A robust structured preconditioner for the time-harmonic parabolic optimal control problem

0495413 - ÚGN 2019 RIV NL eng J - Journal Article
Liang, Z. Z. - Axelsson, Owe - Neytcheva, M.
A robust structured preconditioner for the time-harmonic parabolic optimal control problem.
Numerical Algorithms. Roč. 79, č. 2 (2018), s. 575-596. ISSN 1017-1398. E-ISSN 1572-9265
R&D Projects: GA MŠMT LQ1602
Institutional support: RVO:68145535
Keywords : PDE-constrained optimization * time-harmonic parabolic equation * preconditioning * iterative solution method * spectral analysis
OECD category: Applied mathematics
Impact factor: 2.417, year: 2018
https://link.springer.com/content/pdf/10.1007%2Fs11075-017-0451-5.pdf

We consider the iterative solution of optimal control problems constrained by the time-harmonic parabolic equations. Due to the time-harmonic property of the control equations, a suitable discretization of the corresponding optimality systems leads to a large complex linear system with special two-by-two block matrix of saddle point form. For this algebraic system, an efficient preconditioner is constructed, which results in a fast Krylov subspace solver, that is robust with respect to the mesh size, frequency, and regularization parameters. Furthermore, the implementation is straightforward and the computational complexity is of optimal order, linear in the number of degrees of freedom. We show that the eigenvalue distribution of the corresponding preconditioned matrix leads to a condition number bounded above by 2. Numerical experiments confirming the theoretical derivations are presented, including comparisons with some other existing preconditioners.
Permanent Link: http://hdl.handle.net/11104/0288404