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Optimal control methods are applied in various problems and can be efficient also for solving inverse problems, such as parameter identification and boundary control which arise in many important applications. For boundary optimal control methods one can identify conditions on an inaccessible part of the boundary by letting them play the role of a control variable function and by overimposing boundary conditions at another part of the boundary of the given domain. The paper shows a mathematical formulation of the problem, the arising (regularized) Karush-Kuhn-Tucker (KKT) system and introduces a preconditioner for this system. The spectral analysis of the preconditioner and numerical tests with preconditioning are included.

In this study we consider an efficient implementation of Implicit Runge-Kutta methods for solving large systems of ordinary differential equations that originate from finite element discretization of the heat and similar equations, to be solved on large time intervals. The main contribution of this work is to show how to implement a fully stage-parallel version of the method, utilizing the dominance of the block lower triangular part of the quadrature matrix, and to illustrate it numerically. Its usage for the solution of algebraic-differential equations is also touched.

For the iterative solution of saddle point problems one needs efficient preconditioners to achieve a fast convergence. Three types of preconditioners are presented which are based on regularization by use of an augmented matrix. They are applicable also for problems with a highly singular pivot block matrix. One of the methods is applicable also for nonsymmetric saddle point problems.

This chapter is devoted to the preconditioned iterative solution of a particular type of linear systems, mainly involving matrices of a two-by-two block form with square matrix blocks. Such systems arise in the finite element (FE) solution of optimal control problems for partial differential equations (PDEs) in various applications. Owing to the large scale of the problems, they must be solved by an iterative method. The choice of proper preconditioners is then crucial. We propose several types of preconditioning methods and make an analytical comparison of them, leading to estimates of linear convergence. We also discuss superlinear convergence estimates of such preconditioning methods and mesh-independence.