# Efficient Preconditioning of Optimality Systems with Non Self-Adjoint State operators

Kent-Andre Mardal^{*} and Bjørn Fredrik Nielsen

he construction of preconditioners for optimality systems by using the mapping properties of the involved differential operators in Hilbert spaces has recently received significant attention, see, e.g.~\cite{n-m-12},\cite{n-m-10},\cite{s-z-07},\cite{z-11}. These work all employ a rather general approach, where the preconditioners are constructed as Riesz operators in properly chosen Hilbert spaces. However, the analysis is limited to self-adjoint state operators. In this talk we will extend the abstract framework to operators that are not self-adjoint.\newline Our theoretical results will be illuminated by a number of numerical experiments. \begin{thebibliography}{99} \bibitem{n-m-12}{ B.\ F.\ Nielsen and K.-A.\ Mardal, \textit{Analysis of the minimal residual method applied to ill-posed optimality systems}, SIAM Journal of Scientific Computing, accepted 2012. } \bibitem{n-m-10}{ B.\ F.\ Nielsen and K.-A.\ Mardal, \textit{Efficient preconditioners for optimality systems arising in connection with inverse problems Analysis of the minimal residual method applied to ill-posed optimality systems}, SIAM Journal of Control and Optimization, vol. 48(8), pp. 5143--5177, 2010. } \bibitem{s-z-07}{ J.\ Sch{\"o}berl and W. Zulehner, \textit{Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems}, SIAM Journal of Matrix Analysis and Applications, vol 29(3), pp. 752--773, 2007. } \bibitem{z-11}{ W.\ Zulehner, \textit{Nonstandard norms and robust estimates for saddle point problems}, SIAM Journal of Matrix Analysis and Applications, vol. 32(2), pp. 536--560, 2011. } \end{thebibliography}

Mathematics Subject Classification: 65F08

Keywords: PDE constrained optimization, preconditioning

Minisymposion: Preconditioning for PDE-Constrained Optimization