A fast PDE-constrained optimization solver for nonlinear diffusion-reaction processes

Large-scale nonlinear model predictive control (NMPC) often relies on real-time solution of optimization problems that are constrained by partial differential equations (PDEs). However, the size and complexity of the underlying PDEs present significant computational challenges. In this regard, the development of fast, efficient and scalable PDEconstrained optimization solvers remains central to large-scale NMPC.

As a contribution in this direction, this paper proposes a new efficient preconditioned iterative scheme for optimal control of large-scale time-dependent diffusion-reaction problems with nonlinear reaction kinetics. The scheme combines a custom-made high-order spectral Petrov-Galerkin (SPG) method with a new preconditioner tailored for the linearquadratic control problems that underly Sequential Quadratic Programming (SQP) methods.

The preconditioner is matrixfree and amenable to parallelization. To demonstrate efficiency, a case study applies the SPG scheme to control solid fuel ignition (SFI) processes. In the absence of control, such processes lead to unstable systems that naturally exhibit finite-time blow-up phenomena. Open-loop simulations demonstrate the ability of the SPG scheme to efficiently control SFI processes, independently of the problem size and the model parameters.