This thesis aims to investigate and improve the efficiency of ensemble transform methods for data assimilation, using an application of multilevel Monte Carlo. Multilevel Monte Carlo is an interesting framework to estimate statistics of discretized random variables, since it uses a hierarchy of discretizations with a refinement in resolution. This is in contrast to standard Monte Carlo estimators that only use a discretization at a fine resolution. A linear combination of sub-estimators, on different levels of this hierarchy, can provide new statistical estimators to random variables at the finest level of resolution with significantly greater efficiency than a standard Monte Carlo equivalent. Therefore, the extension to computing filtering estimators for data assimilation is a natural, but challenging area of study. These challenges arise due to the fact that correlation must be imparted between ensembles on adjacent levels of resolution and maintained during the assimilation of data. The methodology proposed in this thesis, considers coupling algorithms to establish this correlation. This generates multilevel estimators that significantly reduce the computational expense of propagating ensembles of discretizations through time and space, in between stages of data assimilation.

An effective benchmark of this methodology is realised by filtering data into high-dimensional spatio-temporal systems, where a high computational complexity is required to solve the underlying partial differential equations. A novel extension of an ensemble transform localisation framework to finite element approximations within random spatio-temporal systems is proposed, in addition to a multilevel equivalent.