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

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F23%3A00363281" target="_blank" >RIV/68407700:21230/23:00363281 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s00454-022-00462-0" target="_blank" >https://doi.org/10.1007/s00454-022-00462-0</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00454-022-00462-0" target="_blank" >10.1007/s00454-022-00462-0</a>

Result language

angličtina

Original language name

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

Original language description

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).

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10102 - Applied mathematics

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2023

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Discrete & Computational Geometry

ISSN

0179-5376

e-ISSN

1432-0444

Volume of the periodical

69

Issue of the periodical within the volume

1

Country of publishing house

US - UNITED STATES

Number of pages

24

Pages from-to

260-283

UT code for WoS article

001179763600001

EID of the result in the Scopus database

2-s2.0-85144669648