Prof. Dr. Matthias Heinkenschloss (Rice University)
Optimal control and optimal design problems governed by partial differential equations (PDEs) arise in many engineering and science applications. In these applications one wants to maximize the performance of the system subject to constraints. When problem data, such as material parameters, are not known exactly but are modeled as random variables/fields, the system performance is a random variable. So-called risk measures are applied to this random variable to obtain the objective function for PDE constrained optimization under uncertainty. Instead of only maximizing expected performance, risk-averse optimization also considers the deviation of actual performance below expected performance. The resulting optimization problems are difficult to solve because a single objective function evaluation requires sampling of the governing PDE at many parameters, risk-averse optimization requires sampling in the tail of the distribution, and many risk measures introduce non-smoothness into the optimization.In this talk I will demonstrate the impact of risk-averse optimization formulations on the solution and illustrate the difficulties that arise in solving risk-averse optimization problems. These difficulties include the inherent non-smoothness of these problems and sampling. I will show that solution methods for several risk-averse optimization problems ultimately lead to sequences of smoothed optimization problems, which can be solved using modifications of Newton's method. Our new modifications address numerical difficulties that arise from the inherent non-smoothness of the underlying risk-averse problem and they reduce the number of samples needed for approximate derivative evaluations. Time permitting, I will introduce a new sampling schemes that exploits the structure of risk measures and use reduced order models to identify the small regions in parameter space which are important for the optimization.
If you wish to attend one (or more) of the talks please send to request for the zoom link to anapde_mathematik.tu-darmstadt.de (please replace _ by @ ). Please include in your mail your full name, status (teacher, professor, student,...) and institution.
Wenn Sie als Zuhörer*in an den Kolloquien teilnehmen möchten, schicken Sie bitte eine Anfrage per Mail an anapde_mathematik.tu-darmstadt.de (ersetzen Sie dabei bitte _ durch @ ). Ihre Mail sollte Ihren vollständigen Namen, Ihren Status (Lehrer*in, Professor*in, Student*in, Doktorand*in , ...) und Ihre Institution enthalten.
27. Oktober 2021, 17:15-19:00
FB Mathematik, AG Numerik und wissenschaftliches Rechnen
Prof. Dr. Jan Giesselmann