Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method

TL;DR

This paper develops a multilevel Monte Carlo approach for robust PDE-constrained optimization under uncertainty, providing efficient algorithms with proven cost bounds and demonstrating their effectiveness on elliptic diffusion problems.

Contribution

It introduces a multilevel Monte Carlo method for PDE optimization with uncertain coefficients, including algorithms that adaptively select samples and analyze their computational cost.

Findings

The proposed MLMC-based algorithms reduce the number of samples needed.

Cost of optimization is proportional to the cost of a gradient evaluation with specified error.

Algorithms are effective on elliptic diffusion problems with lognormal coefficients.

Abstract

This paper addresses optimization problems constrained by partial differential equations with uncertain coefficients. In particular, the robust control problem and the average control problem are considered for a tracking type cost functional with an additional penalty on the variance of the state. The expressions for the gradient and Hessian corresponding to either problem contain expected value operators. Due to the large number of uncertainties considered in our model, we suggest to evaluate these expectations using a multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that this results in the gradient and Hessian corresponding to the MLMC estimator of the original cost functional. Furthermore, we show that the use of certain correlated samples yields a reduction in the total number of samples required. Two optimization methods are investigated: the nonlinear conjugate gradient method and the Newton method. For both, a specific algorithm is provided that dynamically decides which and how many samples should be taken in each iteration. The cost of the optimization up to some specified tolerance $\tau$ is shown to be proportional to the cost of a gradient evaluation with requested root mean square error $\tau$. The algorithms are tested on a model elliptic diffusion problem with lognormal diffusion coefficient. An additional nonlinear term is also considered.