  Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the
description of fluctuations in many-particle systems. In this paper, we develop
and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean--Kawasaki
equation, a pivotal representative of this class of SPDEs. We prove
analytically and demonstrate numerically that our MLMC scheme provides a
significant reduction in computational cost (with respect to a standard Monte
Carlo method) in the simulation of the Dean--Kawasaki equation. Specifically,
we link this reduction in cost to having a sufficiently large average particle
density, and show that sizeable cost reductions can be obtained even when we
have solutions with regions of low density. Numerical simulations are provided
in the two-dimensional case, confirming our theoretical predictions.
  Our results are formulated entirely in terms of the law of distributions
rather than in terms of strong spatial norms: this crucially allows for MLMC
speed-ups altogether despite the Dean--Kawasaki equation being highly singular.
