Optimized Schwarz Methods for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations

摘要

Optimal control of systems governed by elliptic partial differential equations (PDEs) without constraint on the set of controls can be equivalently reformulated as a coupled system of second order elliptic PDEs, which has been considered to solve by a non-overlapping Schwarz domain decomposition method with the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions by Benamou (SIAM J Numer Anal 33:2401-2416, 1996) where a convergence analysis had been performed. Towards fast convergence of the overlapping and non-overlapping Schwarz subdomain iterations, in this paper we firstly perform, for fixed Tikhonov parameter , rigorous analyses based on optimization of the convergence factor of subdomain iterations in Fourier frequency domain to give the optimized transmission parameters involved in the above mentioned transmission conditions, as well as those involved in a Ventcell-like and a two-sided Robin-like transmission condition that we propose to accelerate the Schwarz subdomain iterations, and meanwhile we obtain also the corresponding asymptotic convergence rate estimates. The results show that the Tikhonov parameter occurs in both the optimized transmission parameters and the corresponding convergence rate estimates and affects the performance of the Schwarz domain decomposition methods significantly: when is less than a certain threshold value, with the decreasing of the Tikhonov parameter , the subdomain iteration converges more and more fast, though the regularity of the system deteriorates in this process. We lastly investigate the case where the Tikhonov parameter =h4 that is suggested by Benamou (where h is the mesh size). We obtain as well the optimized transmission parameters involved in the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions, and find that they lead to optimized Schwarz methods that are very robust in the mesh size. The analysis also sheds light on optimizing the Schwarz domain decomposition methods for biharmonic equations, since they can also be reformulated as a system of second order elliptic PDEs. We use various numerical experiments to illustrate the theoretical findings.