Mass Lumping and Numerical Quadrature for Approximation of Fractional Elliptic Differential Equations Driven by Gaussian White Noise

Abstract

Fractional elliptic stochastic partial differential equations (SPDEs) are widely used in statistics and machine learning for computationally efficient and flexible modeling of Gaussian random fields. The computational efficiency of the SPDE approach relies on finite element approximations combined with numerical quadrature and mass lumping, which enable sparse matrix methods during inference. Although many works have studied finite element approximations of fractional SPDEs, the effect of the mass lumping and quadrature approximations used in practice has not been fully analyzed. To fill this gap, we derive convergence rates for numerical approximations of fractional SPDEs based on finite element discretizations combined with numerical quadrature and mass lumping. Specifically, we obtain explicit convergence rates for the mean-squared error of the covariance function in a general framework that covers the main settings where mass lumping is used in the SPDE approach. We also analyze non-stationary variance-control factors of the form $L^\beta(\tau u)=\mathcal{W}$, where $\tau$ is spatially varying, and derive covariance error estimates showing how the regularity of $\tau$ affects the convergence rate. As specific examples, we provide results for random fields on bounded Euclidean domains, Riemannian manifolds, and metric graphs. Numerical experiments are presented that confirm the theoretical results.