Optimizations for Tensorial Bernstein-Based Solvers by Using Polyhedral Bounds - Archive ouverte HAL
Article Dans Une Revue International Journal of Shape Modeling Année : 2010

Optimizations for Tensorial Bernstein-Based Solvers by Using Polyhedral Bounds

Dominique Michelucci

Résumé

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 quadratic number 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.
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Dates et versions

hal-00629036 , version 1 (04-10-2011)

Identifiants

Citer

Fuenfzig Christoph, Dominique Michelucci, Foufou Sebti. Optimizations for Tensorial Bernstein-Based Solvers by Using Polyhedral Bounds. International Journal of Shape Modeling, 2010, 16 (1-2), pp.109-128. ⟨10.1142/S0218654310001304⟩. ⟨hal-00629036⟩
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