Lattice Rules Meet Kernel Cubature

2025/01/16 by Kaarnioja, Vesa, Klebanov, Ilja, Schillings, Claudia +1
#46E22 #65D30 #65D32 #FOS: Mathematics #Numerical Analysis (math.NA) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2501.09500

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.

Related

• Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights
• On the convergence conditions of Laplace importance sampling with randomized quasi-Monte Carlo
• Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients
• Random weights, robust lattice rules and the geometry of the cbcrc algorithm
• A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces