Domain decomposition-based space-time preconditioner for the optimal control problem with constraints imposed by the Cahn-Hilliard equation

This study presents preconditioning methodologies for addressing the optimal control problem (OCP) constrained by the Cahn-Hilliard (CH) equation. PDE-constrained optimization problems are pervasive across various fields but often give rise to large and ill-conditioned linear systems, particularly in time-dependent scenarios. To combat this, we propose a preconditioning approach for the all-at-once space-time discretized system. The preconditioning process involves spatial decomposition, temporal decomposition and a combination of spatial and temporal decomposition to construct various space-time Schwarz preconditioners. This not only mitigates the computational challenges of large linear systems but also harnesses parallel computing capabilities effectively. To validate our approach, we present a series of test cases and numerical examples that demonstrate the effectiveness of the proposed preconditioners.