Abstract: 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 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. In addition, risk averse optimization requires sampling in the tail of the distribution. 

This talk demonstrates the impact of risk averse optimization formulations on the solution and illustrates the difficulties that arise in solving risk averse optimization. New sampling schemes are introduced that exploit the structure of risk measures and use reduced order models to identify the small regions in parameter space which are important for the optimization. It is shown that these new sampling schemes substantially reduce the cost of solving the optimization problems. 

Brief Bio and Research Summary 

Matthias Heinkenschloss joined the Rice faculty in 1996 after serving for three years as an assistant professor in the Department of Mathematics at Virginia Polytechnic Institute and State University. He rose through the ranks at Rice, and is now the Noah G. Harding Chair and Professor of Computational and Applied Mathematics. He served as department chair for six years. Matthias began his academic career at the University of Trier in the Federal Republic of Germany, where he was from 1988 to 1993.

Matthias Heinkenschloss’ research interests are in the design and analysis of mathematical optimization algorithms for nonlinear, large-scale (often infinite dimensional) problems and their applications to science and engineering problems. Research areas include large-scale nonlinear optimization, model order reduction, optimal control of partial differential equations (PDEs), optimization under uncertainty, PDE constrained optimization, iterative solution of KKT systems and domain decomposition in optimization.

Hosts: Yanzhao Cao and Hans-Werner van Wyk