A robust all-at-once multigrid method for the Stokes control problem
A robust all-at-once multigrid method for the Stokes control problem
复制标题
斯托克斯控制问题的鲁棒一次性多重网格方法
作者：
Stefan Takacs
在本文中，我们提出了一个所有在一次多重网格方法的分布式斯托克斯控制问题（速度跟踪问题）。为了解决这样的问题，我们使用的事实，即解决方案的特点是最优性系统（Karush-Kuhn-Tucker-system）。离散化的最优性系统是一个大规模的线性系统，其条件数取决于网格的大小和正则化参数的选择形成的问题的一部分。最近，已经提出了块对角预条件，它允许使用Krylov空间方法来解决问题，该方法的收敛速度在网格大小和正则化参数或成本参数方面都是鲁棒的。在本文中，我们开发了一个所有的一次多重网格方法的Stokes控制问题，并显示出强大的收敛性，更准确地说，我们表明，该方法收敛的速度是有界的一个常数，这是独立的网格大小和选择的正则化或成本参数。
In this paper we present an all-at-once multigrid method for a distributed Stokes control problem (velocity tracking problem). For solving such a problem, we use the fact that the solution is characterized by the optimality system (Karush–Kuhn–Tucker-system). The discretized optimality system is a large-scale linear system whose condition number depends on the grid size and on the choice of the regularization parameter forming a part of the problem. Recently, block-diagonal preconditioners have been proposed, which allow to solve the problem using a Krylov space method with convergence rates that are robust in both, the grid size and the regularization parameter or cost parameter. In the present paper, we develop an all-at-once multigrid method for a Stokes control problem and show robust convergence, more precisely, we show that the method converges with rates which are bounded away from one by a constant which is independent of the grid size and the choice of the regularization or cost parameter.