In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical elliptic equations with log-normal random coefficients. In \cite{Wan_model}, a Wick-type elliptic model was proposed by modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.

中文翻译：

---

具有对数正态随机系数的椭圆问题的数值逼近

在这项工作中，我们考虑了对具有对数正态随机系数的经典椭圆方程的数值逼近的非标准预处理策略。在 \cite{Wan_model} 中，通过对通过 Wick 乘积的随机通量进行建模，提出了 Wick 型椭圆模型。由于不确定性传播器的下三角结构，可以使用概率空间中的维纳混沌展开有效地近似该模型。通常，这种 Wick 型模型根据基础高斯过程的标准偏差提供了经典模型的二阶近似。此外，当底层高斯过程的相关长度趋于无穷大时，Wick 型模型产生与经典模型相同的解。这些观察结果表明，在适当的条件下，Wick 型椭圆方程可以为经典随机椭圆方程提供有效的预处理器。我们使用 Wick 型椭圆模型来加速蒙特卡罗方法和随机伽辽金有限元方法。给出并讨论了数值结果。