On moment problems with holonomic functions

Florent Bréhard 1, 2, 3, 4 Mioara Joldes 1 Jean-Bernard Lasserre 1
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
3 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
4 PLUME - Preuves et Langages
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Many reconstruction algorithms from moments of algebraic data were developed in optimization, analysis or statistics. Lasserre and Putinar proposed an exact reconstruction algorithm for the algebraic support of the Lebesgue measure, or of measures with density equal to the exponential of a known polynomial. Their approach relies on linear recurrences for the moments, obtained using Stokes theorem. In this article, we extend this study to measures with holonomic densities and support with real algebraic boundary. In the framework of holonomic distributions (i.e. they satisfy a holonomic system of linear partial or ordinary differential equations with polynomial coefficients), an alternate method to creative telescoping is proposed for computing linear recurrences for the moments. When the coefficients of a polynomial vanishing on the support boundary are given as parameters, the obtained recurrences have the advantage of staying linear with respect to them. This property allows for an efficient reconstruction method. Given a finite number of numerically computed moments for a measure with holonomic density, and assuming a real algebraic boundary for the support, we propose an algorithm for solving the inverse problem of obtaining both the coefficients of a polynomial vanishing on the boundary and those of the polynomials involved in the holonomic operators which annihilate the density.
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Submitted on : Monday, February 4, 2019 - 5:06:14 PM
Last modification on : Friday, June 14, 2019 - 6:31:29 PM
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Florent Bréhard, Mioara Joldes, Jean-Bernard Lasserre. On moment problems with holonomic functions. ISSAC 2019-44th International Symposium on Symbolic and Algebraic Computation, Jul 2019, Beijing, China. ⟨hal-02006645⟩

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