Multilevel Quasi-Monte Carlo for Optimization under Uncertainty

TL;DR

This paper introduces a multilevel quasi-Monte Carlo approach for efficiently estimating gradients in optimization problems involving elliptic PDEs with uncertain coefficients, demonstrating superior convergence and computational efficiency over traditional methods.

Contribution

The paper develops a novel MLQMC method with regularity analysis for gradient estimation in PDE-constrained optimization under uncertainty, outperforming existing Monte Carlo techniques.

Findings

MLQMC achieves faster error decay than Monte Carlo.

Multilevel approach significantly reduces computational cost.

Numerical results confirm superior convergence rates.

Abstract

This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the CE method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and the single level quasi-Monte Carlo method.