Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities
Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities
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DOI：
10.1137/130916126
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发表时间：
2014-05
期刊：
影响因子：
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通讯作者：
R. Kornhuber;C. Schwab;M. Wolf
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文献类型：
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作者：
R. Kornhuber;C. Schwab;M. Wolf
介绍了求解随机椭圆型变分不等式的多层蒙特卡罗有限元方法（MLMC-FEM），并对其进行了分析和数值研究。在对随机扩散系数、随机强迫函数和确定性障碍的适当假设下，证明了“路径”弱方程解的存在唯一性.确定性，椭圆障碍问题的适当的规律性结果导致统一的路径误差界，提供最优阶误差估计的统计误差和上界相应的计算成本的经典MC方法和新型MLMC-FEM。利用合适的多重网格求解器来解决发生的样本问题，在两个空间维度中，MLMC-FEM可以提供随机解期望的数值逼近，其效率与相应的确定性问题相同，直至对数项。我们的理论研究结果说明了数值实验…
Multilevel Monte Carlo finite element methods (MLMC-FEMs) for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of “pathwise” weak formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for the classical MC method and novel MLMC-FEMs. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FEMs then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experi...