Abstract
In this paper, we propose some preconditioning techniques for reduced saddle point systems arising from linear elliptic distributed optimal control problems. The eigenvalues of preconditioned matrices are analyzed. Moreover, the bounds of these eigenvalues with respect to the mesh size h are also obtained. Some numerical tests are presented to validate the theoretical analysis.
Highlights
We consider the preconditioning techniques for solving the saddle point system arising from linear elliptic distributed optimal control problems: min u,f u – u L ( )+β f )subject to – u = f in, u = g on ∂, ( )where ⊂ R is a connected polygonal domain with a connected boundary ∂
For the elliptic PDE-constrained optimization problem ( ), we take the method based on discretize--optimize [ ] approach
We extend similar preconditioning techniques, which were used in [, ] for system ( ), to the saddle point system ( )
Summary
We consider the preconditioning techniques for solving the saddle point system arising from linear elliptic distributed optimal control problems: min u,f u – u. We use the P conforming finite elements for the approximations of the state variable u and the control variable f , which yields the following corresponding minimization problem [ , ]:. Using the Lagrange multiplier technique for the minimization problem ( ), we can find that f, u and λ satisfy the following linear saddle point system [ ]:. In Section , we propose preconditioning techniques for the case that the Tikhonov parameter β is sufficiently small, and the corresponding spectral analysis is presented. In Section , some numerical results are presented to demonstrate our theoretical analysis
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