论文标题
与参数不确定性的系统线性稳定性分析的随机Galerkin方法
Stochastic Galerkin methods for linear stability analysis of systems with parametric uncertainty
论文作者
论文摘要
我们提出了一种在随机Galerkin框架中使用参数不确定性的系统线性稳定性分析的方法。具体而言，我们假设对于模型部分微分方程，参数以广义多项式膨胀的形式给出。稳定性分析导致了随机特征值问题的解决方案，我们希望表征最右边的特征值。我们尤其将重点放在非对称矩阵算子的问题上，为此，感兴趣的特征值可能是一个复杂的结合对，我们开发了其有效解决方案的方法。这些方法基于不精确的线路搜索牛顿迭代，该迭代需要使用预处理的GMRE。该方法应用于随机粘度的Navier-Stokes方程的线性稳定性分析，将其精度与蒙特卡洛和随机搭配的精度进行了比较，并且通过数值实验来说明效率。
We present a method for linear stability analysis of systems with parametric uncertainty formulated in the stochastic Galerkin framework. Specifically, we assume that for a model partial differential equation, the parameter is given in the form of generalized polynomial chaos expansion. The stability analysis leads to the solution of a stochastic eigenvalue problem, and we wish to characterize the rightmost eigenvalue. We focus, in particular, on problems with nonsymmetric matrix operators, for which the eigenvalue of interest may be a complex conjugate pair, and we develop methods for their efficient solution. These methods are based on inexact, line-search Newton iteration, which entails use of preconditioned GMRES. The method is applied to linear stability analysis of Navier-Stokes equation with stochastic viscosity, its accuracy is compared to that of Monte Carlo and stochastic collocation, and the efficiency is illustrated by numerical experiments.
