Multivariate Interpolation: Preserving and Exploiting Symmetry

Erick Rodriguez Bazan (1, 2) , Evelyne Hubert (1, 2)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
2 UniCA - Université Côte d'Azur

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.

Mots clés: Multivariate Interpolation; Vandermonde matrix; Collocation matrix; Symmetry; Representation theory; Symmetry adapted basis; Block diagonalization; Least interpolation

Domaines: Calcul formel [cs.SC]; Théorie des groupes [math.GR]; Anneaux et algèbres [math.RA]; Analyse numérique [math.NA]

Erick Rodriguez Bazan, Evelyne Hubert. Multivariate Interpolation: Preserving and Exploiting Symmetry. Journal of Symbolic Computation, inPress, ⟨10.1016/j.jsc.2021.01.004⟩. ⟨hal-03123418⟩