Numerical Astronomy 1, Inverse problems
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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Revision 1.3 2005/09/01 14:57:04 norman
Converted to XML
Added a Creative Commons licence.