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Data Assimilation Seminar

Title: Towards an ENKF on a Riemannian Manifold: Tensors and Parallel Vector Fields - Part 2
Speaker(s): Bryan Smith
Affiliation: UCD Department of Mathematical and Statistical Sciences
When: Monday,  February 13, 2012
Time: 12:00 PM  -  2:00 PM
Where: CU building, Room 656

The ENKF attempts to predict the trajectory of a particle by combining uncertain measurements of the particle's state vector (x(t_2)) with uncertain information from a previous state of the state vector (x(t_1)). The particle is assumed to be a random variable undergoing an affine transformation from t_1 to t_2. We wish to extend this method of prediction to particles travelling in a Riemannian or even a more general manifold. It is not immediately clear how we can extend the notion of an affine transformation on a manifold, but it can be done. We consider the simplest case in which the particle undergoes a parallel translation. In R^n, this corresponds to x(t_2) = x(t_1) + w, where w is some vector in R^n. One can visualize this as a vector field over R^n, with all vectors being parallel and equal to w, and each point x(t_1) being translated along this vector field. In a Riemannian manifold, we can extend this notion of parallel translation. This is actually one version of the Fundamental Theorem of Riemannian Geometry. We will explain how this theorem relates to parallel translation. We will also derive transformation laws for the mean and covariance matrix of a random vector on a Riemannian Manifold. We will also explain the concept of tensors, which has many applications in analysis, linear algebra, and pde's.

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