论文标题
一种有效的多模型蒙特卡洛同质化方法，用于随机材料
An efficient multi-modes Monte Carlo homogenization method for random materials
论文作者
论文摘要
在本文中，我们提出并分析了一种新的随机和快速振荡系数扩散方程的随机均质化方法。在提出的方法中，通过两阶段的程序寻求均质的溶液。在第一阶段，对于每个固定的随机样品W，原始振荡扩散方程是通过空间均匀的扩散方程（带有分段恒定系数）近似的，从而导致随机扩散方程。在第二阶段，通过使用有效的多模型Monte Carlo方法对所得的随机扩散方程进行近似并计算，该方法仅需要求解具有恒定扩散系数和随机右侧的扩散方程。提出的方法的主要优点是，它分离了由溶液的空间快速振荡和溶液的随机性引起的计算难度，因此可以使用不同的策略分别克服它们。建立了空间匀浆方程（从第一阶段）到原始随机扩散方程的溶液的溶液的收敛性，并为所提出的多模型蒙特卡洛方法获得了最佳收敛速率。还提出了关于随机复合材料的一些基准测试问题的数值实验，以评估所提出的两阶段随机均质化方法的效率和准确性。
In this paper, we propose and analyze a new stochastic homogenization method for diffusion equations with random and fast oscillatory coefficients. In the proposed method, the homogenized solutions are sought through a two-stage procedure. In the first stage, the original oscillatory diffusion equation is approximated, for each fixed random sample w, by a spatially homogenized diffusion equation with piecewise constant coefficients, resulting a random diffusion equation. In the second stage, the resulted random diffusion equation is approximated and computed by using an efficient multi-modes Monte Carlo method which only requires to solve a diffusion equation with a constant diffusion coefficient and a random right-hand side. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial fast oscillation of the solution and that caused by the randomness of the solution, so they can be overcome separately using different strategies. The convergence of the solution of the spatially homogenized equation (from the first stage) to the solution of the original random diffusion equation is established and the optimal rate of convergence is also obtained for the proposed multi-modes Monte Carlo method. Numerical experiments on some benchmark test problems for random composite materials are also presented to gauge the efficiency and accuracy of the proposed two-stage stochastic homogenization method.
