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：本文研究具有随机系数的随机亥姆霍兹方程解的低秩逼近方法。采用随机伽辽金有限元法对亥姆霍兹问题离散化。当系统矩阵为不定矩阵时，建立了低秩逼近的存在性理论。该低秩算法无需构建大型系统矩阵，从而在计算时间和存储空间方面具有优势。数值结果表明，若高效执行低秩方法中的运算，相较于全秩计算，可在存储和计算时间上获得优势。此外，本文还提出了一种利用低秩格式高效实现预条件器的一般性方法。