Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems,Numerical Linear Algebra with Applications

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Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems
Numerical Linear Algebra with Applications ( IF 1.8 ) Pub Date : 2020-01-02 , DOI: 10.1002/nla.2281
Pieterjan Robbe ₁ , Dirk Nuyens ₁ , Stefan Vandewalle ₁

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In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics and porous media flow, in particular, the uncertain permeability of the material is modeled as a random field. These random fields can be highly anisotropic. Efficient solvers, such as the multiple semicoarsened multigrid (MSG) method are required to compute solutions for various realizations of the uncertain material. The MSG method is an extension of the classic multigrid method, which uses additional coarse grids that are coarsened in only a single coordinate direction. In this sense, it closely resembles the extension of multilevel Monte Carlo to multi‐index Monte Carlo (MIMC). We present an unbiased MIMC method that reuses the MSG coarse solutions. Our formulation of the estimator can be interpreted as the problem of learning the unknown distribution of the number of samples across all indices and unifies the previous work on adaptive MIMC and unbiased estimation. We analyze the cost of this new estimator theoretically and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over unbiased MIMC without sample reuse.

中文翻译：

借助多个半粗化多网格增强的针对各向异性扩散问题的多指标蒙特卡洛

在工程和科学中使用的许多模型中，材料属性是不确定的或在空间上变化的。例如，特别是在地球物理学和多孔介质流动中，材料的不确定的渗透性被建模为随机场。这些随机场可以是高度各向异性的。需要有效的求解器，例如多半粗化多网格（MSG）方法来计算不确定材料的各种实现的解决方案。MSG方法是经典多重网格方法的扩展，该方法使用了仅在单个坐标方向上被粗糙化的其他粗糙网格。从这个意义上讲，它非常类似于将多级蒙特卡洛扩展为多索引蒙特卡洛（MIMC）。我们提出了一种重用MSG粗解的无偏MIMC方法。我们对估计量的表述可以解释为学习样本在所有指标上的未知分布的问题，并将先前关于自适应MIMC和无偏估计的工作统一起来。我们从理论上分析了这种新估计器的成本，并提出了各种各向异性随机场的数值实验，其中协方差模型中的未知系数被视为超参数。我们展示了它在不重用样品的情况下比无偏MIMC的稳健性和优越性。其中协方差模型中的未知系数被视为超参数。我们展示了它在不重用样品的情况下比无偏MIMC的稳健性和优越性。其中协方差模型中的未知系数被视为超参数。我们展示了它在不重用样品的情况下比无偏MIMC的稳健性和优越性。

更新日期：2020-01-02

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