FINITE ELEMENT ERROR ANALYSIS OF ELLIPTIC PDES WITH RANDOM COEFFICIENTS AND ITS APPLICATION TO MULTILEVEL MONTE CARLO METHODS
FINITE ELEMENT ERROR ANALYSIS OF ELLIPTIC PDES WITH RANDOM COEFFICIENTS AND ITS APPLICATION TO MULTILEVEL MONTE CARLO METHODS
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DOI：
10.1137/110853054
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发表时间：
2013-01-01
影响因子：
2.9
通讯作者：
Teckentrup, A. L.
中科院分区：
文献类型：
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作者：
Charrier, J.;Scheichl, R.;Teckentrup, A. L.
我们考虑具有随机系数的椭圆形部分微分方程的有限元近似。例如，在地下流量建模中出现了这种方程式。这些应用中经常使用的随机系数的模型，例如具有指数协方差的对数正态分子随机场，仅具有非常有限的空间规则性，并且导致缺乏相对于随机参数缺乏统一的矫正和界限的变异问题。在我们的分析中，我们几乎可以肯定地在随机参数中对模型问题进行仔细处理，从而克服了这些挑战，这使我们能够证明标准Bochner空间中有限元误差的界限。然后，这些新界限可用于对这些椭圆形问题进行多级蒙特卡洛方法进行严格的分析，这些问题缺乏完全规律性，统一的胁迫和界限。总而言之，我们给出一些数值结果来确认新的界限。
We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modeling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.