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Inexact Methods for Symmetric Stochastic Eigenvalue Problems

机译：不精确的对称随机的方法特征值问题

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摘要

We study two inexact methods for solutions of random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric matrix operator, the methods solve for eigenvalues and eigenvectors represented using polynomial chaos expansions. Both methods are based on the stochastic Galerkin formulation of the eigenvalue problem and they exploit its Kronecker-product structure. The first method is an inexact variant of the stochastic inverse subspace iteration [B. Sousedik and H. C. Elman, SIAM/ASA J. Uncertain. Quantif., 4 (2016), pp. 163-189]. The second method is based on an inexact variant of the Newton iteration. In both cases, the problems are formulated so that the associated stochastic Galerkin matrices are symmetric, and the corresponding linear problems are solved using preconditioned Krylov subspace methods with several novel hierarchical preconditioners. The accuracy of the methods is compared with that of Monte Carlo and stochastic collocation, and the effectiveness of the methods is illustrated by numerical experiments.
机译：我们研究两个不精确解的方法随机特征值问题的上下文中谱随机有限元素。给定一个parameter-dependent,特别对称矩阵算子,求解的方法特征值和特征向量代表使用多项式混沌扩张。基于随机加勒金制定他们利用其特征值问题克罗内克积结构。一个随机的不精确的变体逆子空间迭代(B。暹罗/ ASA j .不确定。163 - 189年)。不精确牛顿迭代的变体。情况下,制定这样的问题随机加勒金相关矩阵对称的,和相应的线性问题解决了使用预先处理这些维子空间吗分级方法和一些小说预调节器。相比之下,蒙特卡罗和随机的搭配,和方法的有效性数值试验所示。

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