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：本文采用带有初始预积分步骤的拟蒙特卡洛方法，对不确定性量化中的累积分布函数与概率密度函数进行估计。这些分布与密度函数对应于涉及椭圆型偏微分方程解的关注量，该方程具有对数正态分布的系数与正态分布的源项。已有大量应用QMC计算UQ中期望值的研究成果，这些方法在解决各类偏微分方程问题上成效显著。然而，本文首次探索将QMC用于UQ问题中的密度估计。相较于期望值计算，密度估计面临更大挑战，因为分布函数与密度函数的积分形式均存在间断性。我们的策略是通过精心选取随机参数进行预积分以消除间断性，进而利用QMC逼近剩余积分。首先，我们建立了偏微分方程关注量的正则性结果，该结果对保证预积分平滑效果至关重要。随后证明，针对分布函数与密度函数对应的被积函数，可构造N点晶格规则，使得经过预积分后，QMC误差阶为$\mathcal{O}(N^{-1+\epsilon})$（$\epsilon>0$可任意小）。这一精度与计算关注量期望值所达到的收敛阶相同。数值实验结果验证了理论分析的正确性。