Mesh-free error integration in arbitrary dimensions: a numerical study of discrepancy functions

TL;DR

This paper conducts a numerical study on mesh-free Monte Carlo methods for multi-dimensional integral approximation using reproducing-kernel spaces, comparing different kernels and strategies to evaluate error bounds and accuracy.

Contribution

It introduces a comprehensive numerical analysis of discrepancy functions for various kernels and localization strategies, providing insights into error bounds and optimal point distributions.

Findings

Periodic kernels with lattice-based Fourier transforms yield predictable discrepancy bounds.

Transport-based kernels offer alternative localization with comparable accuracy.

Numerical experiments validate theoretical error estimates and compare kernel strategies to random sampling.

Abstract

We are interested in mesh-free formulas based on the Monte-Carlo methodology for the approximation of multi-dimensional integrals, and we investigate their accuracy when the functions belong to a reproducing-kernel space. A kernel typically captures regularity and qualitative properties of functions "beyond" the standard Sobolev regularity class. We are interested in the issue whether quantitative error bounds can be a priori guaranteed in applications (e.g. mathematical finance but also scientific computing and machine learning). Our main contribution is a numerical study of the error discrepancy function based on a comparison between several numerical strategies, when one varies the choice of the kernel, the number of approximation points, and the dimension of the problem. We consider two strategies in order to localize to a bounded set the standard kernels defined in the whole Euclidian space (exponential, multiquadric, Gaussian, truncated), namely, on one hand the class of periodic kernels defined via a discrete Fourier transform on a lattice and, on the other hand, a class of transport-based kernels. First of all, relying on a Poisson formula on a lattice, together with heuristic arguments, we discuss the derivation of theoretical bounds for the discrepancy function of periodic kernels. Second, for each kernel of interest, we perform the numerical experiments that are required in order to generate the optimal distributions of points and the discrepancy error functions. Our numerical results allow us to validate our theoretical observations and provide us with quantitative estimates for the error made with a kernel-based strategy as opposed to a purely random strategy.