论文标题
以目标为导向的自适应有限元多级蒙特卡洛（Monte Carlo）
Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates
论文作者
论文摘要
我们提出了一种自适应多级蒙特卡洛（AMLMC）算法，用于近似确定性，实现的，有界的线性函数，该函数取决于在$ \ mathbb的有界界面中具有逻辑差异系数和几何范围的线性椭圆PDE解决方案。 我们的AMLMC算法建立在工作中弱收敛速率的结果上[Moon等，位Numer。 Math。，46（2006），367-407]用于自适应算法，使用等型D线性四边形有限元近似值和确定性设置中的双重加权残差误差表示。我们的AMLMC算法旨在适合解决方案中奇异性的几何性质，使用一系列确定性的，不均匀的辅助网格作为构建块。确定性自适应算法会生成这些网络，对应于几何降低的公差序列。为了实现扩散系数和精度水平的实现，AMLMC使用层次结构中的第一个网格构建其近似样品，以满足相应的偏置精度约束。 这种自适应方法对于此处处理的对数正态病例特别有用，该病例缺乏均匀的牢固性，因此产生的功能输出在采样时会因数量级而变化。 我们讨论迭代求解器并将其效率与直接效率进行比较。为了减少计算工作，我们提出了有关迭代求解器的停止标准，相对于利益量，扩散系数的实现以及AMLMC近似的所需水平。 根据数值实验，基于系数场的傅立叶膨胀，我们观察到效率的提高与标准的蒙特卡洛和标准MLMC相比，具有与裂缝建模裂缝的尖端相似的奇异性问题。
We present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of $\mathbb{R}^d$. Our AMLMC algorithm is built on the results of the weak convergence rates in the work [Moon et al., BIT Numer. Math., 46 (2006), 367-407] for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. For a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. We discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation. From the numerical experiments, based on a Fourier expansion of the coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo and standard MLMC for a problem with a singularity similar to that at the tip of a slit modeling a crack.
