Author:

Keywords:

Uncertainty, Multilevel, Monte Carlo, Robust Optimization

Abstract:

We consider PDE-constrained optimization problems, where the partial differential equation has uncertain coefficients modeled by means of random variables or random fields. The goal of the optimization is to determine a robust optimum, i.e., an optimum that is satisfactory in a broad parameter range, and as insensitive as possible to parameter uncertainties. First, a general overview is given of different deterministic goal functions which achieve the above aim with a varying degree of robustness. The relevant goal functions are deterministic because of the expected value operators they contain. Since the stochastic space is often high-dimensional, a multilevel (Quasi-) Monte Carlo method is presented which allows the efficient calculation of the gradient and the Hessian, whose expressions then also contain expected value operators. The convergence behavior for different gradient and Hessian based methods is illustrated for a model elliptic diffusion problem with lognormal diffusion coefficient. We demonstrate the efficiency of the algorithm, in particular for a large number of optimization variables and a large number of uncertainties.