European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS30: Numerical methods for PDE-constrained and controlled optimization problems with appplications (Part 3)
Chair: Maya Neytcheva
25 mins
PDE Constrained Optimization Problems for the Waste Clearance of the Human Brain
Kent-Andre Mardal, Lars Magnus Valnes, Sebastian Mitusch, Geir Ringstad, Per Kristian Eide, Simon Funke
Abstract: The brain is our most energy expensive organ, but its metabolic cycle is not understood. In particular, its waste disposal system is a mystery because it lacks the lymphatic system that is present elsewhere in our body. Understand how waste is cleared under healthy and diseased condition is important as accumulation of waste is associated with dementia such as Alzheimer’s and Parkinson’s diseases. Novel imaging protocols are under investigation for assessing brain clearance on the long time scales hours. In this talk we will present the newly proposed protocols and discuss the PDE constrained optimization problems that arise. Furthermore, a crucial component in this type of investigations is that the data exist everywhere in space, albeit at coarse resolution, but at certain points in time. Order optimal algorithms are derived for some simplified problems.
25 mins
Hyper-differential sensitivity analysis for PDE-constrained optimization
Bart vanBloemenWaanders, Joey Hart
Abstract: Hyper-differential sensitivities (HDS) analyze the dependence of PDE-constrained optimization solutions to parameter perturbations. Such analysis may be used to prioritize uncertainties in the service of data acquisition, uncertainty quantification, and model development. Low rank structure is exploited through a Singular Value Decomposition which is numerically implemented with randomized algorithms using multi-level parallelism in C++. HDS is demonstrated (1) to prioritize the influence of uncertain boundary conditions and material properties on control strategies, (2) to analyze the stability of optimal solutions under uncertainty, and (3) to augment optimal experimental design for data acquisition.
25 mins
Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks
Nikolaj Takata Mücke, Lasse Hjuler Christiansen, Allan Peter Karup-Engsig, John Bagterp Jørgensen
Abstract: With simulation based decision making playing an increasingly important role in science and engineering the demand for fast and reliable computational schemes is increasing. This is e.g. the case for nonlinear model predictive control (NMPC) where real-time multi-query solutions are essential. However, in cases where the mathematical model is of high dimension in the solution space, e.g. for solution of partial differential equations, black-box optimizers are rarely sufficient to get the required online computational speed. In this talk, I will present a reduced order modeling approach, based on proper orthogonal decomposition (POD) and artificial neural networks (ANN), to address before mentioned problems associated with nonlinear PDE-constrained optimization. The role of POD is to identify a lower dimensional representation of the solution manifold while the ANN is used for approximating a parametrization of the low dimensional manifold. Thus, leading to an equation free online model. I will consider a specific nonlinear time dependent PDE-constrained optimization problem and assess the performance and potential of the proposed strategy.
25 mins
An alternative method to impose state and control constraints in PDE-constained optimization problems
Maya Neytcheva
Abstract: We compare two approaches to impose additional constraints on the state and the control variable in the framework of PDE-constrained optimization problems.