Multivariate Interpolation on Unisolvent Nodes – Lifting the Curse of Dimensionality
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We present generalizations of the classic Newton and Lagrange interpolation schemes to arbitrary dimensions. The core contribution that enables this new method is the notion of unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. We prove that by choosing these nodes in a proper way, the resulting interpolation schemes become generic, while approximating all continuous Sobolev functions. If in addition the function is analytical in the Trefethen domain then, by validation, we achieve the optimal exponential approximation rate given by the upper bound in Trefethen's Theorem. 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 this, we propose an algorithm that can efficiently and numerically stably solve arbitrary-dimensional interpolation problems, and approximate non-analytical functions, with at most quadratic runtime and linear memory requirement.
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