Optimization-Aided Construction of Multivariate Chebyshev Polynomials
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914991459794944 |
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| author | Dressler, Mareike Foucart, Simon Joldes, Mioara de Klerk, Etienne Lasserre, Jean Bernard Xu, Yuan |
| author_facet | Dressler, Mareike Foucart, Simon Joldes, Mioara de Klerk, Etienne Lasserre, Jean Bernard Xu, Yuan |
| contents | This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial $x_1^2 x_2^2 x_3$ on the euclidean ball and for the monomial $x_1^2 x_2 x_3$ on the simplex. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10438 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimization-Aided Construction of Multivariate Chebyshev Polynomials Dressler, Mareike Foucart, Simon Joldes, Mioara de Klerk, Etienne Lasserre, Jean Bernard Xu, Yuan Optimization and Control Numerical Analysis 41A10, 65D15, 90C22 This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial $x_1^2 x_2^2 x_3$ on the euclidean ball and for the monomial $x_1^2 x_2 x_3$ on the simplex. |
| title | Optimization-Aided Construction of Multivariate Chebyshev Polynomials |
| topic | Optimization and Control Numerical Analysis 41A10, 65D15, 90C22 |
| url | https://arxiv.org/abs/2405.10438 |