## EOS550: Course Outline

In the physical sciences we are often confronted with the task of determining the distribution of a physical property (e.g. density, velocity, conductivity, susceptibility etc.) in a volume without sampling the volume directly. An experiment is preformed in which observations are collected. The goal is to "invert" these observations to obtain information about the model the distribution of the physical parameter of interest).

Linear Inverse Theory: This course presents the fundamentals of formulating and solving linear inverse problems when the model to be recovered is either a function or a vector of parameters. There is an emphasis on understanding the basic difficulties involved in solving any inverse problem, and on practical methodologies for obtaining answers. Topics include: constructing models which acceptably fit the data, incorporating various forms of a priori information, singular value decomposition solutions, and assessing non-uniqueness. Additionally there will be an introduction to basic elements of solving nonlinear inverse problems and parameter estimation problems. A majority of the assigned class problems will be solved with MATLAB, and a tomographic inverse problem is used as a theme throughout the course. Practical examples will be drawn from various areas of geophysics. Students are encouraged to bring their own inverse problems to add diversification and relevance.

Prerequisites: Solid background in linear algebra. Students are assumed to have completed a B.Sc in a scientific discipline. Please contact Dr. Doug Oldenburg (Dept of Earth and Ocean Sciences, 822–5406, doug@eos.ubc.ca).

Outline of topics:

1. Overview to linear inverse theory
2. Review of essential mathematics
3. Minimum norm construction using accurate
4. Discretization and Singular Value Decomposition
5. Linear inverse problems with inaccurate
6. Choosing a regularization parameter: (GCV,
7. Basics for solving nonlinear inverse problems
8. Tuning the inversion algorithm: applications
9. Calculation of sensitivities and Frechet derivatives
10. Inverse problems with partial differential
11. General measures of misfit and model norm
12. Parameter estimation and uncertainty anlaysis
13. Resolution analysis and uncertainty in inverse

1. Problem Assignments: 50%
2. Project: 50%

- Feb 21: A 1–page description about proposed project.
- April 11: final project due.

## References

### Inverse Theory

1. R.L.Parker, 1994, Geophysical Inverse Theory, Princeton Academic Press
2. P.C. Hansen, 1998, Rank-Deficient and Discrete Ill-posed Problems, SIAM
3. J.Scales and M.Smith, Geophysical Inverse Theory. (//samizdat.mines.edu/
4. C.L.Lawson and R.J.Hanson, 1974, Solving Least Squares Problems. Prentice_hall, Englewood Cliffs, New Jersey
5. A. Tarantola, 1987, Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation, Elsevier
6. A.N.Tikhonov and V.Y.Arsenin. Solutions of Ill-Posed Problems. John Wiley, New York, 1977.
7. C.Vogel, 2002, Computational Methods for Inverse Problems, SIAM
8. W. Menke, Geophysical Data Analysis: Revised Edition, volume 45 of International Geophysics Series. Academic Press, San Diego, 1989.
9. R. Aster, B. Borchers, and C. Thurber, Parameter Estimation and Inverse Problems, in press (I have a copy)
10. D.W.Oldenburg, Y. Li, F.Jones, Inversion for Applied Geophysics, UBC-Geophysical Inversion Facility, (betatest version available on CD-ROM)
11. E. Haber, 1997, Numerical Strategies for the Solution of Inverse Problems, PhD thesis, Institute for Applied Mathematics, UBC. (www.iam.ubc.ca/theses/index.html)

### Optimization and Numerical Solutions

1. J.E. Dennis and R.B.Schnabel, 1996, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM
2. C.T.Kelley, 1999, Iterative Methods for Optimization, SIAM
3. D. Luenberger, 1969, Optimization by Vector Space Methods. John Wiley and Sons.
4. M. Hestenes., 1980, Conjugate Direction Methods in Optimization. Springer-Verlag.
5. J. Nocedal, and S.J.Wright, 1999, Numerical Optimization, Springer
6. P.Gill, W.Murray and M.Wright., 1981, Practical Optimization: Academic Press.
7. Y.Saad, 1996, Iterative Methods for Sparse Linear Systems, PWS Publishing Co.

### Websites

1. UBC-Geophysical Inversion Facility: (www.eos.ubc.ca/ubcgif)
2. Neos Guide: (www.mcs.anl.gov/otc/Guide/)