Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
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DOI：
10.1007/s00791-011-0160-x
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发表时间：
2011-01-01
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通讯作者：
Teckentrup, A. L.
中科院分区：
文献类型：
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作者：
Cliffe, K. A.;Giles, M. B.;Teckentrup, A. L.
我们考虑具有随机系数的椭圆形部分微分方程的数值解。例如，在地下水流量的不确定性定量中出现了此类问题。我们描述了标准蒙特卡洛法的一种新颖的差异技术，称为多级Carlo方法，并以数字上的优势证明了其优势。通过多级方法解决随机问题的渐近成本总是显着低于标准方法的渐近成本，并且仅与在某些情况下解决确定性问题的成本成正比增长。介绍了该方法在地下水流中产生的一维模型问题的有效性。
We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevelMonte Carlo method, and demonstrate numerically its superiority. The asymptotic cost of solving the stochastic problem with the multilevel method is always significantly lower than that of the standard method and grows only proportionally to the cost of solving the deterministic problem in certain circumstances. Numerical calculations demonstrating the effectiveness of the method for one-and two-dimensional model problems arising in groundwater flow are presented.