| Abstract: |
| Ensemble transform methods provide frameworks for a Bayesian approach to filtering. In the Ensemble Transform Particle Filter (ETPF) for example, a deterministic transform from forecast to analysis ensembles replaces the resampling step in a classical particle filter. This analysis ensemble provides an approximation to the posterior distribution in question. Filtering within high-dimensional systems is of great importance to a wide-range of industries. However, the Bayesian approach to filtering can fail in this setting due to the curse of dimensionality. This work presents two advancements in this area of research, using variants of the ETPF: making the process of propagating ensembles in these filters more efficient, and proposing a localisation scheme for the ETPF when the forecasting system is a spatio-temporal system discretized by Finite Element (FE) methods. The former is achieved via an adaptation of the multilevel Monte Carlo method, a novel variance reduction technique. The latter is achieved by using projection and multi-grid ideas for FE approximations to random spatio-temporal fields. A case-study of the stochastic quasi-geostrophic equations is presented. |
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