标题： 目标导向自适应有限元多级拟蒙特卡洛{M}{C}算法 显示英文标题

标题： Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo

摘要： 对具有对数正态扩散系数的偏微分方程得出的兴趣量的有效近似是不确定性量化中的核心挑战。 在本研究中，我们提出了一种多级准蒙特卡罗框架，以近似依赖于具有对数正态扩散系数的线性椭圆偏微分方程解的确定性、实值、有界线性泛函，{由一个49维的高斯随机向量参数化}以及在$\mathbb{R}^d$有界域中的确定性几何奇异性。 我们分析了参数正则性，并基于一系列自适应网格进行多级实现，这些网格在“目标导向的自适应有限元多级蒙特卡罗及其收敛速率”中开发，\emph{计算方法在工程力学中的应用}，402（2022），第115582页。 为了进一步减少方差，我们结合了重要性抽样，并在多级层次结构中引入了零级控制变量。 {引入这样的控制变量可以改变初始网格的最优选择，进一步突出自适应网格的优势。} 数值实验表明，我们的自适应QMC算法在计算成本显著低于标准多级蒙特卡罗方法的情况下实现了预定的精度。 显示英文摘要

摘要： The efficient approximation of quantity of interest derived from PDEs with lognormal diffusivity is a central challenge in uncertainty quantification. In this study, we propose a multilevel quasi-Monte Carlo framework to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient {parameterized by a 49-dimensional Gaussian random vector} and deterministic geometric singularities in bounded domains of $\mathbb{R}^d$. We analyze the parametric regularity and develop the multilevel implementation based on a sequence of adaptive meshes, developed in "Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates", \emph{CMAME}, 402 (2022), p. 115582. For further variance reduction, we incorporate importance sampling and introduce a level-0 control variate within the multilevel hierarchy. {Introducing such control variate can alter the optimal choice of initial mesh, further highlighting the advantages of adaptive meshes.} Numerical experiments demonstrate that our adaptive QMC algorithm achieves a prescribed accuracy at substantially lower computational cost than the standard multilevel Monte Carlo method.