European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
14:15   Zuiderduinzaal: Keynote: Stefan Vandewalle, KU Leuven, Belgium
45 mins
Multilevel Monte Carlo Methods for the Robust Optimization of Systems Described by Partial Differential Equations
Stefan Vandewalle
Abstract: We consider PDE-constrained optimization problems, where the partial differential equation has uncertain coefficients modelled by means of random variables or random fields. The goal of the optimization is to determine an optimum that is satisfactory in a broad parameter range, and as insensitive as possible to parameter uncertainties. First, an overview is given of different deterministic goal functions which achieve the above aim with a varying degree of robustness. Next, a multilevel Monte Carlo method is presented which allows the efficient calculation of the gradient and the Hessian arising in the optimization method [1]. The convergence and computational complexity for different gradient and Hessian based optimization methods is then illustrated for a model elliptic diffusion problem with lognormal diffusion coefficient [2]. We also explain how the optimization algorithm can benefit from taking optimization steps at different levels of the multilevel hierarchy, in a classical MG/OPT framework [3]. We demonstrate the efficiency of the algorithm, in particular for a large number of optimization variables and a large number of uncertainties. [1] M. B. Giles, Multilevel Monte Carlo Methods. Acta Numerica, 24, pp. 259–328, 2015. [2] A. Van Barel, S. Vandewalle. Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method. SIAM/ASA Journal on Uncertainty Quantification 7 (1), pp. 174-202, 2019. [3] S.G. Nash. A Multigrid Approach to Discretized Optimization Problems. Optimization Methods and Software, 14 (1-2), pp. 99–116, 2000.