Multivariate interpolation: Preserving and exploiting symmetry. (November 2021)

• Record Type: Journal Article
• Authors: Rodriguez Bazan, Erick; Hubert, Evelyne
• Is Part Of: Journal of symbolic computation. Volume 107(2021)
• Page Start: 1
• Page End: 22
• Publication Date: 2021-11
• Subjects: Multivariate interpolation -- Vandermonde matrix -- Collocation matrix -- Block diagonalization -- Symmetry -- Representation theory; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2021.01.004
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. The article shows how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost. We revisit minimal degree and least interpolation with symmetry adapted bases, rather than monomial bases. For a space of linear forms invariant under a group action, we construct bases of invariant interpolation spaces in blocks, capturing the inherent redundancy in the computations. With the so constructed symmetry adapted interpolation bases, the uniquely defined interpolant automatically preserves any equivariance the interpolation problem might have. Even with no equivariance, the computational cost to obtain the interpolant is alleviated thanks to the smaller size of the matrices to be inverted.