We study an inexact inner-outer generalized Golub-Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.
翻译:我们研究了一种不精确的内外通俗的Golub-Kahan算法,用两倍二块结构解决马鞍问题。 在每一个外向循环中,内部系统必须解决,理论上必须完全解决。 但是,当系统变大时,内部精确解答器就不再有效,甚至不可行,必须使用迭代方法。我们把这篇文章的焦点放在一项数字研究上,显示内迭代解决办法的准确性对区块系统解决办法的准确性的影响。我们进一步强调降低计算成本,计算成本的定义是内向循环的总数。我们开发了旨在动态改变每个外向循环的内容技术,以进一步尽量减少内向循环的总数。我们展示了我们关于斯托克斯问题的研究结果,并在Poisson问题的混合配方上验证了这些结果。