1. Problem Formulation
(i) Dynamical Systems: iterated maps, Markov kernels, time-averaging and ergodicity, explicit examples.
(ii) Bayesian Probability: joint, marginal and conditional probabilities; Bayes’ formula.
(iii) Smoothing, Filtering: formulation of these off-line and on-line probability distributions using Bayes’ theorem and the links between them.
(iv) Well-Posedness: introduction of metrics on probability measure and demonstration that smoothing and filtering distributions are Lipschitz with respect to data, using these metrics.
2. Smooting Algorithms
(i) Monte Carlo Markov Chain: Random walk Metropolis, Metropolis-Hastings, proposals tuned to the data assimilation scenario.
(ii) Variational Methods: relationship between maximizing probability and minimizing a cost function; demonstration of multi-modal behaviour.
3. Filtering Algorithms
(i) Kalman filter: derivation using precision matrices and use of Sherman-Woodbury identity to formulate with covariances.
(ii) 3DVAR: derivation as a minimization principle compromising between fit to model and to data.
(iii) Extended and Ensemble Kalman Filter. Generalize 3DVAR to allow for adaptive estimation of (covariance) weights in the minimization principle.
(iv) Particle Filter. Sequential importance sampling, proof of convergence.
The module will form a fourth year option on the MMath Degree. Data Assimilation is concerned with the principled integration of data and dynamial models to produce enhanced predictive capability. As such it finds wide-ranging applications in areas such as weather forecasting, oil reservoir management, macro and micro economic modelling and traffic flow. This module aim is to describe the mathematical and computational tools required to study data assimilation.
By the end of the module the student should be able to understand a range of important subjects in modern applied mathematics, namely:
- Stochastic dynamical systems
- Long-time behaviour of dynamical systems
- Bayesian probability
- Metrics on probability measures
- Monte Carlo Markov Chain
- Matlab programming
Instructor has has own printed lecture notes (draft of a book) which will provide the primary source. These notes have an extensive bibliography and include matlab codes which will be made available to the students.