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首页> 外文期刊>SIAM/ASA Journal on Uncertainty Quantification >A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients
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A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients

机译：椭圆偏微分方程的研究跳扩散系数

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As a simplified model for subsurface ows, elliptic equations may be utilized. Insufficient measure- ments or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given medium. As an extension of this methodology to ows in heterogeneous/fractured/porous media, we incorporate jumps in the diffusion coefficient. These discontinuities then represent transitions in the media. More precisely, we consider a second order elliptic problem where the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. To estimate moments of the solution to the resulting random partial differential equation, we use a pathwise numerical approximation combined with multilevel Monte Carlo sampling. In order to account for the discontinuities and improve the convergence of the pathwise approximation, the spatial domain is decomposed with respect to the jump positions in each sample, leading to path-dependent grids. Hence, it is not possible to create a nested sequence of grids which is suitable for each sample path a priori. We address this issue by an adaptive multilevel algorithm, where the discretization on each level is sample-dependent and fulfills given refinement conditions.
机译：作为地下ows的简化模型,椭圆方程可以利用。在那些通常事件或不确定性建模的随机系数,然后占的不确定的渗透率鉴于媒介。在异构ows /断裂/多孔介质,我们将跳跃扩散系数。在媒体上的转换。考虑二阶椭圆问题的地方的随机系数的总和(连续)高斯随机场和一个(不连续)跳的部分。解决由此产生的随机部分微分方程中,我们使用一个pathwise数值近似与多层次相结合蒙特卡罗抽样。不连续性,提高收敛的pathwise近似,空间域分解的职位每个样本,导致路径依赖网格。因此,它是不可能创建一个嵌套的适合每个序列的网格先验样本路径。自适应多级算法,在每个水平的离散化和满足给定的优化条件。

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