In this talk I will present a new reduced order model (ROM) Hessian approximation for large-scale linear-quadratic optimal control problems where the optimal control is the initial value. Such problems arise in parameter identification, where the parameters to be identified appear in the initial data of the underlying instationary PDE, and as subproblems in multiple shooting formulations of more general optimal control problems constrained by instationary PDEs.
The computation of a Hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the Hessian is applied to, followed by the solution of the second order adjoint equation. Projection based ROMs of these two linear differential equations are used to generate the Hessian approximation. The challenge is that in general no fixed ROM well-approximates neither the application of the Hessian to all possible vectors of initial data nor the solution of the corresponding linear equation for all possible right hand sides. The new approach, after having selected a basic ROM, augments this basic ROM by one vector. This vector is either the right hand side or the vector of initial data to which the Hessian is applied to. Although the size of the ROM increases only by one, this new augmented ROM produces substantially better approximations than the basic ROM. I will also analyze the use of these ROM Hessians in a conjugate gradient (CG) method.
This is joint work with Matthias Heinkenschloss.