Statistical and Computational Inverse Problems, APAM 6901, Fall 2008
This course will present an introduction to inverse problems. We adopt a Bayesian viewpoint, which allows us to deal with ill-posedness. Essentially, an inverse problem is ill-posed if small changes in the measured data result in large changes to the reconstructed function.
Days/Times: Tuesday, Thursday, 11- 12:15
Instructor: Ian Langmore
Instructor's website: www.columbia.edu/~il2176
Textbook: Statistical and Computational Inverse Problems by Kaipio and Somersalo. Software: MATLAB
Prerequisites: Strong linear algebra skills. Ability to write basic scientific code.
Further Reading (in no particular order)
Monte Carlo Statistical Methods by Robert and Casella. A comprehensive introduction and advanced topic text. Well-written, with theorems, proofs, and examples. Assumes graduate level knowledge of probability/statistics.
Draper's online books. In particular, Bayesian Modeling, Inference and Prediction provide an introduction with many examples (in my opinion, the examples are too complex and the theory is way too limited).
Numerical Optimization by Nocedal and Wright. A comprehensive introduction to optimization. Recommended to me by Kui-Ren (an optimization guru).
IP Course Notes By Guillaume Bal. An introduction to deterministic inverse problems, with an emphasis on imaging.
IP Course Notes by Fox and Nicholls. A Bayesian approach to inverse problems. We could have used these for the entire course.
Bayesian Data Analysis by Gelman. A standard statistics introduction. Covers quite a bit.
Weekly Outline
|
Week |
Lecture/Reading |
HW |
Supplimentary Reading |
|
|
Part 1: Basics of Ill-Posed Problems |
|
|
|
1 |
Ill-posed problems and noise. Ch. 1 Truncated SVD Regularization. Ch. 2.1, 2.2 |
Chapter 8 in [4]. |
|
|
2 |
Tikhonov Regularization. Ch. 2.3 Truncated Iterative Methods. Ch. 2.4 |
HW2. |
|
|
3 |
Implementing these methods with MATLAB HW2 explained Conjugate gradient methods explained |
|
Nocedal/ Wright, [3] |
|
|
Part 2: Intro to Statistical Inversion Theory |
|
|
|
4 |
Review of probability, Appendix B, and on line notes.
|
||
|
5 Sept 30, Oct 2 |
Introduction to Bayesian Modeling, Ch. 3.1 |
|
|
|
6 Oct 7, 9 |
Estimators, Ch. 3.1.1 The likelihood function, basic noise models. Ch. 3.2 |
|
|
|
7 Oct 14, 16 |
Prior Models. Ch. 3.3 |
|
|
|
8 Oct 21, 23 |
Gaussian Densities, Ch. 3.4 |
||
|
|
Part 3: MCMC, noise modeling, estimation theory (no more h.w.). Project presentations. |
|
|
|
9 Oct 28, 30 |
Introduction to Markov Chains. Ch. 3.6 |
|
|
|
10 Nov 4,6 No class Tues |
Markov Chain Monte Carlo (MCMC) methods. (continue introduction, introduce algorithms) Ch. 3.6 |
|
|
|
11 Nov 11,13 |
Metropolis-Hastings algorithm Ch. 3.6.1, 3.6.2 Rate of convergence, [1] |
|
Casella/Robert [1], and Draper [2] Chapters 7-9 from Fox/Nicholls [5] |
|
12 Nov 18, 20 |
Diagnosing convergence (ch. 9 in [5]) Importance Sampling, [1] |
|
Gelman's "Bugs" package (in R), see also [6]. |
|
13 Nov 25, No class Thursday |
Noise Modeling Ch. 5.8, 7.6 |
|
|
|
14 Dec 2, 4 |
Project Presentations: 10 minute presentation, 5 minute setup, Q&A A projector will be provided, but chalkboard presentations are just fine. Make your presentation interesting and use the same notation we have been using in clas. |
|
|
|
|
---------No class finals week-------------- |
|
|