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
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
PDE-constrained optimization: Matrix structures and preconditioners
Ivo Dravins, Maya Neytcheva
Abstract: \documentclass[12pt,a4paper]{article} \title{PDE-constrained optimization: \\ Matrix structures and preconditioners} \author{Ivo Dravins, Maya Neytcheva } \begin{document} \maketitle Optimization and control of processes governed by partial differential equations are central to many important applications. The importance of their numerical simulation steadily grows, together with the need for robust and efficient numerical techniques to solve the arising large-scale algebraic problems. In this presentation we the minimization of a cost functional of the form $$ J(y, u) = \frac{1}{2} \|y-y_d\|^2_{L^2 (\Omega_0)}+\frac{\alpha}{2}\|u\|^2_{L^2(\Omega)} + \beta\|u\|^2_{L^1(\Omega)}, $$ optimizing for both a state and a control variable, $y$ and $u$, correspondingly. In addition we include constraints on $u$ and or $y$ to be within certain bounds and on $u$ - to be sparse, which is achieved by including the $L^1$ norm term in $J$. Here $y_0$ is some desired state, $0<\alpha<1$ and $0<\beta<1$ are regularization parameters. The arising algebraic problems in the above setting are nonlinear, solved using the so-called semi-smooth Newton method. The focus in this presentation is on the structure of the matrices in the arising linear systems to be solved at each nonlinear iteration and approaches to construct numerically and computationally efficient preconditioners. We illustrate the performance of the involved nonlinear and linear solvers with some numerical experiments. The implementation is done in \texttt{Julia}. \end{document}
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.