Abstract
In this paper, we present numerical methods suitable for solving convex quadratic fractional differential equation (FDE) constrained optimization problems, with box constraints on the state and/or control variables. We develop an alternating direction method of multipliers (ADMM) framework, which uses preconditioned Krylov subspace solvers for the resulting subproblems. The latter allows us to tackle a range of partial differential equation (PDE) optimization problems with box constraints, posed on space-time domains, that were previously out of the reach of state-ofthe- art preconditioners. In particular, by making use of the powerful generalized locally Toeplitz (GLT) sequences theory, we show that any existing GLT structure present in the problem matrices is preserved by ADMM, and we propose some preconditioning methodologies that could be used within the solver, to demonstrate the generality of the approach. Focusing on convex quadratic programs with time-dependent 2-dimensional FDE constraints, we derive multilevel circulant preconditioners, which may be embedded within Krylov subspace methods, for solving the ADMM subproblems. 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). We manage to keep the storage requirements linear, with respect to the grid size N, while ensuring an order N logN 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.
Original language | English |
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Pages (from-to) | 1443-1476 |
Number of pages | 34 |
Journal | SIAM Journal on Matrix Analysis and Applications |
Volume | 41 |
Issue number | 3 |
Early online date | 23 Sept 2020 |
DOIs | |
Publication status | Published - 2020 |
Keywords
- Alternating direction method of multipliers
- Circulant preconditioners
- Fractional differential equation optimization
- Generalized locally Toeplitz sequences
- Preconditioned iterative methods
- Toeplitz matrices
ASJC Scopus subject areas
- Analysis