Multigrid methods for saddle point problems: Optimality systems

Abstract

We construct multigrid methods for an elliptic distributed optimal control problem that are robust with respect to a regularization parameter. We prove the uniform convergence of the W -cycle algorithm and demonstrate the performance of V -cycle and W -cycle algorithms in two and three dimensions through numerical experiments.Date: September 22, 2018. 1991 Mathematics Subject Classification. 49J20, 65N30, 65N55, 65N15. Key words and phrases. elliptic distributed optimal control problem, saddle point problem, P 1 finite element method, multigrid methods.

Convergence analysis of the Schwarz alternating method for unconstrained elliptic optimal control problems

“…In [5,21,38,51], the authors used preconditioned Krylov subspace methods to solve the first order optimality system by constructing some block preconditioners. In [6,7,8,37,40,46], the authors used mutigrid methods to design fast solvers. Another strategy is to use domain decomposition methods to deal with optimal control problems (see e.g., [2,3,4,12,11,17,25,24,34,43,42]).…”

“…It is even harder to obtain robust theoretical convergence results of the methods with respect to the regularization parameter α. In this paper, we will prove the robust convergence results of the Schwarz alternating method for the elliptic optimal control problem by defining proper error merit functions (or vectors in the discrete case) which are related to the proper norms that are used in [8,21,22,43,38,51] and using the maximum principle of the elliptic operator. We will also give a uniform upper bound of the convergence rate.…”

A scalable preconditioning framework for stabilized contact mechanics with hydraulically active fractures

“…Algebraic multigrid (AMG, [72]) is one of the most effective multilevel approaches and consists of the complementary use of: (i) a smoother that reduces high frequency errors, (ii) a coarse grid correction that reduces low frequency errors, and (iii) restriction and interpolation operators, to move from one grid to another. Starting from the original works, e.g., [73], a wide range of multigrid approaches has appeared in the literature, extending the applicability of this method, originally designed for elliptic PDEs, to both non-symmetric [74,75] and block matrices [76][77][78][79][80][81]. Nonetheless, robustness and efficiency is still an open issue for AMG whenever used as a black-box tool in problems with these algebraic properties.…”

Multigrid Methods for Elliptic Optimal Control Problems

“…Multigrid methods for (1.8) based on continuous Galerkin methods are intensively studied in the literature, for example, in [6,13,37,35,36] and the references therein. In [13], based on the approaches in [11,14,12], the authors developed multigrid methods for (1.8) using a continuous P 1 finite element method.…”

“…Multigrid methods for (1.8) based on continuous Galerkin methods are intensively studied in the literature, for example, in [6,13,37,35,36] and the references therein. In [13], based on the approaches in [11,14,12], the authors developed multigrid methods for (1.8) using a continuous P 1 finite element method. Besides the robustness of the multigrid methods with respect to β, the estimates in [13] are established in a natural energy norm and the multigrid methods have a standard O(m −1 ) performance where m is the number of smoothing steps.…”