Distributed Solution of Optimal Control Problems Governed by Parabolic Equations

We present a spatial domain decomposition (DD) method for the solution of discretized parabolic linear–quadratic optimal control problems. Our DD preconditioners are extensions of Neumann-Neumann DD methods, which have been successfully applied to the solution of single elliptic partial differential equations and of linear–quadratic optimal control problems governed by elliptic equations. We use a decomposition of the spatial domain into non-overlapping subdomains. The optimality conditions for the parabolic linear–quadratic optimal control problem is split into smaller problems restricted to spatial subdomain-time cylinders. These subproblems correspond to parabolic linear–quadratic optimal control problems on subdomains with Dirichlet data on interfaces. The coupling of these subdomain problems leads to a Schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces in space and time. The Schur complement system is solved using a preconditioned Krylov subspace method. The preconditioner is obtained from the solution of appropriate subdomain parabolic linear– quadratic optimal control problems. The dependence of the performance of these preconditioners on mesh size and subdomain size is studied numerically. Our tests indicate that their dependence on mesh size and subdomain size is similar to that of its counterpart applied to elliptic equations only. Our tests also suggest that the preconditioners are insensitive to the size of the control regularization parameter.