Optimizations for tensorial Bernstein-based solvers by using polyhedral bounds

Christoph Fünfzig, Dominique Michelucci, Sebti Foufou

Research output: Contribution to journalArticle

Abstract

The tensorial Bernstein basis for multivariate polynomials in n variables has a number 3n of functions for degree 2. Consequently, computing the representation of a multivariate polynomial in the tensorial Bernstein basis is an exponential time algorithm, which makes tensorial Bernstein-based solvers impractical for systems with more than n = 6 or 7 variables. This article describes a polytope (Bernstein polytope) with a number O((n,2)) of faces, which allows to bound a sparse, multivariate polynomial expressed in the canonical basis by solving several linear programming problems. We compare the performance of a subdivision solver using domain reductions by linear programming with a solver using a change to the tensorial Bernstein basis for domain reduction. The performance is similar for n = 2 variables but only the solver using linear programming on the Bernstein polytope can cope with a large number of variables. We demonstrate this difference with two formulations of the forward kinematics problem of a Gough-Stewart parallel robot: a direct Cartesian formulation and a coordinate-free formulation using Cayley-Menger determinants, followed by a computation of Cartesian coordinates. Furthermore, we present an optimization of the Bernstein polytope-based solver for systems containing only the monomials xi and xi2. For these, it is possible to obtain even better domain bounds at no cost using the quadratic curve (xi, xi2) directly.

Original languageEnglish (US)
Pages (from-to)109-128
Number of pages20
JournalInternational Journal of Shape Modeling
Volume16
Issue number1-2
DOIs
StatePublished - Jun 1 2010

Fingerprint

Bernstein Basis
Polytope
Linear programming
Multivariate Polynomials
Polynomials
Optimization
Cartesian
Formulation
Parallel Robot
Canonical Basis
Cayley
Exponential time
Kinematics
Subdivision
Robots
Determinant
Face
Curve
Costs
Computing

Keywords

  • algebraic systems
  • Bernstein polynomials
  • linear programming
  • simplex algorithm
  • subdivision solver

ASJC Scopus subject areas

  • Software
  • Modeling and Simulation
  • Computer Vision and Pattern Recognition
  • Geometry and Topology
  • Computer Science Applications
  • Applied Mathematics

Cite this

Optimizations for tensorial Bernstein-based solvers by using polyhedral bounds. / Fünfzig, Christoph; Michelucci, Dominique; Foufou, Sebti.

In: International Journal of Shape Modeling, Vol. 16, No. 1-2, 01.06.2010, p. 109-128.

Research output: Contribution to journalArticle

Fünfzig, Christoph ; Michelucci, Dominique ; Foufou, Sebti. / Optimizations for tensorial Bernstein-based solvers by using polyhedral bounds. In: International Journal of Shape Modeling. 2010 ; Vol. 16, No. 1-2. pp. 109-128.
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