Summary
We consider a linear-quadratic elliptic control problem (LQECP). For the problem we consider here, the control variable corresponds to the Neumann data on the boundary of a convex polygonal domain. The optimal control unknown is the one for which the harmonic extension approximates best a specified target in the interior of the domain. We propose a multilevel preconditioner for the reduced Hessian resulting from the application of the Schur complement method to the discrete LQECP. In order to derive robust stabilization parameters-free preconditioners, we first show that the Schur complement matrix is associated to a linear combination of negative Sobolev norms and then propose preconditioner based on multilevel methods. We also present numerical experiments which agree with the theoretical results.
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Gonçalves, E., Sarkis, M. (2013). Robust Parameter-Free Multilevel Methods for Neumann Boundary Control Problems. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_11
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DOI: https://doi.org/10.1007/978-3-642-35275-1_11
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