This page contains the lecture notes for a 10-lecture, honours level, course on Inverse Problem theory. The course introduces Inverse Problem Theory, the theory of integral equations, and describes both classical and non-classical inverse problem theory.

`http://purl.org/nxg/text/numerical-astronomy`

-- please
use this Persistent URL when citing this page.

Numerical Astronomy 1, Inverse Problems is part of the honours course offered by the Department of Physics and Astronomy in the University of Glasgow. It is presented in alternate years. I taught this option in session 1998--9, when it ran from 16 October to 11 December 1998. In session 2000--1, it was presented by Dr Richard Barrett, after which it was taken over by Dr Graham Woan.

I produced (rather compressed) course notes. These were intended to be a support to the lectures, rather than an independent exposition, but I hope they might be of some use to others, and you can download them if you wish.

If you find the notes useful (and certainly if you find any inaccuracies), please do let me know.

Other web-based material on this subject includes:

- Geophysical Inverse Theory (draft) by John Scales and Martin Smith, at Samizdat Press (there are other similar sources there), and
- Inverse Theory (Geophysics 529) by Richard Aster and Brian Borchers at New Mexico Institute of Mining and Technology, Socorro, New Mexico.

Course contents:

- Part 1: Introduction
(pdf,
ps)
- 1. Types of inverse problems, and examples; 1.1: The `interpretation problem'; 1.2: The `instrument problem'; 1.3: The `synthesis problem'; 1.4: The `control problem';
- 2. Background and reading; 2.1: Further reading.

- Part 2: Integral equations, inverse problems, and the loss of information
(pdf,
ps)
- 1. Integral equations;
- 2. Examples; 2.1: Electron spectra from bremsstrahlung photon spectra; 2.2: Instrument convolutions - spectrometer;
- 3. Error amplification and ill-posedness;
- 4. Stability and the Riemann-Lebesgue lemma; 4.1: The Riemann-Lebesgue lemma; 4.2: The null space and nearly-singular opearators.

- Part 3: Classical inversions and instability
(pdf,
ps)
- 1. Classical solutions; 1.1: Quadrature; 1.2: Product integration; 1.3: Polynomial expansion; 1.4: Singular value decomposition - SVD;
- 2. The instability; 2.1: Norms of vectors and matrices; 2.2: Instability in IP inversion; 2.3: Examples;
- 3. Example: Inversion of Abel's equation.

- Part 4: Non-classical inversion
(pdf,
ps)
- 1. Overview;
- 2. Regularisation;
- 3. Other non-classical techniques; 3.1: Backus-Gilbert; 3.2: Maximum entropy; 3.3: Bayes theorem.

- Part 5: Conclusion
(pdf,
ps)
- 1. Modelling;
- 2. Review: Choosing an IP algorithm

These notes are Copyright, 1998-2000, Norman Gray.

These notes are made available under the Attribution-NonCommercial-NoDerivs 2.5 licence, of the Creative Commons. The summary
and full text of the licence are available at `http://creativecommons.org/licenses/by-nc-nd/2.5/`

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