841 research outputs found
Practical Volume Estimation by a New Annealing Schedule for Cooling Convex Bodies
We study the problem of estimating the volume of convex polytopes, focusing
on H- and V-polytopes, as well as zonotopes. Although a lot of effort is
devoted to practical algorithms for H-polytopes there is no such method for the
latter two representations. We propose a new, practical algorithm for all
representations, which is faster than existing methods. It relies on
Hit-and-Run sampling, and combines a new simulated annealing method with the
Multiphase Monte Carlo (MMC) approach. Our method introduces the following key
features to make it adaptive: (a) It defines a sequence of convex bodies in MMC
by introducing a new annealing schedule, whose length is shorter than in
previous methods with high probability, and the need of computing an enclosing
and an inscribed ball is removed; (b) It exploits statistical properties in
rejection-sampling and proposes a better empirical convergence criterion for
specifying each step; (c) For zonotopes, it may use a sequence of convex bodies
for MMC different than balls, where the chosen body adapts to the input. We
offer an open-source, optimized C++ implementation, and analyze its performance
to show that it outperforms state-of-the-art software for H-polytopes by
Cousins-Vempala (2016) and Emiris-Fisikopoulos (2018), while it undertakes
volume computations that were intractable until now, as it is the first
polynomial-time, practical method for V-polytopes and zonotopes that scales to
high dimensions (currently 100). We further focus on zonotopes, and
characterize them by their order (number of generators over dimension), because
this largely determines sampling complexity. We analyze a related application,
where we evaluate methods of zonotope approximation in engineering.Comment: 20 pages, 12 figures, 3 table
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
Pruning Algorithms for Pretropisms of Newton Polytopes
Pretropisms are candidates for the leading exponents of Puiseux series that
represent solutions of polynomial systems. To find pretropisms, we propose an
exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton
polytopes. We prefer exact arithmetic not only because of the exact input and
the degrees of the output, but because of the often unpredictable growth of the
coordinates in the face normals, even for polytopes in generic position. We
provide experimental results with our preliminary implementation in Sage that
compare favorably with the pruning method that relies only on cone
intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning,
accepted for presentation at Computer Algebra in Scientific Computing, CASC
201
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