# Data assimilation seminar

## Contents |

## Coordinates

- The seminar currently meets Wednesday at 2:00pm-3:30pm in Room AB1-4018, Student Commons Building, University of Colorado Denver Downtown Campus.
- Planned future seminars are at the events page and additional material sometimes linked from Jan's blog
- List of all seminars to date

Conducted by Jan Mandel and Loren Cobb.

## Purpose

The purpose of this seminar series is to help us on our learning journey and to write papers. We are currently interested in probability, statistics, and filtering in very-high-dimensional spaces, including infinite dimensional function spaces. We apply these theories to problems in:

- data assimilation and sequential filtering in discrete time
- data assimilation methods in epidemiology, wildfire modeling, and numerical weather prediction
- stochastic spatial epidemic models
- software development to support the above

Anyone is welcome, and attendance or registration of any kind is not required. However, UCD PhD students may get Readings class credit on request.

## Format

The meetings usually are highly informal work sessions, scheduled for two hours. You will not find here many prepared formal presentations where nobody understands what is going on. We actually expect to learn something and torture our speakers until they explain what they do - in sufficient detail so that we can understand and use what they say. Anyone can interrupt at any time and divert the discussion. We are convinced that *it is always good to review the basics* so we may dive into the background or digress at any time. We often bring notes or manuscripts in progress. We often run out of time; then the presentation and the discussions provide the entertainment for several more seminar sessions, until we are satisfied that we understand.

Our purpose is:

- to educate ourselves in those parts of mathematics that will help us to understand advanced data assimilation,
- to write research papers at or beyond the current frontiers of the theory and algorithm development,
- to write research proposals and then do the work

## Projects

We are moving forward on several parallel and mutually reinforcing tracks:

### Asymptotics

Our 2009 paper, *On the Convergence of the Ensemble Kalman Filter*, has been accepted for publication in *Applications of Mathematics*. We are attempting to recast this paper 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.

### Bayesian spatio-temporal statistics

The current state of the art in the statistical literature on Bayesian spatial statistics is fundamentally static, with temporal change occurring only in naïve ways (e.g. models with time as a categorical or linear variable). We foresee that the gap between Bayesian spatial statistics and the theory of modern data assimilation will be closed over the next few years, and we would like to participate in that pioneering effort with research results and papers.

### Practical applications

Practical applications of very-high-dimensional data assimilation can be found in epidemiology, wildfire modeling, and meteorology (rain). We think there are many more applications to be explored, especially in economics, health, engineering, and ecosystem dynamics.