783 research outputs found
Recommended from our members
Convexity of Bẻzier nets on sub-triangles
This note generalizes a result of Goodman[3], where it is shown that the convexity of Bèzier nets defined on a base triangle is preserved on sub-triangles obtained from a mid-point subdivision process. Here we show that the convexity of Bèzier nets is preserved on and only on sub-triangles that are "parallel" to the base triangle
A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations
We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods
Geometry of Log-Concave Density Estimation
Shape-constrained density estimation is an important topic in mathematical
statistics. We focus on densities on that are log-concave, and
we study geometric properties of the maximum likelihood estimator (MLE) for
weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the
optimal log-concave density is piecewise linear and supported on a regular
subdivision of the samples. This defines a map from the space of weights to the
set of regular subdivisions of the samples, i.e. the face poset of their
secondary polytope. We prove that this map is surjective. In fact, every
regular subdivision arises in the MLE for some set of weights with positive
probability, but coarser subdivisions appear to be more likely to arise than
finer ones. To quantify these results, we introduce a continuous version of the
secondary polytope, whose dual we name the Samworth body. This article
establishes a new link between geometric combinatorics and nonparametric
statistics, and it suggests numerous open problems.Comment: 22 pages, 3 figure
Asymptotically maximal families of hypersurfaces in toric varieties
A real algebraic variety is maximal (with respect to the Smith-Thom
inequality) if the sum of the Betti numbers (with coefficients)
of the real part of the variety is equal to the sum of Betti numbers of its
complex part. We prove that there exist polytopes that are not Newton polytopes
of any maximal hypersurface in the corresponding toric variety. On the other
hand we show that for any polytope there are families of hypersurfaces
with the Newton polytopes that are
asymptotically maximal when tends to infinity. We also show that
these results generalize to complete intersections.Comment: 18 pages, 1 figur
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
Convergent adaptive hybrid higher-order schemes for convex minimization
This paper proposes two convergent adaptive mesh-refining algorithms for the
hybrid high-order method in convex minimization problems with two-sided
p-growth. Examples include the p-Laplacian, an optimal design problem in
topology optimization, and the convexified double-well problem. The hybrid
high-order method utilizes a gradient reconstruction in the space of piecewise
Raviart-Thomas finite element functions without stabilization on triangulations
into simplices or in the space of piecewise polynomials with stabilization on
polytopal meshes. The main results imply the convergence of the energy and,
under further convexity properties, of the approximations of the primal resp.
dual variable. Numerical experiments illustrate an efficient approximation of
singular minimizers and improved convergence rates for higher polynomial
degrees. Computer simulations provide striking numerical evidence that an
adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even
with empirical higher convergence rates
- …