An economic implementation of the optimal rotated block-diagonal preconditioning method

The numerical discretization of the optimal control problems constrained with certain kind of time-dependent fractional diffusion equations leads to a class of highly structured block two-by-two linear systems. We present a different and economic implementation of the approximated rotated block diagonal (ARBD) preconditioner, denoted briefly as the ARBDe preconditioner, for solving this class of linear systems effectively by making use of the correspondingly preconditioned Krylov subspace iteration methods such as the ARBDe-preconditioned flexible GMRES (FGMRES) method, or the ARBDe-FGMRES method. Compared with the ARBD-GMRES method constructed and analyzed by Bai and Lu in 2021 (Appl. Numer. Math. 163:126–146), the ARBDe-FGMRES method requires a lower computational complexity and can achieve much higher computational efficiency in practical applications. With numerical experiments, we have examined and confirmed the robustness, accuracy, and effectiveness of the ARBDe-FGMRES method in solving this class of discrete optimal control problems.