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Fast Interpolation and Fourier Transform in High-Dimensional Spaces

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Intelligent Computing (SAI 2018)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 857))

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Abstract

In scientific computing, the problem of finding an analytical representation of a given function \(f: \Omega \subseteq \mathbb {R}^m \longrightarrow \mathbb {R},\mathbb {C}\) is ubiquitous. The most practically relevant representations are polynomial interpolation and Fourier series. In this article, we address both problems in high-dimensional spaces. First, we propose a quadratic-time solution of the Multivariate Polynomial Interpolation Problem (PIP), i.e., the N(mn) coefficients of a polynomial Q, with \(\deg (Q)\le n\), uniquely fitting f on a determined set of generic nodes \(P\subseteq \mathbb {R}^m\) are computed in \(\mathcal {O}(N(m,n)^2)\) time requiring storage in \(\mathcal {O}(mN(m,n))\). Then, we formulate an algorithm for determining the N(mn) Fourier coefficients with positive frequency of the Fourier series of f up to order n in the same amount of computational time and storage. Especially in high dimensions, this provides a fast Fourier interpolation, outperforming modern Fast Fourier Transform methods. We expect that these fast and scalable solutions of the polynomial and Fourier interpolation problems in high-dimensional spaces are going to influence modern computing techniques occurring in Big Data and Data Mining, Deep Learning, Image and Signal Analysis, Cryptography, and Non-linear Optimization.

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Correspondence to Michael Hecht .

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Hecht, M., Sbalzarini, I.F. (2019). Fast Interpolation and Fourier Transform in High-Dimensional Spaces. In: Arai, K., Kapoor, S., Bhatia, R. (eds) Intelligent Computing. SAI 2018. Advances in Intelligent Systems and Computing, vol 857. Springer, Cham. https://doi.org/10.1007/978-3-030-01177-2_5

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