Multilevel Monte Carlo Methods for Robust Optimization of Partial Differential Equations

Many complex systems and physical phenomena can be modeled by a partial differential equation (PDE). Methods to obtain a numerical solution for a given input are well established. An important engineering problem is to find the input such that the solution is optimal. This requires the definition and minimization of a cost which depends on the control input and the PDE solution. However, models of real-world systems often contain uncertainties in their parameters. The goal is then to find a robust optimum, i.e., one that performs well for a wide range of parameter realizations. The uncertainties often manifest themselves as stochastic fields or stochastic processes, which require many individual random numbers to represent. To deal with the high dimensional stochastic spaces, Monte Carlo (MC) is often the method of choice. Several techniques have been developed to remedy its slow convergence rate. Multilevel MC (MLMC) considers multiple discretization grids and moves as much of the work as possible to the coarser grids. Quasi-Monte Carlo (QMC) methods consider special 'better than random' points. This thesis applies those techniques in an optimization context. The MLMC method is used to estimate cost functionals, gradients and Hessian vector products. We present optimization methods based on those quantities that choose the precision of the MLMC estimators optimally. The cost of reaching a given gradient tolerance can then be made proportional to the cost of calculating a gradient with MLMC at a precision equal to that tolerance. Furthermore, we show how coupling of samples in the adjoint equation caused by certain cost functionals can be dealt with efficiently. Multigrid optimization techniques based on the MG/OPT framework are combined with MLMC to speed up the optimization process itself. Numerical results are presented for elliptic and also hyperbolic PDEs. Finally, we provide a rigorous analysis to prove the increased efficiency of the multilevel QMC method for the calculation of gradients.

Publication year:2021

Accessibility:Open