  We provide a new approach for the efficient matrix-free application of the
transpose of the Jacobian for the spectral element method for the adjoint based
solution of partial differential equation (PDE) constrained optimization. This
results in optimizations of nonlinear PDEs using explicit integrators where the
integration of the adjoint problem is not more expensive than the forward
simulation. Solving PDE constrained optimization problems entails combining
expertise from multiple areas, including simulation, computation of
derivatives, and optimization. The Portable, Extensible Toolkit for Scientific
computation (PETSc) together with its companion package, the Toolkit for
Advanced Optimization (TAO), is an integrated numerical software library that
contains an algorithmic/software stack for solving linear systems, nonlinear
systems, ordinary differential equations, differential algebraic equations, and
large-scale optimization problems and, as such, is an ideal tool for performing
PDE-constrained optimization. This paper describes an efficient approach in
which the software stack provided by PETSc/TAO can be used for large-scale
nonlinear time-dependent problems. While time integration can involve a range
of high-order methods, both implicit and explicit. The PDE-constrained
optimization algorithm used is gradient-based and seamlessly integrated with
the simulation of the physical problem.
