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Symmetry Preserving Interpolation

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GCS - Geometry, compatibility and structure preservation in computational differential equations

In this talk I choose to present the PhD work of Erick Rodriguez Bazan. We address multivariate interpolation in the presence of symmetry as given by a finite group. 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. Beside its preservation, symmetry shall also be exploited to alleviate the computational cost.

We revisit minimal degree and least interpolation spaces [de Boor & Ron 1990] with symmetry adapted bases (rather than the usual monomial bases). In these bases, the multivariate Vandermonde matrix  (a.k.a colocation matrix) is block diagonal as soon as the set of nodes is invariant. These blocks capture the inherent redundancy in the computations. Furthermore any equivariance an interpolation problem might have will be automatically preserved : the output interpolant will have the same equivariance property.

The special case of multivariate Hermite interpolation leads us to question the representation of polynomial ideals. Gröbner bases, the preferred tool for algebraic computations, breaks any kind of symmetry. The prior notion of H-Bases, introduced by Macaulay, appears as more suitable.

Reference:
  https://dl.acm.org/citation.cfm?doid=3326229.3326247
  https://hal.inria.fr/hal-01994016   Joint work with  Erick Rodriguez Bazan




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