European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   MS18: Model order reduction in optimisation, control, and data assimilation (Part 2)
Chair: Martin Grepl
25 mins
Reduced order methods for parametrized nonlinear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences
Maria Strazzullo, Zakia Zainib, Francesco Ballarin, Gianluigi Rozza
Abstract: We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are reliably suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we exploit POD-Galerkin reduction over a parametrized optimality system, derived from Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.
25 mins
Reduced order models for risk measure estimation in robust design
Boris Kramer, Matthias Heinkenschloss, Timur Takhtaganov, Karen Willcox
Abstract: We present a reduced-order-model-based approach for the efficient and accurate evaluation of the Conditional-Value-at-Risk (CVaR) of quantities of interest (QoI) in engineering systems with uncertain parameters. CVaR is used to model objective or constraint functions in risk-averse engineering design and optimization applications under uncertainty. Estimating the CVaR of the QoI requires sampling in the tail of the QoI distribution and typically requires many solutions of an expensive full-order model of the engineering system. Our reduced-order model approach substantially reduces this computational expense. We present error bounds that relate the ROM error to the CVaR error, and show how adaptively refining the surrogate models can further reduce the computational cost of CVaR estimation.
25 mins
Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method
Stefan Banholzer, Bennet Gebken, Sebastian Peitz, Michael Dellnitz, Stefan Volkwein
Abstract: Many optimization problems in applications can be formulated using several objective functions, which are conflicting with each other. This leads to the notion of multiobjective or multicriterial optimization problems. \\ In this talk we present a homotopy method for solving general multiobjective optimization problems, which also exploits the hierarchical structure of the solution set. This is then applied to the multiobjective optimization of an elliptic convection-diffusion-reaction equation with the cost functions either being of tracking type or measuring the parameter costs. \\ In the course of the homotopy method, numerous scalar KKT-systems have to be solved, which involves the repeated solution of the elliptic PDE and its adjoint equation. Thus, the Reduced Basis (RB) method is introduced as an approach for model-order reduction. By modifying the KKT-systems according to the desired exactness of the reduced-order model, a covering of the original solution set can be computed. Theoretical convergence results as well as practical examples are presented to validate our numerical algorithm.
25 mins
RB method for control constrained optimal control problems
Ahmad Ali, Michael Hinze
Abstract: Using the variational discretization concept, we establish a reduced model for the underlying parameter dependent optimal control problem. This approach allows us to establish sharp  posterior error estimators that do not have residuals for the control variables, which leads to minimal selection of the snapshots in the greedy algorithm. Furthermore, we show theoretically the convergence of the reduced model to the truth one as the number of snapshots increases.