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arXiv · 2010.10824

Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality

Michael Hecht·Krzysztof Gonciarz·Jannik Michelfeit·Vladimir Sivkin·Ivo F. Sbalzarini
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Abstract

We extend Newton and Lagrange interpolation to arbitrary dimensions. The core contribution that enables this is a generalized notion of non-tensorial unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. By validation, we reach the optimal exponential Trefethen rates for a class of analytic functions, we term Trefethen functions. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on these results, we propose an algorithm to efficiently and numerically stably solve arbitrary-dimensional interpolation problems, with at most quadratic runtime and linear memory requirement.

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Michael Hecht, Krzysztof Gonciarz, Jannik Michelfeit, Vladimir Sivkin, Ivo F. Sbalzarini. 2020-10-21. Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality. https://arxiv.org/abs/2010.10824

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