Sequential Monte Carlo methods for applications in high dimensions.

Sequential Monte Carlo (SMC) methods are nowadays routinely employed across a wide range of disciplines to calibrate mathematical models and carry out forecasting about complex non-linear stochastic systems, using information from incoming data. SMC methods have been successfully applied in such diverse areas as econometrics, communications, target tracking, computer vision, roboting and biology. They will infer about unknown parameters of stochastic systems and unobserved states of the systems (the "signal") given streams of data.

However, it is a common knowledge that standard SMC methods cannot tackle important high-dimensional problems, arising in fields such as atmospheric sciences, oceanography, hydrology and signal processing, as their computational cost has been found to increase exponentially fast with the dimension of the state space of the system.

The proposed research will investigate and develop advanced SMC methods of improved algorithmic efficiency in high dimensions, rendering SMC methodology practically relevant in such contexts. This is of high importance as alternative methods currently used in high dimensions cannot fully capture non-linear model dynamics arising in applications, and can give inaccurate estimates of uncertainty or forecasts in such non-linear scenarios.

Filtering and signal-denoising methods are used in a wide range of important applications for the economy and the society ranging from target tracking and missile guidance, to meteorological and oceanographical forecasting. Some indicative main beneficiaries that could be mentioned are the following:

- Meteorology
- Oceanography

A potential development of computationally efficient SMC methods for data assimilation procedures in such areas will be of great significance, since it will allow practitioners to fully calibrate non-linear stochastic models used in these contexts and provide fully justified measures of uncertainty in such non-linear scenarios..

More analytically, current data assimilation methods at the Met Office, the European Centre for Medium-Range Weather Forecasts and oceanographic centres have been combining carefully formulated Partial Differential Equations capturing physical laws with data arriving from satellites, in situ data and data from many other sources to infer about present states of the atmosphere and parts of the ocean and make forecasts about future ones. Recent efforts is such organisations have looked at Kalman-Filter sequential methods that deliver 'ensembles' of possible future states, thus providing important indications about the uncertainly and the level of confidence at the stated predictions. However, Kalman-Filter methods will have to approximate non-linear model dynamics with linear ones with an effect at the accuracy of the stated forecasts and uncertainly estimates. Thus, discussions have been made in the literature on extending the feasibility of SMC methods in such high-dimensional problems to fully accommodate non-linear model dynamics and increase confidence at stated estimates. The proposed project is expected to make important contributions at this direction.