Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems

Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems

“…In this paper the PMHSS iteration method is applied to a class of KKT linear systems arising from a finite element discretization of a class of distributed control problems (Lions, 1968;Lass et al, 2009;Bai, 2011). In order to apply the method, we first eliminate the Lagrange multiplier from the original KKT system to obtain a block two-by-two system, which is then diagonally scaled so as to obtain linear systems of the form (1.1).…”

“…It frequently arises from finite element discretizations of elliptic partial differential equation (PDE)-constrained optimization problems such as distributed control problems (Lions, 1968;Lass et al, 2009;Rees & Stoll, 2010;Bai, 2011) and also from real equivalent formulations of complex symmetric linear systems (Axelsson & Kucherov, 2000;Day & Heroux, 2001). Based on the Hermitian and skew-Hermitian splitting (HSS) 1 A = H + S of the matrix A ∈ R 2n×2n , with…”

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

“…Here, the function u * (the "desired state") is known, and we want to find u which satisfies the PDE problem and is as close to u * as possible in the L 2 -norm sense. For recent references on this topic, see, e.g., [6,23]. When the PDE-constrained optimization problem (3.1)-(3.3) is treated with the discretize-then-optimize approach [29] through application of a Galerkin finite-element method to its weak formulation, we obtain a KKT system in the following saddle-point form:…”

“…This class of linear systems can be formally regarded as a special case of the generalized saddle-point problem [4,5,16] and the skew-Hamiltonian linear system [25,32]. It frequently arises from finite-element discretization and firstorder linearization of the two-phase flow problems based on Cahn-Hilliard equation [3,17,18], order-reduction and sinc discretization of the third-order linear ordinary differential equations [30], finite-element discretizations of elliptic PDE-constrained optimization problems such as the distributed control problems [6,23,24,29], real equivalent formulations of complex symmetric linear systems [2,15,20], linear quadratic control problems [22,25], and H ∞ control problems [28,32,34]. being the symmetric and the skew-symmetric parts and A T being the transpose of the matrix A; see [1,14].…”

“…The exact and the inexact additive block diagonal preconditioners are applied to precondition a class of KKT (Karush-Kuhn-Tucker) linear systems arising from a finite-element discretization of a class of distributed control problems [6,23,24,29]. Numerical results show that the exact and the inexact ABD-preconditioned MINRES or GMRES methods [19] lead to rapid convergence and outperform the exact and the inexact PMHSS-preconditioned GMRES methods, respectively; see [9].…”