Spectral properties of primal-based penalty preconditioners for saddle point problems
Journal of Computational and Applied Mathematics
Preconditioning Saddle-Point Systems with Applications in Optimization
SIAM Journal on Scientific Computing
Optimal Solvers for PDE-Constrained Optimization
SIAM Journal on Scientific Computing
Efficient Preconditioners for Optimality Systems Arising in Connection with Inverse Problems
SIAM Journal on Control and Optimization
SIAM Journal on Matrix Analysis and Applications
A Robust Multigrid Method for Elliptic Optimal Control Problems
SIAM Journal on Numerical Analysis
Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations
SIAM Journal on Scientific Computing
Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs
Computational Optimization and Applications
Preconditioning for Allen-Cahn variational inequalities with non-local constraints
Journal of Computational Physics
SIAM Journal on Scientific Computing
All-at-once solution of time-dependent Stokes control
Journal of Computational Physics
Computational Optimization and Applications
A robust finite element solver for a multiharmonic parabolic optimal control problem
Computers & Mathematics with Applications
Preconditioning of complex symmetric linear systems with applications in optical tomography
Applied Numerical Mathematics
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We consider large scale sparse linear systems in saddle point form. A natural property of such indefinite 2-by-2 block systems is the positivity of the (1,1) block on the kernel of the (2,1) block. Many solution methods, however, require that the positivity of the (1,1) block is satisfied everywhere. To enforce the positivity everywhere, an augmented Lagrangian approach is usually chosen. However, the adjustment of the involved parameters is a critical issue. We will present a different approach that is not based on such an explicit augmentation technique. For the considered class of symmetric and indefinite preconditioners, assumptions are presented that lead to symmetric and positive definite problems with respect to a particular scalar product. Therefore, conjugate gradient acceleration can be used. An important class of applications are optimal control problems. It is typical for such problems that the cost functional contains an extra regularization parameter. For control problems with elliptic state equations and distributed control, a special indefinite preconditioner for the discretized problem is constructed, which leads to convergence rates of the preconditioned conjugate gradient method that are not only independent of the mesh size but also independent of the regularization parameter. Numerical experiments are presented for illustrating the theoretical results.