In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute expected values in UQ, which have proven very successful in tackling a range of different PDE problems. However, the use of QMC for density estimation applied to UQ problems will be explored here for the first time. Density estimation presents a more difficult challenge compared to computing the expected value due to discontinuities present in the integral formulations of both the distribution and density. Our strategy is to use preintegration to eliminate the discontinuity by integrating out a carefully selected random parameter, so that QMC can be used to approximate the remaining integral. First, we establish regularity results for the PDE quantity of interest that are required for smoothing by preintegration to be effective. We then show that an $N$-point lattice rule can be constructed for the integrands corresponding to the distribution and density, such that after preintegration the QMC error is of order $\mathcal{O}(N^{-1+\epsilon})$ for arbitrarily small $\epsilon>0$. This is the same rate achieved for computing the expected value of the quantity of interest. Numerical results are presented to reaffirm our theory.

翻译：本文应用结合初始预积分步骤的拟蒙特卡洛（QMC）方法，对不确定性量化（UQ）中的累积分布函数与概率密度函数进行估计。该分布与密度函数对应于一个包含椭圆型偏微分方程（PDE）解的感兴趣量，该方程具有对数正态分布的系数与正态分布的源项。先前已有大量研究采用QMC计算UQ中的期望值，并在处理各类不同PDE问题上取得显著成功。然而，将QMC应用于UQ问题中的密度估计，在本文中尚属首次探索。由于分布与密度函数的积分表达式中存在间断性，密度估计相较于期望值计算提出了更为严峻的挑战。我们的策略是采用预积分方法，通过积出精心选取的随机参数以消除间断性，从而可利用QMC近似剩余积分。首先，我们建立了PDE感兴趣量所需的正则性结果，这是预积分平滑处理能够生效的前提。随后证明，针对分布与密度对应的被积函数，可构建$N$点格点法则，使得经过预积分后QMC误差阶为$\mathcal{O}(N^{-1+\epsilon})$（其中$\epsilon>0$可任意小）。该收敛速率与计算感兴趣量期望值时达到的速率相同。数值实验结果进一步验证了理论分析。