Statistical Inverse Problems: Data Assimilation and Predictability of Nonlinear Systems
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 reentry 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
 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, lognormal, 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.
 Review of probability and statistics continued
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 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 selfadjoint matrix argument;
Probability densities and maximum likelihood estimation revisited;
The singular value decomposition;
The Jordan canonical form and its geometric interpretation
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 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
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 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;
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 Review of differential equations
 Kalman filtering, extended Kalman filtering
 Point mass dynamics, numerical methods, RungeKutta integration
 Extended Kalman filtering continued; the CramerRao lower bound
 The effects of nonlinear measurements and nonlinear dynamics on estimation
 Examples of complex dynamics: Lorenz (1963), Lorenz (1996), the double pendulum
 Adjoint formulation for numerical minimization of the loglikelihood function
 Compensating for model mismatch with process noise
 Methods for filtering of nonGaussian processes: unscented Kalman filtering, sigma point filtering, ensemble Kalman filtering, particle filtering
 Spatially correlated systems, partial differential equations
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 Remote sensing application: computing the observation operator and tangent linear map
 Remote sensing application continued: computing the adjoint equation
 Application of the method to simple spatial systems: the heat equation, the diffusive Burger's equation
 Application of the method to more complex spatial systems: the shallow water equation
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 Ensemble Kalman filtering for spatially coupled systems
 Maximum likelihood (unscented) Kalman filtering
 Overview of advanced applications: numerical weather prediction, climate modeling
