# Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

## TL;DR

This paper introduces an efficient ADMM-based numerical framework with specialized preconditioners for solving large-scale convex quadratic fractional differential equation constrained optimization problems, enabling solutions on complex space-time domains.

## Contribution

The paper develops a novel ADMM approach combined with Krylov solvers and multilevel circulant preconditioners tailored for FDE-constrained optimization, preserving matrix structures and achieving linear storage and N log N complexity.

## Key findings

- Method demonstrates scalability and efficiency in various FDE problem setups.
- Preconditioners effectively handle large dense linear systems.
- Approach extends to PDE optimization problems previously out of reach.

## 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 sub-problems. 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-of-the-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 sub-problems. 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 $N \log N$ 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.