Abstract for the talk on 02.05.2017 (11:00 h)Seminar on Nonlinear Algebra
Cordian Riener (Universität Konstanz)
An optimization view on the quadrature problem
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Let d and k be positive integers and μ be a positive Borel measure on ℝ2 possessing finite moments up to degree 2d − 1. Using methods from convex optimization, we study the question of the minimal m ∈ ℕ such that there exists a quadrature rule for μ with m nodes which is exact for all polynomials of degree at most 2d− 1 We show that if the support of μ is contained in an algebraic curve of degree k, then there exists a quadrature rule for μ with at most dk many nodes all placed on the curve (and positive weights). This generalizes Gauss quadrature where the curve is a line and (the odd case of) Szegö quadrature where the curve is a circle to arbitrary plane algebraic curves. In the even case, i.e., 2d instead of 2d − 1 this result generalizes to compact curves. We use this result to show that, any plane measure μ has a quadrature rule with at most 3∕2d(d − 1) many nodes, which is exact up to degree 2d1. All our results are obtained by minimizing a certain linear functional on the polynomials of degree 2d and our proof uses both results from convex optimisation and from real algebraic geometry.
(Joint work with Markus Schweighofer)