On the construction of general cubature formula by flat extensions

Marta Abril Bucero (1) , Chandrajit Bajaj (2) , Bernard Mourrain (1)

1 GALAAD - Geometry, algebra, algorithms
2 University of Texas at Austin [Austin]

We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with at extensions. We then analyse the algebraic properties associated to at extensions and show how to recover the cubature points and weights from the truncated Hankel operator. We next present an algorithm to test the at 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 cu-bature 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.

Mots clés: SDP; truncated hankel matrices; Cubature formulas; convex optimization; flat extension

Domaines: Géométrie algébrique [math.AG]

Marta Abril Bucero, Chandrajit Bajaj, Bernard Mourrain. On the construction of general cubature formula by flat extensions. 2015. ⟨hal-01158099v1⟩