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Lattice Rules Meet Kernel Cubature
[Submitted on 16 Jan 2025 (v1), last revised 30 Jun 2026 (this v · 2025-01-16 · via math.ST updates on arXiv.org

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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.

Submission history

From: Vesa Kaarnioja [view email]
[v1] Thu, 16 Jan 2025 12:20:42 UTC (543 KB)
[v2] Thu, 5 Jun 2025 10:49:20 UTC (316 KB)
[v3] Tue, 30 Jun 2026 04:27:38 UTC (316 KB)