  To solve problems in domains such as filtering, optimization, and posterior
sampling, interacting-particle methods have recently received much attention.
These parallelizable and often gradient-free algorithms use an ensemble of
particles that evolve in time, based on a combination of well-chosen dynamics
and interaction between the particles. For computationally expensive dynamics
-- for example, dynamics that solve inverse problems with an expensive forward
model -- the cost of attaining a high accuracy quickly becomes prohibitive. We
exploit a hierarchy of approximations to this forward model and apply
multilevel Monte Carlo (MLMC) techniques, improving the asymptotic
cost-to-error relation. More specifically, we use MLMC at each time step to
estimate the interaction term within a single, globally-coupled ensemble. This
technique was proposed by Hoel et al. in the context of the ensemble Kalman
filter; the goal of the present paper is to study its applicability to a
general framework of interacting-particle methods. After extending the
algorithm and its analysis to a broad set of methods with fixed numbers of time
steps, we motivate the application of the method to the class of algorithms
with an infinite time horizon, which includes popular methods such as ensemble
Kalman algorithms for optimization and sampling. Numerical tests confirm the
improved asymptotic scaling of the multilevel approach.
