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.