TY - JOUR
T1 - Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization
AU - Langer, Ulrich
AU - Steinbach, Olaf
AU - Yang, Huidong
N1 - Publisher Copyright:
© 2021 Walter de Gruyter GmbH, Berlin/Boston 2022.
PY - 2022/1/1
Y1 - 2022/1/1
N2 - 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 Qh to the state u and the desired state u ¯ in terms of the mesh-size h and the regularization parameter Q. The choice Q = 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, Q, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.
AB - 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 Qh to the state u and the desired state u ¯ in terms of the mesh-size h and the regularization parameter Q. The choice Q = 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, Q, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.
KW - A Priori Error Estimates
KW - Elliptic Optimal Control Problems
KW - Energy Regularization
KW - Fast Solvers
KW - Finite Element Discretization
KW - Parallelization
UR - http://www.scopus.com/inward/record.url?scp=85117257640&partnerID=8YFLogxK
U2 - 10.1515/cmam-2021-0169
DO - 10.1515/cmam-2021-0169
M3 - Article
AN - SCOPUS:85117257640
SN - 1609-4840
VL - 22
SP - 97
EP - 111
JO - Computational Methods in Applied Mathematics
JF - Computational Methods in Applied Mathematics
IS - 1
ER -