Fast iterative solver for the optimal control of time?dependent PDEs with Crank–Nicolson discretization in time

In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all-at-once of optimal control problems with time-dependent PDEs as constraints, including heat equation non-steady convection–diffusion equation. After applying an optimize-then-discretize approach, one is faced continuous first-order optimality conditions consisting coupled system PDEs. As opposed to most work in preconditioning resulting discretized systems, where (first-order accurate) backward Euler method used discretization time derivative, employ (second-order Crank–Nicolson time. We apply carefully tailored invertible transformation symmetrizing matrix, then preconditioner saddle-point obtained. The key components are accurate mass matrix approximation, good approximation Schur complement, appropriate multigrid process latter approximation—these constructed using our transforming system. prove complement through bounds on eigenvalues, test solver against widely-used linear arising from discretization. These demonstrate effectiveness robustness respect mesh-sizes, regularization parameter, diffusion coefficient.