A Method for Dimensionally Adaptive Sparse Trigonometric Interpolation of Periodic Functions

TL;DR

This paper introduces a novel dimensionally adaptive sparse trigonometric interpolation method for periodic functions, effectively handling unknown anisotropy and preserving periodicity, with applications to molecular potential energy surfaces.

Contribution

It presents the first dimensionally adaptive sparse interpolation algorithm using a trigonometric basis, with an anisotropic rate estimation and iterative refinement strategy.

Findings

Successfully recovers theoretical convergence rates in numerical tests.

Effectively approximates periodic potential energy surfaces.

Open-source implementation available in Tasmanian UQ library.

Abstract

We present a method for dimensionally adaptive sparse trigonometric interpolation of multidimensional periodic functions belonging to a smoothness class of finite order. This method targets applications where periodicity must be preserved and the precise anisotropy is not known a priori. To the authors' knowledge, this is the first instance of a dimensionally adaptive sparse interpolation algorithm that uses a trigonometric interpolation basis. The motivating application behind this work is the adaptive approximation of a multi-input model for a molecular potential energy surface (PES) where each input represents an angle of rotation. Our method is based on an anisotropic quasi-optimal estimate for the decay rate of the Fourier coefficients of the model; a least-squares fit to the coefficients of the interpolant is used to estimate the anisotropy. Thus, our adaptive approximation strategy begins with a coarse isotropic interpolant, which is gradually refined using the estimated anisotropic rates. The procedure takes several iterations where ever-more accurate interpolants are used to generate ever-improving anisotropy rates. We present several numerical examples of our algorithm where the adaptive procedure successfully recovers the theoretical "best" convergence rate, including an application to a periodic PES approximation. An open-source implementation of our algorithm resides in the Tasmanian UQ library developed at Oak Ridge National Laboratory.