Reduced order modeling for time-dependent optimization problems with initial value controls
This paper presents a new reduced order model (ROM) Hessian approximation for linear-quadratic optimal control problems where the optimal control is the initial value. Such problems arise in parameter identification and data assimilation, where the parameters to be identified appear in the initial data, and also as subproblems in multiple shooting formulations of more general optimal control problems. The new ROM Hessians can provide a substantially better approximation than the underlying basic ROM approximation, and thus can substantially reduce the computing time needed to solve these optimal control problems. 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, 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 while the objective function and gradient are computed exactly using the full model. The challenge is that in general no fixed ROM well-approximates the application of the Hessian to all possible vectors of initial data. 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. It is shown that although the size of the ROM increases only by one, this new augmented ROM produces substantially better approximations of the true Hessian vector products as well as of the optimal solution than the basic ROM. The use of these ROM Hessians in a conjugate gradient method is analyzed.