This is a course I taught in the Georgia Tech Deparment of Mathematics in Spring 2008

This course addresses topics in the estimation of nonlinear dynamical systems from noisy measurements. Necessary concepts from linear algebra, dynamics, probability, and statistical information theory are reviewed; these concepts are then be applied to the estimation of physical processes whose dynamics are characterized by a system of first order differential equations. Examples of this type of problem include estimation of touch down in satellite re-entry and predictability in the Lorenz equations. These concepts are then extended to spatially correlated systems, such as those arising from the discretization of PDE used in the modeling of weather and associated atmospheric processes. These methods are illustrated in the context of the assimilation of satellite remote sensing measurements into a forecasting model. A number of estimation approaches for nonlinear systems are discussed, including variants on least squares and Kalman filtering, Gaussian mixture filtering, and filtering using discrete density approximations.

Lecture Schedule

1. Introduction to statistical inverse problems and data assimilation; review of probability and statistics
   • Exaples of inverse problems and statistical inverse problems
   • Past and current research and future directions; Discussion of rigorous results, numerical results, and missing theorems; Definition of a probability space and working with a probability measure; Transformation of density under change of coordinates and general differentiable mappings; Gaussian, log-normal, and exponential densities; Maximum likelihood estimation Bayesian statistics
     Follow this link to a visual explanation of sequential estimation with accompanying Matlab code, courtesy of Steve Conover.
2. Review of probability and statistics continued
3. (continued)
4. Review of matrix analysis
   The spectral theorem for self adjoint matricies; The eigenvector/eigenvalue geometry of self adjoint matricies; Extention of scalar functions of a scaler arguement to matrix functions of a self-adjoint matrix argument; Probability densities and maximum likelihood estimation revisited; The singular value decomposition; The Jordan canonical form and its geometric interpretation
5. (continued)
6. Classical inverse problems
   Examples Solvability, ill posed problems; Example in depth: Herglotz inversion of seismic travel time data; Example in depth: Gas concentration retrieval from multispectral/hyperspectral measurments
7. (continued)
8. Measurement uncertainty and statistical inverse problems
   Set theoretic description of uncertainty; Probabalistic description of uncertainty; Maximum likelihood estimation, Bayesian statistics revisited; Example in depth: constant velocity estimation;
9. (continued)
10. Review of differential equations
11. Kalman filtering, extended Kalman filtering
12. Point mass dynamics, numerical methods, Runge-Kutta integration
13. Extended Kalman filtering continued; the Cramer-Rao lower bound
14. The effects of nonlinear measurements and nonlinear dynamics on estimation
15. Examples of complex dynamics: Lorenz (1963), Lorenz (1996), the double pendulum
16. Adjoint formulation for numerical minimization of the log-likelihood function
17. Compensating for model mismatch with process noise
18. Methods for filtering of non-Gaussian processes: unscented Kalman filtering, sigma point filtering, ensemble Kalman filtering, particle filtering
19. Spatially correlated systems, partial differential equations
20. (continued)
21. (continued)
22. Remote sensing application: computing the observation operator and tangent linear map
23. Remote sensing application continued: computing the adjoint equation
24. Application of the method to simple spatial systems: the heat equation, the diffusive Burger's equation
25. Application of the method to more complex spatial systems: the shallow water equation
26. (continued)
27. Ensemble Kalman filtering for spatially coupled systems
28. Maximum likelihood (unscented) Kalman filtering
29. Overview of advanced applications: numerical weather prediction, climate modeling