一种计算多项式混沌逼近中刚度矩阵的蒙特卡洛方法
A Monte Carlo approach to computing stiffness matrices arising in polynomial chaos approximations
AI总结：
本文提出一种基于蒙特卡洛方法构建有限元矩阵的策略，用于处理具有随机系数的椭圆方程的多项式混沌逼近，通过稀疏块对角矩阵替代块满矩阵，从而求解耦合的椭圆方程组。
摘要：
我们使用蒙特卡洛方法来构建多项式混沌逼近中椭圆方程随机系数有限元矩阵。在此方法中，所有所需期望均通过蒙特卡洛方法近似。所提出的方法需要处理稀疏块对角矩阵而非块满矩阵。这导致求解一个耦合的椭圆方程组，其中耦合由涉及多项式评估矩阵的克罗内克积矩阵给出。该方法扩展了经典蒙特卡洛近似和置信度方法用于近似这些方程解的功能。
英文摘要：
We use a Monte Carlo method to assemble finite element matrices for polynomial Chaos approximations of elliptic equations with random coefficients. In this approach, all required expectations are approximated by a Monte Carlo method. The resulting methodology requires dealing with sparse block-diagonal matrices instead of block-full matrices. This leads to the solution of a coupled system of elliptic equations where the coupling is given by a Kronecker product matrix involving polynomial evaluation matrices. This generalizes the Classical Monte Carlo approximation and Collocation method for approximating functionals of solutions of these equations.
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