Math 577 Inference from Data and Models in Geosciences and Engineering

Section 1

Spring 2006

Tu 2:00-3:15, Th 2:00-3:15

PAS 416

Instructor: Prof. Juan M. Restrepo

Office: Math 707

Office Hours: Tu 3:30 - 5:00 in Math707. Other hours, by appointment

Phone: 621-4367

E-mail: restrepo@math.arizona.edu

CLASS URL: http://www.physics.arizona.edu/~restrepo/577/577.html

Demand for scientists/engineers with skills in estimation, data assimilation, and inverse modeling is increasing. The kinds of problems in which this type of mathematics technology plays an important role come from economic and metereological forecasting, tomography in seismic and medical problems, optimization of parameters in models and in control devices.

This course is directed at making scientifically sensible deductions from the combination of observations with dynamics and kinematics represented, generically, as "models". The present course will focus on data assimilation, via Kalman, and Variational/Adjoint and their weakly nonlinear/non-Gaussian generalizations. We will also present particle methods and Monte Carlo methods that can handle the nonlinear/non-Gaussian cases. We will also cover singular value decomposition, regression, objective mapping.

The course can be considered a continuation of Prof. Richardson's course Geos/Phys/Atmo 567, Inverse Methods in Geophysics

Book: Discrete Inverse and State Estimation Problems by Carl Wunsch. The new edition of this book is not yet in print. I will be providing material to the class members.

book

Pre-requisites: Calculus and linear algebra, basic elementary probability theory. The course will be non-rigorous and accessible to geoscience/hydrology/atmospheric science graduate students with a solid background in the above prerequisites.

577 Listserv Subscription Instructions



ASSIGNMENTS: There will be about 5 sets.

  • Least Squares

    Homework 1 (PDF)

    Homework 1 Solution (matlab)

  • SVD

    Homework 2 (PDF)

    dna1.dat

    dna2.dat

  • Gauss-Markov and Kalman Filters

    Homework 3 (PDF)

  • Monte Carlo image denoising and Monte Carlo Parameter estimation

    Homework 4 (PDF)

  • READING ASSIGNMENT

    Homework 5 (PDF)



    TOPICS AND ASSIGMENT SCHEDULE:

  • Linear Algebra Review. This, you need to do on your own. I would strongly suggest getting a book that emphasizes geometry. A very good example is Gil Strang's Linear Algebra and its Applications .
  • Statistics Review. See for a list of topics . Download the jpg files from the lectures by clicking here. Download the pdf file on the derivation of the Kolmogorov Forward Equation by by clicking here.

    PART I BACKGROUND

  • Least Squares
  • Singular Value Decomposition
  • Assignment II
  • Least Squares and Adjoints
  • Gauss/Markov Estimation, Interpolation
  • Recursive Improvements
  • EOF's
  • Assignment III

    PART II DATA ASSIMILATION

  • Kalman Filter
  • Pontryagin Principle, the Adjoint, 3 and 4D-var
  • Assignment IV
  • The Representer Method (optional)
  • Markov Chains, transition probabilities, the Fokker Planck Equation
  • Sampling Techniques and Monte Carlo
  • Particle Methods (Monte Carlo)
  • Nonlinear Techniques: enKF, EKF, Path integral method.
  • Assignment V
  • A little matlab script that finds the minimizer. The underdetermined case.
  • Schematic MCMC in the form of pseudo-code

    Prof. R. Richardson, Geosci U. Arizona, has lent me the following notes which you will find extremely useful. They cover the deterministic portion of this course.

    Some notes on Markov Chain Monte Carlo and Gibbs sampling. These were obtained from the web (see notes for author).

    Some notes on distribution functions and their empirical derivation, from B. Van Zeghbroeck. These were obtained from the web (see notes for author).

    The path integral presentation: click here

    The PR workshop codes click here