Time-domain decomposition preconditioners for the solution of discretized parabolic optimal control problems
Doctor of Philosophy
Optimal control problems governed by time-dependent partial differential equations (PDEs) lead to large-scale optimization problems. While a single PDE can be solved by marching forward in time, the optimality system for time-dependent PDE constrained optimization problems introduces a strong coupling in time of the governing PDE, the so-called adjoint PDE, which has to be solved backward in time, and the gradient equation. This coupling in time introduces huge storage requirements for solution algorithms. We study a time-domain decomposition based method that addresses the problem of storage and additionally introduces parallelism into the optimization algorithm. The method reformulates the original problem as an equivalent constrained optimization problem using ideas from multiple shooting methods for PDEs. For convex linear-quadratic problems, the optimality conditions of the reformulated problem lead to a linear system in state and adjoint variables at time-domain interfaces and in the original control variables. This linear system is solved using a preconditioned Krylov subspace method. Block Gauss-Seidel preconditioners have been introduced by Heinkenschloss (2000) for a suitable permutation of the optimality system. Unfortunately, the number of Krylov subspace iterations significantly increases when these preconditioners are parallelized. This problem has motivated the study of a preconditioner based on an approximate factorization of the optimality system, used by Biros, et. al. (1999) in another context. The factorization-based preconditioner requires approximate state and adjoint solves as well as a preconditioner for the so-called reduced Hessian. The approximate state and adjoint solves are derived from the parareal algorithm of Maday, et. al. (2001). New results on the spectrum of the reduced Hessian for a class of optimal control problems show that a simple 'scaling' preconditioner for the reduced Hessian suffices for this problem class. Our analysis of the spectrum of the optimality system preconditioned by the factorization-based preconditioner shows that we can replace the preconditioner for the reduced Hessian by a preconditioner for a less expensive Hessian-type operator. For problems of control in the initial condition, we apply this result to modify the multigrid preconditioner for the reduced Hessian introduced by Draganescu (2004) with no effects on the multigrid preconditioner performance.