How to Choose the Shift in the Shifted Laplace Preconditioner for the Helmholtz Equation Combined with Deflation
How to Choose the Shift in the Shifted Laplace Preconditioner for the Helmholtz Equation Combined with Deflation
复制标题
如何选择结合紧缩的亥姆霍兹方程的移位拉普拉斯预条件子中的移位
DOI：
--
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
C. Vuik
中科院分区：
文献类型：
--
作者：
D. Lahaye;C. Vuik
在最近的工作中，我们表明，离散亥姆霍兹方程的复杂的移位拉普拉斯预处理器的性能可以显着提高相结合的乘法与通缩过程，采用多重网格矢量。在这一章中，我们认为，在这种组合的预条件，提高了外克雷洛夫加速通过一个新的机制的收敛性。这种机制允许更大的阻尼，并有助于预处理器的近似求解。外Krylov加速度的收敛没有明显延迟，偶尔甚至加速。为了提供这些索赔的基础，我们分析了一个一维问题的两个层次的变体的方法，其中的预处理后，通货紧缩，其中的预处理和粗网格问题被完全倒置。我们表明，在网格是足够精细的情况下，解决波长，放气后的频谱由一个集群包围的两个尾巴，沿沿着的真实的轴在两个方向上延伸。逆预条件子的作用是收缩尾部的长度，同时旋转尾部并将簇的中心移向原点。可以使用比没有放气的算法中大得多的阻尼参数。
In recent work we showed that the performance of the complex shifted Laplace preconditioner for the discretized Helmholtz equation can be significantly improved by combining it multiplicatively with a deflation procedure that employs multigrid vectors. In this chapter we argue that in this combination the preconditioner improves the convergence of the outer Krylov acceleration through a new mechanism. This mechanism allows for a much larger damping and facilitates the approximate solve with the preconditioner. The convergence of the outer Krylov acceleration is not significantly delayed and occasionally even accelerated. To provide a basis for these claims, we analyze for a one-dimensional problem a two-level variant of the method in which the preconditioner is applied after deflation and in which both the preconditioner and the coarse grid problem are inverted exactly. We show that in case that the mesh is sufficiently fine to resolve the wave length, the spectrum after deflation consists of a cluster surrounded by two tails that extend in both directions along the real axis. The action of the inverse of the preconditioner is to shrink the length of the tails while at the same time rotating them and shifting the center of the cluster towards the origin. A much larger damping parameter than in algorithms without deflation can be used.