论文标题
波数中有不确定性的helmholtz方程
The Helmholtz equation with uncertainties in the wavenumber
论文作者
论文摘要
我们研究了波数中有合适的边界条件和不确定性的Helmholtz方程。因此，波数被建模为随机变量或随机场。我们使用空间中有限差异将Helmholtz方程离散，这导致了代数方程的线性系统，包括随机变量。随机盖尔金方法产生代数方程的确定性线性系统。这种线性系统是高维，稀疏和复杂的对称的，但通常不是冬宫。因此，我们使用GMRES迭代解决该系统，并提出了两个预处理：一个复杂的移动拉普拉斯预处理和一个平均值预处理。两个预处理都减少了迭代步骤的数量以及数值实验中的计算时间。
We investigate the Helmholtz equation with suitable boundary conditions and uncertainties in the wavenumber. Thus the wavenumber is modeled as a random variable or a random field. We discretize the Helmholtz equation using finite differences in space, which leads to a linear system of algebraic equations including random variables. A stochastic Galerkin method yields a deterministic linear system of algebraic equations. This linear system is high-dimensional, sparse and complex symmetric but, in general, not hermitian. We therefore solve this system iteratively with GMRES and propose two preconditioners: a complex shifted Laplace preconditioner and a mean value preconditioner. Both preconditioners reduce the number of iteration steps as well as the computation time in our numerical experiments.
