7,104 research outputs found
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
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
Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere
We study the convergence rate of a hierarchy of upper bounds for polynomial
minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp.
864-885], for the special case when the feasible set is the unit (hyper)sphere.
The upper bound at level r of the hierarchy is defined as the minimal expected
value of the polynomial over all probability distributions on the sphere, when
the probability density function is a sum-of-squares polynomial of degree at
most 2r with respect to the surface measure.
We show that the exact rate of convergence is Theta(1/r^2), and explore the
implications for the related rate of convergence for the generalized problem of
moments on the sphere.Comment: 14 pages, 2 figure
Star Integrals, Convolutions and Simplices
We explore single and multi-loop conformal integrals, such as the ones
appearing in dual conformal theories in flat space. Using Mellin amplitudes, a
large class of higher loop integrals can be written as simple
integro-differential operators on star integrals: one-loop -gon integrals in
dimensions. These are known to be given by volumes of hyperbolic simplices.
We explicitly compute the five-dimensional pentagon integral in full generality
using Schl\"afli's formula. Then, as a first step to understanding higher
loops, we use spline technology to construct explicitly the hexagon and
octagon integrals in two-dimensional kinematics. The fully massive hexagon
and octagon integrals are then related to the double box and triple box
integrals respectively. We comment on the classes of functions needed to
express these integrals in general kinematics, involving elliptic functions and
beyond.Comment: 23 page
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