  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.
