Discrete Inverse Problems

## Discrete Inverse Problems: Insight and Algorithms

This book is
published by SIAM in the series Fundamentals of Algorithms.
It has 8 chapters, 46 exercises, and 213 pages. It was on SIAM's
bestseller list three years in a row in
2010,
2011 and
2012.
The book uses the software from
Regularization Tools.

Misprints as of April 15, 2015:

- Page 16, line 14 from bottom: "and the two types" should be "the two types"
- Page 24, line 9 from bottom (displayed equation):
*x*_{j} should be *t*_{j}
- Page 27, last line: "we have no information"
- Page 31, line 5: "note that"
- Page 49, third paragraph in Exercise 3.5 should read:

"Then compute three different exact solutions and the corresponding
right-hand sides by means of the three examples provided in
`gravity`. Notice that ..."
- Page 49, line 12 from bottom: after "for
*n* = 32" add
"and *d* = 0.25.
- Page 49, line 9 from bottom: at the end of the exercise add
"Start with ||
*e* ||_{2} = 10^{-10}."
- Page 80, line 10 from bottom: after "
*n* = 32" add "and
example = 3".
- Page 81, third paragraph of Exercise 4.5: "use
`loglog` to
plot the corresponding L-curve."
- Page 92, line 5 from botom (displayed equation): in the denominator
change λ
^{2} to λ^{4}.
- Page 96, first displayed equation:
*I*_{n-1} should be
*I*_{n}.
- Page 99, Eq. (5.13): subscript
*k* should be λ
- Page 106, Example 5.2: use
*n* = 64 and η = 0.01.
- Page 123, middle of the page: in the displayed equation for
*φ*_{i}^{(k)},
*σ*_{1}^{4} and
*σ*_{1}^{6} should be
*σ*_{i}^{4} and
*σ*_{i}^{6}, respectively.
- Page 131, line 8 from bottom should be:
`x = x - 0.5*max(x);`
- Page 133, line 11 from top: change "50" to "
*n*".
- Page 133, lines 17-18 from bottom: "the previous exercise" should be
"exercise 6.3"
- Page 136, below Eq. (7.2): "larger" should be "smaller"
- Page 137, after Eq. (7.5):
*g*_{-n+1} should be
*h*_{-n+1}
- Page 138, Eq. (7.6):
*h*_{n/2-1},..,*n*-1
should be *i* = 0,1,...,*n*-1
- Page 139, below the displayed equation: delete "blue" (twice)
- Page 168, lines 2 and 3 from bottom: superscript "(
*x*)" should
be "(*k*)" (twice).

### Videos with My Lectures

Videos with my lectures are available at
DTU Podcasts and
YouTube.
### Suggestion for a Short Course

**Lecture 1: Discrete Inverse Problems**
overheads
+ overheads
- Chapter 1
- Chapter 2: sections 2.1, 2.2, 2.3
- Chapter 3: sections 3.1.1, 3.2.1, 3.3, 3.5
- Exercises: 2.2, 2.4, 3.4, 3.5, 3.6

**Lecture 2: Regularization Methods** overheads
- Chapter 4: sections 4.1, 4.2, 4.3, 4.4, 4.7
- Exercises: 4.1, 4.5, 4.6, 4.7

**Lecture 3: Regularization Parameters** overheads
- Chapter 5: sections 5.1, 5.2, 5.3, 5.4, 5.5, 5.6
- Exercises: 4.2, 5.1, 5.2

**Lecture 4: Solving "Real" Problems**
overheads
- Chapter 7: sections 7.1, 7.2, 7.3, 7.4, 7.10
- Exercises: 7.2, 7.3 (TSVD and Tikhonov only), 7.4

**Lecture 5: Iterative Regularization**
overheads
- Chapter 6: sections 6.1, 6.2, 6.3
- Chapter 7: section 7.7
- Exercises: 6.2, 6.3, 6.5, 7.3 (iterative methods), 7.8

### Additional Material for a Longer Course

**Lecture 6: Image Deblurring** (currently no overheads)
- Chapter 7: sections 7.5, 7.6
- Exercises: 7.5, 7.6, 7.7

**Lecture 8: Beyond the 2-Norm**
overheads
- Chapter 8: sections 8.1, 8.2, 8.6
- Exercises: 8.1, 8.3, 8.4