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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