MTH 656 - Sec 010
Numerical Analysis
(Computational Methods for Inverse Problems)

MWF 9:00-9:50
STAG 412
Spring 2022


Professor:

Dr. Nathan Louis Gibson  

Office:

Kidd 056

Office Hours:

MW 12:30-1:20

Course Website:

http://math.oregonstate.edu/~gibsonn/Teaching/MTH656-010S22

Text Book:

Curtis R. Vogel, Computational Methods for Inverse Problems
($54.95 with SIAM member discount).

Optional Text:

C. T. Kelley, Iterative Methods for Optimization (download from here for free)

Optional Text:

Johnathan M. Bardsley, Computational Uncertainty Quantification for Inverse Problems

Note:

All above texts are published by SIAM, and current SIAM members get 30% off list price.
So be sure to sign up for your free membership!

Course Description

This course is concerned with the numerical solution of inverse problems, including numerical optimization of cost functionals. Inverse problems, for example parameter estimation problems, attempt to determine certain quantities in a model based on measurements of related quantities. These types of problems are ubiquitous in many applications, including biomedical or seismic imaging, optimal design of systems or circuits, and ecological or environmental modeling. Specific topics covered in the course will include ill-posedness and regularization, image de-noising, numerical optimization methods (including convergence analysis, implementation concerns, and dealing with noisy functionals), maximum likelihood estimation, parameter identification, sensitivity analysis, uncertainty quantification, discrete optimal control, and constrained optimization. The course will provide an introduction to optimization theory, a working understanding of several numerical solution methods, and MATLAB sample solutions to examples of applications. The content of the course is largely self-contained. For general pre-requisites, students should have some familiarity with linear algebra (including SVD) and numerical analysis. Course requirements will be two or three homework assignments, which will consist of a mixture of analytical work and numerical computations with MATLAB, and a term project on a topic chosen by the student.

Grades

Grade Distribution

Homework 50%
Final Project50%
Total 100%

Grades will be posted online on Canvas.

Matlab

Matlab is required for this course. Matlab is preferred due to the integration of computation and visualization, and the fact that the text book authors provide support. Codes for the optional text are also available.

Oregon State University has subscribed to a Total Academic Headcount (TAH) Site License for MATLAB. This new licensing includes many, but not all MATLAB toolboxes. OSU faculty, staff and students can install on up to 4 personally-owned devices or computers. For more information visit Information Services -- MATLAB or matlab.mathworks.com.

The following are online resources for learning Matlab:

Homework

Homework is required for this course. There will be three short assignments, and they will be posted on Canvas. Problems will reinforce theoretical and computational concepts from lecture. Students are encouraged to work together, but must turn in individual papers.

Note: [V] means Kurt Vogel's book, [K] means Tim Kelley's book, [B] is Bardsley, [TB] is Trefethan-Bau, and [D] is Demmel.

Final Project

A computational project is required for this course. Students must work individually on a topic/problem of their choice involving inverse problems (can be from research/thesis work). Students must submit a typed (less than or equal to two pages) research proposal, including questions to be answered midway through the course. Final papers will be submitted as a typed report including tables or graphs as figures with captions, and references to them inside the body of the text, and a bibliography. Students will give brief (10 minute) presentations on results during the last days of classes in lieu of a final exam.

Supplements

Introduction to Numerical Analysis (from Atkinson-Han):
Sec 1.1
Sec 1.2
Sec 1.3
Math Modeling

Presentations:
Gradient-based Methods for Optimization: Part 1 & Part 2
Approximating Dispersive Mechanisms Using the Debye Model with Distributions of Dielectric Parameters
Finite Difference, Finite Element and Finite Volume Methods for the Numerical Solution of PDEs
Electromagnetic characterization of damage in Space Shuttle foam
PDE Constrained Optimization with Uncertain Data by Matthias Heinkenschloss (Rice University).
Estimation of Distributed Parameters in Permittivity Models by H.T. Banks (North Carolina State University)

Samples:
Files for creating plots in Optimization presentation
Script demonstrating action on unit ball
Script demonstrating SVD image compression
Fancier version from Cleve Moler
Sample Beamer Presentation Sample Latex Report