Finite element analysis of density estimation using preintegration for elliptic PDE with random input

This paper analyses the finite element component of the error when using preintegration to approximate the cdf and pdf for uncertainty quantification (UQ) problems involving elliptic PDEs with random inputs. It is a follow up to Gilbert, Kuo, Srikumar, SIAM J. Numer. Anal. 63 (2025), pp. 1025-1054, which introduced a method of density estimation for a class of UQ problems, based on computing the integral formulations of the cdf and pdf by performing an initial smoothing preintegration step and then applying a quasi-Monte Carlo quadrature rule to approximate the remaining high-dimensional integral. That paper focussed on the quadrature aspect of the method, whereas this paper studies the spatial discretisation of the PDE using finite element methods. First, it is shown that the finite element approximation satisfies the required assumptions for the preintegration theory, including the important monotonicity condition. Then the finite element error is analysed and finally, the combined finite element and quasi-Monte Carlo error is bounded. It is shown that under similar assumptions, the cdf and pdf can be approximated with the same rate of convergence as the much simpler problem of computing expected values.