In this paper, we present numerical methods suitable for solving convex Fractional Differential Equation (FDE) optimization problems, with potential box constraints on the state and control variables. First we derive powerful multilevel circulant preconditioners, which may be embedded within Krylov subspace methods, for solving equality constrained FDE optimization problems. We then develop an Alternating Direction Method of Multipliers (ADMM) framework, which uses preconditioned Krylov solvers for the resulting subproblems. The latter allows us to tackle FDE optimization problems with box constraints, posed on space–time domains, that were previously out of the reach of state–of–the–art preconditioners. Discretized versions of FDEs involve large dense linear systems. In order to overcome this difficulty, we design a recursive linear algebra, which is based on the fast Fourier transform (FFT), and our proposed multilevel circulant preconditioners are suitable for approximating discretized FDEs of arbitrary dimension. Focusing on time-dependent 2–dimensional FDEs, we manage to keep the storage requirements linear, with respect to the grid size N , while ensuring an order N log N computational complexity per iteration of the Krylov solver. We implement the proposed method, and demonstrate its scalability, generality, and efficiency, through a series of experiments over different setups of the FDE optimization problem.
Submitted for publication