General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. In the present work, we consider a random elliptic partial differential equation (PDE) where the discontinuous subordinated Gaussian random fields occur in the diffusion coefficient. Problem specific multilevel Monte Carlo (MLMC) Finite Element methods are constructed to approximate the mean of the solution to the random elliptic PDE. We prove a-priori convergence of a standard MLMC estimator and a modified MLMC - Control Variate estimator and validate our results in various numerical examples.

翻译：具有空间不连续扩散系数的普通椭圆方程式可以用作多元或裂开多孔介质中地表下流的简化模型。在这种模型中,数据宽度和测量错误往往通过对椭圆方程式扩散系数的随机化而得到考虑,这种随机化表明有必要构建灵活、空间不连续随机字段。从子高斯随机字段是高维参数域的随机函数,有不连续的样本路径和很大的分布灵活性。在目前的工作中,我们考虑的是随机的椭圆部分偏差方程式(PDE),在这个方程式中,不连续的从属高斯随机字段出现在扩散系数中。问题特定的多层次蒙特卡洛(MLMC)固定元素方法的构建接近随机椭圆方块解决方案的平均值。我们证明标准 MLMC 估计符和修改的 MLMC - 控制 Variate 估量器的优先组合,并在各种数字实例中验证我们的结果。