On multilevel Monte Carlo methods for deterministic and uncertain hyperbolic systems

TL;DR

This paper investigates the effectiveness of multilevel Monte Carlo methods for hyperbolic systems, revealing that MLMC may not always provide computational advantages over standard Monte Carlo due to the complex interplay of variance, bias, and cost.

Contribution

The paper characterizes conditions under which MLMC can or cannot accelerate Monte Carlo sampling for hyperbolic systems, providing analytical and numerical insights.

Findings

MLMC may not always outperform MC in hyperbolic systems.

Performance depends on variance, bias, and computational cost parameters.

Three regimes identified where MLMC is beneficial or not.

Abstract

In this paper, we evaluate the performance of the multilevel Monte Carlo method (MLMC) for deterministic and uncertain hyperbolic systems, where randomness is introduced either in the modeling parameters or in the approximation algorithms. MLMC is a well known variance reduction method widely used to accelerate Monte Carlo (MC) sampling. However, we demonstrate in this paper that for hyperbolic systems, whether MLMC can achieve a real boost turns out to be delicate. The computational costs of MLMC and MC depend on the interplay among the accuracy (bias) and the computational cost of the numerical method for a single sample, as well as the variances of the sampled MLMC corrections or MC solutions. We characterize three regimes for the MLMC and MC performances using those parameters, and show that MLMC may not accelerate MC and can even have a higher cost when the variances of MC solutions and MLMC corrections are of the same order. Our studies are carried out by a few prototype hyperbolic systems: a linear scalar equation, the Euler and shallow water equations, and a linear relaxation model, the above statements are proved analytically in some cases, and demonstrated numerically for the cases of the stochastic hyperbolic equations driven by white noise parameters and Glimm's random choice method for deterministic hyperbolic equations.