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Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

  • Ulrich Langer ORCID logo , Olaf Steinbach ORCID logo EMAIL logo and Huidong Yang
Published/Copyright: October 10, 2021

Abstract

We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 -norm regularization is replaced by the H - 1 -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ h to the state u and the desired state u ¯ in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

Funding source: Austrian Science Fund

Award Identifier / Grant number: NFN S117-03

Funding statement: The third author was partly supported by the Austrian Science Fund (FWF), Grant No. NFN S117-03.

Acknowledgements

We would like to thank Fredi Tröltzsch (TU Berlin) for fruitful discussions during his visits at Linz and via ZOOM during the corona time. Special thanks goes to Volker John (WIAS Berlin) who drew our attention to the results known for differential filters.

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Received: 2021-09-10
Accepted: 2021-09-16
Published Online: 2021-10-10
Published in Print: 2022-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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