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UID:news1352@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20220516T103313
DTSTART;TZID=Europe/Zurich:20220520T110000
SUMMARY:Seminar in Numerical Analysis: Jens Saak (Max Planck Institute for
Dynamics of Complex Technical Systems)
DESCRIPTION:Optimal control problems subject to constraints given by partia
l differential equations are a powerful tool for the improvement of many t
asks in science an technology. Classic optimization today is applicable on
various problems and tackling nonlinear equations and inclusion of box co
nstraints on the solutions is flexible. However\, especially for non-stati
onary problems\, small perturbations along the trajectories can easily lea
d to large deviations in the desired solutions. Consequently\, optimality
may be lost just as easily. On the other hand\, the linear-quadratic regul
ator problem in system theory is an approach to make a dynamical system re
act to perturbation via feedback controls that can be expressed by the sol
utions of matrix Riccati equations. It’s applicability is limited by the
linearity of the dynamical system and the efficient solvability of the qu
adratic 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 approximabi
lity.\\r\\nFor further information about the seminar\, please visit this w
ebpage [t3://page?uid=1115].
X-ALT-DESC:Optimal control problems subject to constraints given by part
ial 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-sta
tionary problems\, small perturbations along the trajectories can easily l
ead to large deviations in the desired solutions. Consequently\, optimalit
y may be lost just as easily.

On the other hand\, the linear-quadrat
ic regulator problem in system theory is an approach to make a dynamical s
ystem react to perturbation via feedback controls that can be expressed by
the solutions of matrix Riccati equations. It’s applicability is limite
d by the linearity of the dynamical system and the efficient solvability o
f the quadratic matrix equation.

In this talk\, we discuss how certa
in classes of non-stationary PDEs can be reformulated (after spatial semi-
discretization) into structured linear dynamical systems that allow the Ri
ccati feedback to be computed. This allows us to combine both approaches a
nd thus steer solutions of perturbed PDEs back to the optimized trajectori
es. 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 solu
tions usually feature a strong singular value decay\, and thus good low-ra
nk approximability.

\nFor further information about the seminar\, pl
ease visit this webpage.

DTEND;TZID=Europe/Zurich:20220520T120000
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