Multilevel Monte Carlo Approaches for Numerical Homogenization
Multilevel Monte Carlo Approaches for Numerical Homogenization
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DOI：
10.1137/130905836
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发表时间：
2013-01
期刊：
影响因子：
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通讯作者：
Y. Efendiev;Cornelia Kronsbein;F. Legoll
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文献类型：
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作者：
Y. Efendiev;Cornelia Kronsbein;F. Legoll
在这篇文章中，我们研究了多层蒙特卡罗（MLMC）方法的数值随机均匀化的应用。我们的目标是计算齐次系数或齐次解的某些泛函的期望。这是通过考虑不同大小的代表性体积（RVE）在MLMC内实现的。具有最小RVE大小的许多廉价计算与在较大RVE上执行的较少昂贵计算相结合。同样，当涉及到均匀化的解决方案，不同级别的粗网格网格被用来解决均匀化方程。我们表明，通过仔细选择在每个级别的实现的数量，我们可以实现一个加速的计算相比，标准的蒙特卡罗方法。一维和二维的测试情况下，说明了该方法的效率的数值结果。
In this article, we study the application of multilevel Monte Carlo (MLMC) approaches to numerical random homogenization. Our objective is to compute the expectation of some functionals of the homogenized coefficients, or of the homogenized solutions. This is accomplished within MLMC by considering different sizes of representative volumes (RVEs). Many inexpensive computations with the smallest RVE size are combined with fewer expensive computations performed on larger RVEs. Likewise, when it comes to homogenized solutions, different levels of coarse-grid meshes are used to solve the homogenized equation. We show that, by carefully selecting the number of realizations at each level, we can achieve a speed-up in the computations in comparison to a standard Monte Carlo method. Numerical results are presented for both one-dimensional and two-dimensional test-cases that illustrate the efficiency of the approach.