  Solving optimization problems with transient PDE-constraints is
computationally costly due to the number of nonlinear iterations and the cost
of solving large-scale KKT matrices. These matrices scale with the size of the
spatial discretization times the number of time steps. We propose a new two
level domain decomposition preconditioner to solve these linear systems when
constrained by the heat equation. Our approach leverages the observation that
the Schur-complement is elliptic in time, and thus amenable to classical domain
decomposition methods. Further, the application of the preconditioner uses
existing time integration routines to facilitate implementation and maximize
software reuse. The performance of the preconditioner is examined in an
empirical study demonstrating the approach is scalable with respect to the
number of time steps and subdomains.
