The seminar is not for credit, attendance is optional, and there are no formal requirements. Our purpose is twofold: (1) to educate ourselves in those parts of mathematics that will help us to understand advanced data assimilation, and (2) to write research papers at or beyond the current frontiers of the theory.
One of our projects is to recast our previous paper On the Convergence of the Ensemble Kalman Filter into the coordinate-free language of Hilbert spaces, with precise bounds on the rate of convergence as the ensemble size increases to infinity, and as the number of dimensions increases to infinity. If we can achieve this, we believe it will mark a significant step forward. The Ensemble Kalman Filter (EnKF) is already one of the most powerful and popular filters for use in problems of very high dimensionality -- one million dimensions is not unusual in atmospheric models -- but there is a glaring lack of theory on its asymptotic behavior.
To do this, we will need to restate many of the usual tools of probability theory in Hilbert space terminology. Some of these tools may not survive the transition at all, others may need new conditions and assumptions. We need to look very closely at every inequality and every convergence theorem, to make sure that we understand exactly how it works (or fails to work) in Hilbert space.
There is rich literature on probability on Banach and Hilbert spaces. Sadly, it is mostly aimed at the specialist. Much of the work in the seminar is to choose and understand results that may be useful in our quest.
The seminar currently meets Thursdays 10-12am in Room 626.
Conducted by Jan Mandel and Loren Cobb
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