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An Introduction to Risk-Averse PDE-Constrained Optimization: Theory, Numerical Solution, and Open Problems

Prof. Dr. Thomas M. Surowiec

PDE-constrained optimization under uncertainty presents a number of exciting challenges from theory to computation. It provides a modeling framework with which complex systems subject to uncertainty can be controlled or optimized by solutions that are resilient to outlier events. The course is split into two parts with three sections each:

Part I: Modeling and Basic Theory | (9:30 –11:00) | Slides Part 1 | Video |
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Part II: Modeling and Basic Theory | (11:15 –12:45) | Slides Part 2 | Video |

The purpose of **Part I** is to provide a broad, but gentle overview of the field with a number of motivating examples. Otherwise, a running "canonical example" will be used to illustrate the main results. The three sections are:

- Introducing Uncertainty into PDE-Constrained Optimization Problems
- Risk-Averse Decision Making
- Existence of Solutions and Optimality Conditions

**Part II** will focus on algorithmic and computational aspects. In particular the canonical example from Part I will be used to introduce the primal-dual risk minimization algorithm. Additional issues such as treating uncertainty, computation of derivatives, and convergence arguments will be presented. The three sections in Part II are:

- Infinite-Dimensional Stochastic Optimization
- The Primal-Dual Risk Minimization Algorithm
- Implementation and Numerical Solution

Prof. Dr. Thomas M. Surowiec is a professor in the Department of Mathematics and Computer Science at Philipps-Universität Marburg in Germany, where he is head of the research group "Mathematical Optimization." Before being appointed professor in Marburg in the fall of 2016, he was an assistant professor from 2014 to 2016 in the Department of Mathematics at Humboldt-Universität zu Berlin (HUB) in Berlin, Germany, where he led the junior research group on non-smooth optimization and set-valued analysis. He received his PhD in Mathematics in 2010 from HUB. Dr. Surowiec is one of the leading experts in nonsmooth optimization, stochastic optimization, equilibrium and optimal control problems. Together with his collaborators, in the recent years, he has developed novel theories and algorithms to solve risk averse optimization problems with random PDE constraints. These concepts were previously limited to applications in economics, finance, and logistics, but are now being widely used in engineering design and optimization.