2015/05/30 by Marta Abril Bucero, Chandrajit Bajaj, Bucero, Marta Abril +3 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Sparse and Compressive Sensing Techniques #math.AG
paper · pdf · doi:10.48550/arxiv.1506.00085
openalex publication_date 2015/05/30 · arxiv created 2015/06/09 · arxiv updated 2015/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 analyse 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.