Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

TL;DR

This paper introduces a fast kernel-based lattice-point interpolation method for approximating multivariate periodic functions, with applications in uncertainty quantification for PDEs, providing error analysis and supporting numerical experiments.

Contribution

It applies lattice-based kernel interpolation to uncertainty quantification in PDEs with periodic random coefficients, including detailed error analysis and lattice construction.

Findings

Error bounds independent of dimension

Numerical experiments confirm theoretical results

Method achieves fast approximation without linear solvers

Abstract

This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng, Kritzer, Hickernell (Constr. Approx., 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja, Kuo, Sloan (SIAM J. Numer. Anal., 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.