一种有效的随机材料多模态蒙特卡罗均匀化方法
An efficient multi-modes Monte Carlo homogenization method for random materials
摘要：
本文提出并分析了一种新的随机均匀化方法，用于求解具有随机快速振荡系数的扩散方程。在所提出的方法中，均质化的解决方案是通过一个两阶段的过程。在第一阶段中，对于每个固定的随机样本w，通过具有分段常系数的空间均匀化扩散方程来近似原始振荡扩散方程，得到随机扩散方程。在第二阶段，得到的随机扩散方程近似和计算使用一个有效的多模式蒙特卡罗方法，只需要解决一个扩散方程与一个常数的扩散系数和随机的右手边。该方法的主要优点是将解的空间快速振荡引起的计算困难与解的随机性引起的计算困难分离开来，可以分别采用不同的策略来克服。建立了空间均匀化方程（从第一阶段开始）的解到原始随机扩散方程的解的收敛性，并得到了所提出的多模蒙特卡罗方法的最佳收敛速率。对随机复合材料的一些基准测试问题进行了数值实验，以衡量所提出的两阶段随机均匀化方法的效率和精度。
英文摘要：
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.
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