PDE-constrained optimization problems arise in a wide range of subject areas within numerical mathematics and applied science. It is therefore highly desirable to develop fast and robust solvers for the matrix systems which result from such problems. In this talk we present a framework for constructing preconditioned iterative methods for a variety of PDE-constrained optimization problems. In order to derive suitable preconditioners we exploit the saddle point structure of the matrices, and use this to construct effective approximations of the (1,1)-block and Schur complement. For each problem that we consider we motivate and derive our recommended preconditioners, and present numerical results to demonstrate the performance of our methods.
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