Year: 2026
Type: article
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 (Numerische Mathematik 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 $$\varepsilon >0$$ ε > 0 with cost $$\mathcal {O}(\varepsilon ^{-\eta -\nu -\theta })$$ O ( ε - η - ν - θ ) for positive constants $$\eta , \nu , \theta $$ 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 $$\varepsilon $$ ε accuracy at a reduced cost $$\mathcal {O}(\varepsilon ^{-\eta -\max (\nu ,\theta )})$$ 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.
Source: Numerische Mathematik
Institutions: UNSW Sydney, University of Oxford
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4414633426
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