20,046 research outputs found
Inner approximation of convex cones via primal-dual ellipsoidal norms
We study ellipsoids from the point of view of approximating convex sets. Our focus is
on finding largest volume ellipsoids with specified centers which are contained in certain
convex cones. After reviewing the related literature and establishing some fundamental
mathematical techniques that will be useful, we derive such maximum volume ellipsoids
for second order cones and the cones of symmetric positive semidefinite matrices. Then we
move to the more challenging problem of finding a largest pair (in the sense of geometric
mean of their radii) of primal-dual ellipsoids (in the sense of dual norms) with specified
centers that are contained in their respective primal-dual pair of convex cones
The Offset Filtration of Convex Objects
We consider offsets of a union of convex objects. We aim for a filtration, a sequence of nested simplicial complexes, that captures the topological evolution of the offsets for increasing radii. We describe methods to compute a filtration based on the Voronoi partition with respect to the given convex objects. The size of the filtration and the time complexity for computing it are proportional to the size of the Voronoi diagram and its time complexity, respectively. Our approach is inspired by alpha-complexes for point sets, but requires more involved machinery and analysis primarily since Voronoi regions of general convex objects do not form a good cover. We show by experiments that our approach results in a similarly fast and topologically more stable method for computing a filtration compared to approximating the input by a point sample
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Deconstructing Approximate Offsets
We consider the offset-deconstruction problem: Given a polygonal shape Q with
n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance,
as the Minkowski sum of another polygonal shape P with a disk of fixed radius?
If it does, we also seek a preferably simple-looking solution P; then, P's
offset constitutes an accurate, vertex-reduced, and smoothened approximation of
Q. We give an O(n log n)-time exact decision algorithm that handles any
polygonal shape, assuming the real-RAM model of computation. A variant of the
algorithm, which we have implemented using CGAL, is based on rational
arithmetic and answers the same deconstruction problem up to an uncertainty
parameter \delta; its running time additionally depends on \delta. If the input
shape is found to be approximable, this algorithm also computes an approximate
solution for the problem. It also allows us to solve parameter-optimization
problems induced by the offset-deconstruction problem. For convex shapes, the
complexity of the exact decision algorithm drops to O(n), which is also the
time required to compute a solution P with at most one more vertex than a
vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011,
submitted to DC
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
We provide a characterization of the radii minimal projections of polytopes
onto -dimensional subspaces in Euclidean space \E^n. Applied on simplices
this characterization allows to reduce the computation of an outer radius to a
computation in the circumscribing case or to the computation of an outer radius
of a lower-dimensional simplex. In the second part of the paper, we use this
characterization to determine the sequence of outer -radii of regular
simplices (which are the radii of smallest enclosing cylinders). This settles a
question which arose from the incidence that a paper by Wei{\ss}bach (1983) on
this determination was erroneous. In the proof, we first reduce the problem to
a constrained optimization problem of symmetric polynomials and then to an
optimization problem in a fixed number of variables with additional integer
constraints.Comment: Minor revisions. To appear in Advances in Geometr
Linear Scaling Density Matrix Real Time TDDFT: Propagator Unitarity \& Matrix Truncation
Real time, density matrix based, time dependent density functional theory
proceeds through the propagation of the density matrix, as opposed to the
Kohn-Sham orbitals. It is possible to reduce the computational workload by
imposing spatial cut-off radii on sparse matrices, and the propagation of the
density matrix in this manner provides direct access to the optical response of
very large systems, which would be otherwise impractical to obtain using the
standard formulations of TDDFT. Following a brief summary of our
implementation, along with several benchmark tests illustrating the validity of
the method, we present an exploration of the factors affecting the accuracy of
the approach. In particular we investigate the effect of basis set size and
matrix truncation, the key approximation used in achieving linear scaling, on
the propagator unitarity and optical spectra. Finally we illustrate that, with
an appropriate density matrix truncation range applied, the computational load
scales linearly with the system size and discuss the limitations of the
approach.Comment: Accepted for publication in J. Chem. Phy
Approximations of the Wiener sausage and its curvature measures
A parallel neighborhood of a path of a Brownian motion is sometimes called
the Wiener sausage. We consider almost sure approximations of this random set
by a sequence of random polyconvex sets and show that the convergence of the
corresponding mean curvature measures holds under certain conditions in two and
three dimensions. Based on these convergence results, the mean curvature
measures of the Wiener sausage are calculated numerically by Monte Carlo
simulations in two dimensions. The corresponding approximation formulae are
given.Comment: Published in at http://dx.doi.org/10.1214/09-AAP596 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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