Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients

2025/04/22 by Alexander D. Gilbert, Michael B. Giles, Gilbert, Alexander D. +7
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2504.15810

openalex publication_date 2025/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coefficients. Building upon the work of Kaarnioja, Kazashi, Kuo, Nobile, Sloan (Numer. Math., 2022) on kernel interpolation with quasi-Monte Carlo (QMC) lattice point sets, we leverage multilevel techniques to enhance computational efficiency while maintaining a given level of accuracy. In the function space setting with product-type weight parameters, the single-level approximation can achieve an accuracy of ε>0 with cost O(ε^-η-ν-θ) for positive constants η, ν, θ depending on the rates of convergence associated with dimension truncation, kernel approximation, and finite element approximation, respectively. Our multilevel approximation can achieve the same ε accuracy at a reduced cost O(ε^{-η-max(ν,θ)}). Full regularity theory and error analysis are provided, followed by numerical experiments that validate the efficacy of the proposed multilevel approximation in comparison to the single-level approach.

Citations

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Related

• Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions
• Combining sparse grids, multilevel MC and QMC for elliptic PDEs with\n random coefficients
• Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients
• Constructing QMC finite element methods for elliptic PDEs with random\n coefficients by a reduced CBC construction
• Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients