Interpolation in Polynomial Spaces of p-Degree

TL;DR

This paper analyzes the efficiency of the Fast Newton Transform (FNT) for multivariate polynomial interpolation in specific polynomial spaces, demonstrating significant complexity reductions and practical applications in sensitivity analysis.

Contribution

The paper introduces a detailed complexity analysis of FNT in polynomial spaces defined by $ extit{l}^p$ norms, highlighting super-exponential reductions compared to tensor product spaces.

Findings

FNT performs with $ ext{O}(|A_{m,n,p}|mn)$ complexity.

Choosing $ extit{l}^p$ spaces reduces complexity by a super-exponentially decaying factor.

FNT effectively computes activity scores in sensitivity analysis.

Abstract

We recently introduced the Fast Newton Transform (FNT), an hierarchical algorithm for performing multivariate Newton interpolation in arbitrary downward closed polynomial spaces of spatial dimension $m$. Here, we analyze the FNT in the context of a specific family of downward closed sets $A_{m,n,p}$, defined as all multi-indices with $\ell^p$ norm less than $n$ with $p \in [0,\infty]$. The FNT performs with time complexity $\mathcal{O}(|A_{m,n,p}|mn)$ on the induced downward closed polynomial spaces $\Pi_{m,n,p}$. We show that the $\Pi_{m,n,p}$ choice compared to the tensor product spaces $\Pi_{m,n,\infty}$, reduces time complexity by a factor of $\rho_{m,n,p}$, decaying super exponentially with spatial dimension when $m \lesssim n^p$. We showcase the efficiency of the FNT by computing activity scores in sensitivity analysis.