1,168 research outputs found
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Exploiting Polyhedral Symmetries in Social Choice
A large amount of literature in social choice theory deals with quantifying
the probability of certain election outcomes. One way of computing the
probability of a specific voting situation under the Impartial Anonymous
Culture assumption is via counting integral points in polyhedra. Here, Ehrhart
theory can help, but unfortunately the dimension and complexity of the involved
polyhedra grows rapidly with the number of candidates. However, if we exploit
available polyhedral symmetries, some computations become possible that
previously were infeasible. We show this in three well known examples:
Condorcet's paradox, Condorcet efficiency of plurality voting and in Plurality
voting vs Plurality Runoff.Comment: 14 pages; with minor improvements; to be published in Social Choice
and Welfar
An exact general remeshing scheme applied to physically conservative voxelization
We present an exact general remeshing scheme to compute analytic integrals of
polynomial functions over the intersections between convex polyhedral cells of
old and new meshes. In physics applications this allows one to ensure global
mass, momentum, and energy conservation while applying higher-order polynomial
interpolation. We elaborate on applications of our algorithm arising in the
analysis of cosmological N-body data, computer graphics, and continuum
mechanics problems.
We focus on the particular case of remeshing tetrahedral cells onto a
Cartesian grid such that the volume integral of the polynomial density function
given on the input mesh is guaranteed to equal the corresponding integral over
the output mesh. We refer to this as "physically conservative voxelization".
At the core of our method is an algorithm for intersecting two convex
polyhedra by successively clipping one against the faces of the other. This
algorithm is an implementation of the ideas presented abstractly by Sugihara
(1994), who suggests using the planar graph representations of convex polyhedra
to ensure topological consistency of the output. This makes our implementation
robust to geometric degeneracy in the input. We employ a simplicial
decomposition to calculate moment integrals up to quadratic order over the
resulting intersection domain.
We also address practical issues arising in a software implementation,
including numerical stability in geometric calculations, management of
cancellation errors, and extension to two dimensions. In a comparison to recent
work, we show substantial performance gains. We provide a C implementation
intended to be a fast, accurate, and robust tool for geometric calculations on
polyhedral mesh elements.Comment: Code implementation available at https://github.com/devonmpowell/r3
Decomposition Methods for Nonlinear Optimization and Data Mining
We focus on two central themes in this dissertation. The first one is on
decomposing polytopes and polynomials in ways that allow us to perform
nonlinear optimization. We start off by explaining important results on
decomposing a polytope into special polyhedra. We use these decompositions and
develop methods for computing a special class of integrals exactly. Namely, we
are interested in computing the exact value of integrals of polynomial
functions over convex polyhedra. We present prior work and new extensions of
the integration algorithms. Every integration method we present requires that
the polynomial has a special form. We explore two special polynomial
decomposition algorithms that are useful for integrating polynomial functions.
Both polynomial decompositions have strengths and weaknesses, and we experiment
with how to practically use them.
After developing practical algorithms and efficient software tools for
integrating a polynomial over a polytope, we focus on the problem of maximizing
a polynomial function over the continuous domain of a polytope. This
maximization problem is NP-hard, but we develop approximation methods that run
in polynomial time when the dimension is fixed. Moreover, our algorithm for
approximating the maximum of a polynomial over a polytope is related to
integrating the polynomial over the polytope. We show how the integration
methods can be used for optimization.
The second central topic in this dissertation is on problems in data science.
We first consider a heuristic for mixed-integer linear optimization. We show
how many practical mixed-integer linear have a special substructure containing
set partition constraints. We then describe a nice data structure for finding
feasible zero-one integer solutions to systems of set partition constraints.
Finally, we end with an applied project using data science methods in medical
research.Comment: PHD Thesis of Brandon Dutr
On the equivalence between the cell-based smoothed finite element method and the virtual element method
We revisit the cell-based smoothed finite element method (SFEM) for
quadrilateral elements and extend it to arbitrary polygons and polyhedrons in
2D and 3D, respectively. We highlight the similarity between the SFEM and the
virtual element method (VEM). Based on the VEM, we propose a new stabilization
approach to the SFEM when applied to arbitrary polygons and polyhedrons. The
accuracy and the convergence properties of the SFEM are studied with a few
benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined
with the scaled boundary finite element method to problems involving
singularity within the framework of the linear elastic fracture mechanics in
2D
- …