Lattice Rules Meet Kernel Cubature

TL;DR

This paper investigates how replacing equal weights in rank-1 lattice rules with kernel-based optimized weights can significantly improve convergence rates in quasi-Monte Carlo integration, especially in one dimension.

Contribution

It introduces a method to optimize lattice rule weights using reproducing kernels, doubling convergence rates in 1D and demonstrating benefits in higher dimensions.

Findings

Doubled convergence rate in 1D with kernel-optimized weights

Numerical evidence of improved convergence in higher dimensions

Effective application to uncertainty quantification in PDEs

Abstract

Rank-1 lattice rules are a class of equally weighted quasi-Monte Carlo methods that achieve essentially linear convergence rates for functions in a reproducing kernel Hilbert space (RKHS) characterized by square-integrable first-order mixed partial derivatives. In this work, we explore the impact of replacing the equal weights in lattice rules with optimized cubature weights derived using the reproducing kernel. We establish a theoretical result demonstrating a doubled convergence rate in the one-dimensional case and provide numerical investigations of convergence rates in higher dimensions. We also present numerical results for an uncertainty quantification problem involving an elliptic partial differential equation with a random coefficient.