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

TL;DR

This paper generalizes multivariate polynomial interpolation to unisolvent nodes, achieving near-exponential efficiency in high dimensions and providing a stable, efficient algorithm that mitigates the curse of dimensionality for analytic functions.

Contribution

It introduces a generalized notion of unisolvent nodes for multivariate interpolation, enabling optimal rates and a practical, stable algorithm for high-dimensional problems.

Findings

Achieves exponential convergence rates for Trefethen functions.

Requires sub-exponential number of nodes relative to dimension.

Develops an algorithm with quadratic runtime and linear memory usage.

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.