$hp$-Multilevel Monte Carlo Methods for Uncertainty Quantification of Compressible Flows

TL;DR

This paper introduces an $hp$-multilevel Monte Carlo method combining mesh refinement and polynomial degree increase for efficient uncertainty quantification in compressible flows, with proven complexity and practical validation.

Contribution

The paper presents a novel $hp$-multilevel Monte Carlo approach with complexity analysis and a confidence bound estimator, optimized for large-scale, queue-based computing systems.

Findings

Complexity of the method is quadratic with respect to accuracy.

The approach effectively handles complex engineering problems like cavity flow.

Numerical experiments confirm theoretical efficiency and robustness.

Abstract

We propose a novel $hp$-multilevel Monte Carlo method for the quantification of uncertainties in the compressible Navier-Stokes equations, using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes while simultaneously increasing the polynomial degree of the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the $hp$-multilevel Monte Carlo method to compute the mean field with prescribed accuracy is, in best-case, of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overestimating these quantities. The method is in particular designed for application on queue-based computing systems, where it is desirable to compute a large number of samples during one iteration, without overestimating the optimal number of samples. Our theoretical results are verified by numerical experiments for the two-dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.