In the course of this thesis, several different applications of data assimilation will\ud be looked at. In each case, a rigorous mathematical framework will be constructed,\ud in a Bayesian context, to enable the use of various types of data to infer on various\ud infinite dimensional parameters of the system that has been observed. After careful\ud consideration of the forward problem, well-defined posterior distributions on function\ud space are constructed. Using MCMC methods which are defined on these function\ud spaces themselves, we can construct Markov chains whose invariant measures are the\ud posterior of interest. From this point, we can implement these methods on a computer,\ud having finally discretised the problem.\ud The philosophy that we adhere to throughout, is the idea that numerical methods\ud formulated on function space are robust under discretisation, and do not suffer from\ud the curse of dimensionality typically suffered by sampling methods formulated after\ud disretisation.\ud The first few chapters (after the introductory chapter) will focus on various\ud aspects of data assimilation of observations of Stokes \ud ow dynamics. Chapter 2 will\ud focus on Eulerian data where direct observations of the velocity of the \ud fluid at various\ud points in time and space will be made. Chapter 3 will concentrate on data assimilation\ud of indirect observations of the field, in the form of the positions of passive tracers in the flow. In these two chapters we will assume that the forcing of the system is known and\ud that we are merely trying to recover the initial condition of the flow field.\ud In chapter 4 we will consider both Eulerian and Lagrangian data assimilation,\ud with the added complexity of trying to use the data to not only infer on the initial\ud condition but also on the space-time dependant forcing of the system.\ud In chapter 5 we will try to show how these smoothing methods could be adapted\ud into a filtering algorithm, and a simple example will be presented.\ud In the final chapter, 6, this Bayesian framework on function space will be applied\ud to a shape matching problem with applications in the biomedical sciences
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