Saddle-point systems arise quite often in many areas of scientific computing. For instance, this type of system can be found in computational fluid dynamics, constrained optimization, finance, image reconstruction, and many more scientific applications. Due to their convenient structure and high dimensionality, preconditioned iterative methods are usually employed in order to solve this class of problems. By making use of saddle-point theory, one is able to derive optimal preconditioners for a general saddle-point system. However, the major drawback is that the cost of applying their inverse operator is almost as costly as inverting the original system. For this reason, one would rather find easy-to-invert approximation of the main blocks of the preconditioner considered.
In this talk, we will present the “matching strategy” derived in  for approximating the Schur complement of a saddle-point system arsing from optimal control problems with PDE as constraints. We will mainly focus on time-dependent PDE, when employing a Crank–Nicolson discretization in time. After applying an optimize-then-discretize approach, one is faced with continuous first-order optimality conditions consisting of a coupled system of PDEs. After discretizing with Crank–Nicolson, one has to solve for a non-symmetric system. We apply a carefully tailored invertible transformation for symmetrizing the latter, and derive an ideal preconditioner by making use of saddle-point theory. The transformation is then employed within the preconditioner in order to derive optimal approximations of the (1,1)-block and of the Schur complement. We prove the latter through bounds on the eigenvalues, and test our solver against the widely-used preconditioner derived in  for the linear system arising from a backward Euler discretization in time. These demonstrate the effectiveness and robustness of our solver with respect to all the parameters involved in the problem considered.
This talk is based on the work in .
 S. Leveque and J. W. Pearson, Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank–Nicolson Discretization in Time, Numer. Linear Algebra Appl. 29, e2419, 2022.
 J. W. Pearson, M. Stoll and A. J. Wathen, Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems, SIAM J. Matrix Anal. Appl. 33, 1126–1152, 2012.
 J. W. Pearson and A. J. Wathen, A New Approximation of the Schur Complement in Preconditioners for PDE-Constrained Optimization, Numer. Linear Algebra Appl. 19, 816–829, 2012.