**Computational Methods for Linear Inverse Problems**

**Fall 2017**- Prerequisites: mathematical analysis, advanced linear algebra, numerical linear algebra, matlab, latex
- Inverse problems are mathematical problems that arise when our goal is to recover "interior" or "hidden" information from "outside" (or otherwise available) noisy data. -- Per Christian Hansen, 2009

## Contents

## Instructor & Tutor

- Kui Du School of Mathematical Sciences, Xiamen University Email: kuidu@xmu.edu.cn Tel: 0592-2580672
- Name School of Mathematical Sciences, Xiamen University Email:

## Textbook & Reference books

- (textbook) Discrete Inverse Problems: Insight and Algorithms, Per Christian Hansen
- Computational Methods for Inverse Problems, Curtis R. Vogel
- Numerical Linear Algebra, Lloyd N. Trefethen and David Bau, III
- Deblurring Images: Matrices, Spectra, and Filtering, Per Christian Hansen, James G. Nagy, and Dianne P. O'Leary
- Rank-Deficient and Discrete ill-Posed Problems: Numerical Aspects of Linear Inversion, Per Christian Hansen
- Matrix Methods in Data Mining and Pattern Recognition, Lars Elden
- Introduction to Scientific Computing and Data Analysis, Mark H. Holmes
- MATLAB Primer, Timothy A. Davis

## Lecture contents

- Chapter 0: NLA fundamentals, Singular Value Decomposition (SVD), Least Squares
- Chapter 1: Introduction and Motivation
- Chapter 2: Meet the Fredholm Integral Equation of the First Kind, Chebfun
- Chapter 3: Getting to Business: Discretizations of Linear Inverse Problems
- Chapter 4: Computational Aspects: Regularization Methods
- Chapter 5: Getting Serious: Choosing the Regularization Parameter
- Chapter 6: Toward Real-World Problems: Iterative Regularization
- Chapter 7: Regularization Methods at Work: Solving Real Problems
- Chapter 8: Beyond the 2-Norm: The Use of Discrete Smoothing Norms
- Chapter 9: Other topics depending on time: Data mining, Pagerank, Principal Component Analysis, ...

## Grading policy

- Assignment & Programming 40% + Project 20% + Final exam 40%
- Bonus 5% (overall performance: classroom and discussion participation, learning attitude, ...)
- Assignment & Programming + Project + Bonus <= 60%

## Homework

- Write solutions/programming reports in
**english and latex/matlab**(you can use matlab's 'publish'). Only one pdf-file should be submitted with filename studentnumberAx.pdf for Assignment x or studentnumberPx.pdf for Programming x. For example, if your student number is 19020140000000, then your submission for Assignment 1 is 19020140000000A1.pdf, and your submission for Programming 1 is 19020140000000P1.pdf. Submissions in other file formats are unacceptable! - Discussion is encouraged. However, transcribed solutions and copied programs are both unacceptable!
- Submit your homework to: xmulip@163.com. Late submissions get only
**half**the score. Unaccepted submissions get**0**score.**No exceptions!**

- Assignment 1.
- Assignment 2.
- Assignment 3.
- Assignment 4.
- Programming 1.
- Programming 2.
- Programming 3.
- Programming 4.

## Reading project (tentative)

- Martin Fuhry and Lothar Reichel, A new Tikhonov regularization method, NA, 2012
- Silvia Noschese and Lothar Reichel, A modified truncated singular value decomposition method for discrete ill-posed problems, NLAA, 2014
- Michiel E. Hochstenbach, Lothar Reichel, and Giuseppe Rodriguez, Regularization parameter determination for discrete ill-posed problems, JCAM, 2015

## Others

- Chebfun-numerical computing with functions by Trefethen's team
- Regularization Tools Version 4.1 by Hansen