A science and technology workshop on data assimilation for industrial inverse problems was held in January 2009. The aim of the workshop was to explore the potential for wider and improved use of data assimilation in industrial applications. Data assimilation is well developed in some applications, notably tracking, weather prediction and oil reservoir simulation history matching. This workshop presented the state of the art in data assimilation and identified further opportunities for its application to the solution of industrial inverse problems.

In order to make useful predictions of system behaviour, it is necessary to build models that incorporate information available from observations of the system. Given a system model and a dataset of measurements of the system, the inverse problem is how to estimate the model parameter values. Inverse problems are normally solved by a least squares fit of the model to the dataset using an objective function. The objective function often includes extra regularisation terms to provide a set of stable solutions. The best model is the one that gives the best fit by minimising the objective function.

From a Bayesian perspective, the regularised least squares method is equivalent to choosing the solution that maximises the posterior density. If the measurements of the system are statistically independent then solving the whole inverse problem is equivalent to solving the Bayesian filtering equations. Solving the problem in a Bayesian framework enables proper account to be made of uncertainty in the solution and opens up the possibility of solving the inverse problem sequentially. Sequential methods can have significant computational benefits over the least squares approach, sometimes transforming an intractable problem into a tractable one.

Sequential methods of particular interest are particle filters and the Ensemble Kalman Filter (EnKF). The particle filter is a rigorous Monte Carlo method that can be proved to converge to the solution of the exact Bayesian filtering equations. It is usually said that particle filters are only practical for low-dimensional problems. The EnKF has proved to be effective in high dimensional problems such as weather forecasting and history matching. However, the EnKF is a somewhat ad hoc approximation and does not converge to the exact solution of the filtering equations.

The workshop reviewed the situation, helping people to exploit existing knowledge. Key problems for further research were determined. Of significant industrial interest would be the development of a rigorous sequential approach to practical high-dimensional problems.