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

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 >0 with cost O (^---) for positive constants, , depending on the rates of convergence associated with dimension truncation, kernel approximation, and finite element approximation, respectively. Our multilevel approximation can achieve the same accuracy at a reduced cost O (^-- (, ) ). 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.