基于共轭梯度法的蒙特卡洛随机有限元方法

A Monte-Carlo Stochastic FEM Based on Conjugate Gradients Method

将共轭梯度法引入蒙特卡洛随机有限元法,建立基于多项式预处理共轭梯度法的蒙特卡洛随机有限元方法。求解某一特征样本,对于其余样本,采用把特征样本作为预处理阵的多项式预处理共轭梯度法。将该方法与基于Neumann法的随机有限元方法作比较,从理论上证明了Neumann法是基于多项式预处理共轭梯度随机有限元方法的一个退化算法。最后算例比较也验证了该方法有更高的求解效率。

An improved MonteCarlo stochastic FEM based on polynomial preconditioners is established through introduction of the conjugate gradients method into the traditional MonteCarlo stochastic FEM. In the improved FEM, a certain characteristic sample is obtained first; then, using it as a preconditioner, the other samples are solved by polynomial preconditioners for conjugate gradients method. A comparison between the improved MonteCarlo stochastic FEM and the Neumann expansionbased MonteCarlo stochastic FEM shows that the latter is a retrogression of the former, and that the former is more efficient in dealing with stochastic problems. Finally, a numerical example is given to illustrate the advantage of the new method proposed in this paper.