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Mon, 06.11.2017, 17:15

End:
Mon, 06.11.2017, 18:15
Reduced Order Modeling for Time-Dependent Optimal Control Problems with Variable Initial Values
AG-Seminar Optimierung

Speaker: Dörte Jando
Room: S4|10 1

In this talk I will present a new reduced order model (ROM) Hessian approximation for large-scale linear-quadratic optimal control problems (OCPs) with variable initial values. Such problems arise, for example, as subproblems in multiple shooting formulations of OCPs constrained by instationary PDEs. Apart from the controls also the auxiliary initial values introduced by multiple shooting are optimization variables.

The computation of a Hessian vector product requires the solution of the linearized state equation for the vector of controls and initial data to which the Hessian is applied to, followed by the solution of the second order adjoint equation. To speed up computations, projection based ROMs of these two equations are used to generate Hessian approximations for the state variables. The challenge is that in general no fixed ROM well-approximates neither the application of the Hessian to all possible initial values nor the solution of the corresponding linear equation for all possible right hand sides. Our recently proposed approach, after having selected a basic ROM, augments this basic ROM by one vector which is either the right hand side or the initial value. This augmentation improves the ROM quality significantly. We use these Hessian approximations in a conjugate gradient method to approximate the optimal initial value. Although the augmented ROM substantially improves the accuracy of the computed initial value, this accuracy may still not be enough. Thus, I will also present a new sequential approach based on the ROM augmentation which allows to compute an approximate initial value with the same accuracy as the one obtained using the expensive full order model, but at a fraction of the cost. Finally, we will come back to the problem with control variables and apply the presented ROM approaches to efficiently solve this kind of problems.


Contact

Fachbereich Mathematik
Technische Universität Darmstadt

Schlossgartenstraße 7
64289 Darmstadt

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