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.

翻译：本文分析了在涉及随机输入的椭圆型偏微分方程不确定性量化问题中，使用预积分近似累积分布函数和概率密度函数时误差的有限元分量。本文是 Gilbert、Kuo、Srikumar 在 SIAM J. Numer. Anal. 63 (2025), pp. 1025-1054 所发表论文的后续研究，该论文针对一类不确定性量化问题，提出了一种基于计算累积分布函数和概率密度函数积分公式的密度估计方法。该方法首先执行初始平滑预积分步骤，然后应用拟蒙特卡洛积分法则来近似剩余的高维积分。前述论文侧重于该方法的积分方面，而本文则研究使用有限元方法对偏微分方程进行空间离散化。首先，证明了有限元近似满足预积分理论所需的假设，包括重要的单调性条件。随后分析了有限元误差，最后对有限元与拟蒙特卡洛的组合误差进行了界定。研究表明，在相似假设下，累积分布函数和概率密度函数的近似可获得与计算期望值这一更简单问题相同的收敛速率。