Multivariate moment problems : applications of the reconstruction of linear forms on the polynomial ring

Mathieu Collowald 1
1 GALAAD2 - Géométrie , Algèbre, Algorithmes
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This thesis deals with the reconstruction of linear forms on the polynomial ring and its applications. We propose theoretical and algorithmic tools to solve multivariate moment problems: the reconstruction of convex polytopes from their moments (shape-from-moments) and the search for cubatures. The numerical algorithm we propose to reconstruct polytopes uses numerical methods previously known in the case of polygons, and also Brion's identities that relate directional moments and projected vertices. A polyhedron with 57 vertices – a diamond cut – is thus reconstructed. Concerning the search for cubatures, we adapt the univariate Prony's method into a multivariate method thanks to Hankel operators. A matrix completion problem is then solved with a basis-free version of Curto-Fialkow's flat extension theorem. We explain thus the moment matrix approach to cubatures, known in the litterature. Symmetry is here a natural ingredient and reduces the algorithmic complexity. We show that a block diagonalisation of the involved matrices is possible. Those blocs and the matrix of multiplicities of a finite group provide necessary conditions on the existence of cubatures. Given a measure, a degree and a number of nodes, our algorithm first certify the existence of cubatures and then compute the weights and nodes. New cubatures have been found: either by completing the ones known for a given measure and degree, or by adding cubatures with a higher degree for a given measure.
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Mathieu Collowald. Multivariate moment problems : applications of the reconstruction of linear forms on the polynomial ring. General Mathematics [math.GM]. Université Nice Sophia Antipolis, 2015. English. ⟨NNT : 2015NICE4139⟩. ⟨tel-01293281⟩

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