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.

翻译：我们将牛顿插值和拉格朗日插值推广到任意维度。这一成果的核心贡献在于提出了一种广义的非张量积非孤立节点概念，即在这些节点上，函数的多元多项式插值具有唯一性。通过验证，我们针对一类解析函数（称为Trefethen函数）达到了最优的指数级Trefethen收敛速率。计算最优插值所需插值节点数量随维度呈次指数增长，从而有效克服了维度灾难。基于这些结果，我们提出了一种高效且数值稳定的算法，可求解任意维度的插值问题，其运行时复杂度至多为二次阶，内存需求为线性阶。