In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtain an advantage in terms of storage and CPU time compared to computations in full rank. We also propose a general approach to implement a preconditioner using the low-rank format efficiently.
翻译:在本文中,我们考虑的是随机系数的低位近似值,用于解决随机系数的随机螺旋螺旋形螺旋藻等式的解决方案。使用Stochastic Galerkin 有限元素法将Helmholtz问题分开处理。当系统矩阵无限期时,就建立了低位近似理论。低位算法并不要求构建一个大型系统矩阵,从而在CPU时间和存储方面产生优势。数字结果显示,当低位操作效率高时,有可能在储存和CPU时间方面获得优势,而与全位计算相比,我们还提出了高效使用低位格式实施前提条件的通用方法。