Seminar in Numerical Analysis: Jens Saak (Max Planck Institute for Dynamics of Complex Technical Systems)
Optimal control problems subject to constraints given by partial differential equations are a powerful tool for the improvement of many tasks in science an technology. Classic optimization today is applicable on various problems and tackling nonlinear equations and inclusion of box constraints on the solutions is flexible. However, especially for non-stationary problems, small perturbations along the trajectories can easily lead to large deviations in the desired solutions. Consequently, optimality may be lost just as easily.
On the other hand, the linear-quadratic regulator problem in system theory is an approach to make a dynamical system react to perturbation via feedback controls that can be expressed by the solutions of matrix Riccati equations. It’s applicability is limited by the linearity of the dynamical system and the efficient solvability of the quadratic matrix equation.
In this talk, we discuss how certain classes of non-stationary PDEs can be reformulated (after spatial semi-discretization) into structured linear dynamical systems that allow the Riccati feedback to be computed. This allows us to combine both approaches and thus steer solutions of perturbed PDEs back to the optimized trajectories. The key to efficient solvers for the Riccati equations is the usage of the specific structure in the problems and the fact that the Riccati solutions usually feature a strong singular value decay, and thus good low-rank approximability.
For further information about the seminar, please visit this webpage.
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