Coordinators Prof. S. Lakshmivarahan IIT Madras(USA)

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Much of the human activity is controlled by prediction of various phenomena that affect our life - be it of the man-made or natural type. Examples include prediction of weather – hurricanes, tornados, snow/ice showers, heat waves, short-term climate scenarios, prediction of revenue by local/state and national governments to develop budget priorities for the next fiscal year, prediction of the growth in GDP, prediction of IBM stock price on the day of equinox, to name a few. These predictions are generated by running a relevant class of models which may be either causality based (as in hurricane prediction) or empirically derived (as in the prediction of revenue or unemployment or GDP growth, etc).

Besides being causality based or empirically derived, models, in general, occur in various shapes and forms. A model can be (a) static or dynamic (b) deterministic or stochastic (c) operate in discrete/continuous time and (d) may be defined on continuous space or discrete space. Irrespective of its origin, models in general have several unknowns – initial conditions (IC), boundary conditions (BC), and parameters. The solutions of these models, in general, constitute the basis for generating predictions.

However, to compute the solution we need to know or estimate the values of the unknown IC, BC and/or parameters. This estimation is enabled by using the observations of the phenomenon in question – using the observed pressure distribution around the eye of the hurricane, data from satellites or ground based radars, from the time series of data on unemployment, etc. Data assimilation in large measure relates to the process of “fusing” data with the model for the singular purpose of estimating the unknowns. Once these estimates are available, we obtain an instantiation of the model which is then run forward in time to generate the requisite forecast products for public consumption.

Data assimilation as stated is intimately related to the "inverse" problems in mathematics or regression problems in statistics or retrieval problems in Geosciences. Finding the sea surface temperature distribution near the equatorial Pacific from Satellite measurements or distribution of the aerosol from satellite observations or estimating the amount of rain from radar observations are but a few of the examples of inverse problems that lie at the heart of data assimilation.

Some phenomena are intrinsically predictable (lunar/solar eclipses for the next 50 years) but certain other phenomena are only predictable for a short horizon – weather at most for a week. The length of the predictable horizon is related to predictability limit of the model. This is closely related to understanding the growth of prediction errors. This topic is closely related to analysis of chaotic systems.

Our aim is to provide a broad based background on the mathematical principles and tools from linear algebra, multivariate calculus and finite dimensional optimization theory, estimation theory, non-linear dynamics and chaos that constitute the basis for dynamic data assimilation as we know today. Our aim is to present the ideas at the level of a first year graduate/final year undergraduate student aspiring to enter this exciting area.

Good facility with Calculus, Linear Algebra, basic Probability Theory and Statistics

Books

1. J.M. Lewis, S. Lakshmivarahan and S. K. Dhall (2006) Dynamic Data Assimilation: a least squares approach, Cambridge University Press, 654 pages + Appendices A through F
2. E. Kalnay (2003) Atmospheric Modeling, Data Assimilation, and Predictability, Cambridge University Press