Approximation of the stochastic Galerkin matrix in the low‐rank canonical tensor format
Approximation of the stochastic Galerkin matrix in the low‐rank canonical tensor format
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低秩正则张量格式的随机伽辽金矩阵的近似
DOI：
10.1002/pamm.201210380
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
W. Hackbusch
中科院分区：
文献类型：
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作者：
Philipp Wähnert;A. Litvinenko;Mike Espig;H. Matthies;W. Hackbusch
在这篇文章中，我们描述了随机Galerkin矩阵的一种有效的近似，它源于一个平稳的扩散方程。假定不确定渗透系数为对数正态随机场，具有给定的协方差和均值函数。该近似是以正则张量格式进行的，然后与张量串和分层张量格式进行了数值比较。在附加假设下，逼近误差仅取决于协方差函数的光滑性，而不依赖于随机变量的个数和多元Hermite多项式的阶数。(2012Wiley-VCH Verlag GmbH&Co.KGaA，Weinheim)
In this article we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log‐normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)