Parareal-Based Preconditioners for Linear-Quadratic Optimal Control Problems
Master of Arts
This thesis introduces and analyzes an efficient two time-grid parallel algorithm that reduces time and storage requirements of traditional gradient-based methods for time-dependent optimal control problems. Each iteration of a classical gradient-based method requires solving a forward-in-time state equation followed by a backward-in-time adjoint equation. This process is sequential in time and memory intensive because the solution of the state equation enters the adjoint equation. To introduce parallelism, time domain decomposition splits the entire time domain into subdomains and introduces auxiliary state and adjoint variables at the time subdomain boundaries. Given the auxiliary variables, the optimality conditions, which include the subdomain state and the adjoint equations, on the time subdomains can be solved in parallel. The auxiliary state and adjoint variables are then determined from coupling conditions that ensure continuity of the state and adjoint equation over the global time interval. This is accomplished by a two-grid approach based on ‘parareal’, which uses fine grid solvers in parallel and a computationally inexpensive sequential coarse grid solver that propagates information across all subdomains. The overall discretization accuracy is that of the fine grid solver, but parallelization across time subdomains substantially reduces the overall computing time and memory requirements.
Linear quadratic optimal control problems; Parallel-in-time method; Domain decomposition