PDE constrained optimization with uncertain data
Department of Computational and Applied Mathematics
Optimization problems governed by partial differential equations (PDEs) arise in many science and engineering applications, e.g., in the form of optimal control or optimal design problems. In these applications, system parameters are often not known precisely and are modeled as random variables or as random fields. System outputs, which are the objective function or constraints in the deterministic case, are random variables when system parameters are uncertain, and are included into the optimization via their expected values or via so-called risk measures, which measure deviation from expectation or occurrence extreme events.
In this talk I will illustrate challenges that arise in the numerical solution of PDE constrained optimization with uncertain data, with an emphasis on risk-averse optimization problems. These optimization problems are typically solved using derivative based optimization algorithms combined with some sort of sampling of the random inputs. However, as I will illustrate, the choice of risk measures impacts not only the optimal control/design, but also the numerical solution of the optimization problem. In particular some efficient sampling methods that have recently been proposed for quantification of uncertainty in systems, cannot directly be used in the risk-averse optimization context. Instead it is necessary to carefully use the structure of the optimization problem. I will present our current approach for optimization structure exploiting adaptive sampling, sketch their convergence results, and illustrate their performance numerically.