Robust optimization of PDE constrained systems using a multilevel Monte Carlo method

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. Many goal functions can be defined that attempt to solve this problem. They vary in computational cost and in the robustness of the solution. In this talk, we focus on optimizing the expected value of a tracking type objective with an additional penalty on the variance of the state. The gradient and Hessian corresponding to such a cost functional also contain expected value operators. Since the stochastic space is often high dimensional, a multilevel (quasi-) Monte Carlo method is presented to efficiently calculate the gradient and the Hessian. If one is careful, the resulting estimated quantities are the exact gradient and Hessian of the estimated cost functional, which is important in practice for some optimization algorithms. The convergence behavior is illustrated using a gradient and a Hessian based optimization method for a model elliptic diffusion problem with lognormal diffusion coefficient and optionally an additional nonlinear reaction term. The evolution of the variances on each of the levels during the optimization procedure leads to a practical strategy for determining how many and which samples to use. We also investigate the necessary tolerances on the mean squared error of the estimated quantities. Finally, a practical algorithm is presented and tested on a problem with a large number of optimization variables and a large number of uncertainties.​status: publishe