title: On the construction of general cubature formula by flat extensions.
abstract: We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyze the algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated Hankel operator. We next present an algorithm to test the flat extension property and to additionally compute the decomposition. To generate cubature formulae with a minimal number of points, we propose a new relaxation hierarchy of convex optimization problems minimizing the nuclear norm of the Hankel operators. For a suitably high order of convex relaxation, the minimizer of the optimization problem corresponds to a cubature formula. Furthermore cubature formulae with a minimal number of points are associated to faces of the convex sets. We illustrate our method on some examples, and for each we obtain a new minimal cubature formula.
authors: Inria Sophia Antipolis Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France.
authors: Marta Abril
authors: MA
authors: Bucero
authors: Department of Computer Science, The Institute of Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th Street, POB 2.324A, 1 University Station, C0200, Austin, TX 78712-0027, USA.
authors: Chandrajit
authors: C
authors: Bajaj
authors: Inria Sophia Antipolis Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France.
authors: Bernard
authors: B
authors: Mourrain
keywords: Border basis; Cubature formula; Flat extension; Hankel matrix; Orthogonal polynomials; Semidefinite programming