4.1 Article

Stokes, Gibbs, and Volume Computation of Semi-Algebraic Sets

Journal

DISCRETE & COMPUTATIONAL GEOMETRY
Volume -, Issue -, Pages -

Publisher

SPRINGER
DOI: 10.1007/s00454-022-00462-0

Keywords

Numerical methods for multivariate integration; Real algebraic geometry; Convex optimization; Stokes' theorem; Gibbs phenomenon

Funding

  1. French company Reseau de Transport d'Electricite
  2. Swiss National Science Foundation [51NF40_180545]
  3. AI Interdisciplinary Institute ANITI through the French Investing for the Future PIA3 program [ANR-19-PI3A-0004]
  4. Swiss National Science Foundation (SNF) [51NF40_180545] Funding Source: Swiss National Science Foundation (SNF)

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The article discusses the problem of computing the Lebesgue volume of compact basic semi-algebraic sets and presents an improved linear programming method that uses pseudo-moments and polynomials to approximate the volume. The research finds that the convergence speed of this method is significantly accelerated when the set is the smooth super-level set of a single polynomial.
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 infinite-dimensional linear program (LP). At each step one solves a semidefinite 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 discontinuous 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 refined 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 refined 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 recent results on Poisson's partial differential equation (PDE).

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