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Stokes, Gibbs and volume computation of semi-algebraic sets

Abstract : We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain innite-dimensional linear program (LP). At each step one solves a semidenite relaxation of the LP which involves pseudo-moments up to a certain degree. Its dual computes a polynomial of same degree which approximates from above the discon-tinuous indicator function of the set, hence with a typical Gibbs phenomenon which results in a slow convergence of the associated numerical scheme. Drastic improvements have been observed by introducing in the initial LP additional linear moment constraints obtained from a certain application of Stokes' theorem for integration on the set. However and so far there was no rationale to explain this behavior. We provide a rened version of this extended LP formulation. When the set is the smooth super-level set of a single polynomial, we show that the dual of this rened LP has an optimal solution which is a continuous function. Therefore in this dual one now approximates a continuous function by a polynomial, hence with no Gibbs phenomenon, which explains and improves the already observed drastic acceleration of the convergence of the hierarchy. Interestingly, the technique of proof involves classical results on Poisson's partial dierential equation (PDE).
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https://hal.laas.fr/hal-02947268
Contributor : Didier Henrion <>
Submitted on : Wednesday, September 23, 2020 - 6:37:16 PM
Last modification on : Thursday, June 10, 2021 - 3:07:13 AM
Long-term archiving on: : Friday, December 4, 2020 - 5:06:34 PM

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  • HAL Id : hal-02947268, version 1
  • ARXIV : 2009.12139

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Matteo Tacchi, Jean Lasserre, Didier Henrion. Stokes, Gibbs and volume computation of semi-algebraic sets. 2020. ⟨hal-02947268⟩

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