as our experiments suggest
Squeezing a Matrix into Half Precision, with an Application to Solving Linear Systems
Full text linkMotivated by the demand in machine learning, modern computer hardware is increas- ingly supporting reduced precision floating-point arithmetic, which provides advantages in speed, energy, and memory usage over single and double precision. Given the availability of such hardware, mixed precision algorithms that work in single or double precision but carry out part of a compu- tation in half precision are now of great interest for general scientific computing tasks. Because of the limited range of half precision arithmetic, in which positive numbers lie between 6 × 10−8 and 7 × 104, a straightforward rounding of single or double precision data into half precision can lead to overflow, underflow, or subnormal numbers being generated, all of which are undesirable. We develop an algorithm for converting a matrix from single or double precision to half precision. It first applies two-sided diagonal scaling in order to equilibrate the matrix (that is, to ensure that every row and column has ∞-norm 1), then multiplies by a scalar to bring the largest element within a factor θ ≤ 1 of the overflow level, and finally rounds to half precision. The second step ensures that full use is made of the limited range of half precision arithmetic, and θ must be chosen to allow sufficient headroom for subsequent computations. We apply the new algorithm to GMRES-based iterative re- finement (GMRES-IR), which solves a linear system Ax = b with single or double precision data by LU factorizing A in half precision and carrying out iterative refinement with the correction equations solved by GMRES preconditioned with the low precision LU factors. Previous implementations of this algorithm have used a crude conversion to half precision that our experiments show can cause slow convergence of GMRES-IR for badly scaled matrices or failure to converge at all. The new conversion algorithm computes ∞-norms of rows and columns of the matrix and its cost is negligible in the context of LU factorization. We show that it leads to faster convergence of GMRES-IR for badly scaled matrices and thereby allows a much wider class of problems to be solved
Bridging the gap between flat and hierarchical low-rank matrix formats: the multilevel BLR format
No full textNotes on Histotomography
No full textIn many tomographic imaging problems the data consists of integrals along lines or curves. Increas-
ingly we are seeing “rich tomography” problems where the quantity imaged is higher dimensional than
a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and
in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data
contains not just integrals along rays but the distribution of values along the ray. If this is discretized
into bins we can think of this as a histogram. In this talk we introduce the concept of “histotomogra-
phy”. For scalar problems with histogram data this holds the possibility of reconstruction with fewer
rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null
space of related integral transforms. We will illustrate with examples from scalar spectral attenuation
tomography and tensor tomography methods for strain using neutrons, electrons and x-rays