Tutorial on Inverse Problems: Theory, Algorithms, and Hands-on Experience

Tuesday, June 7, 2016 - 2:50pm - 4:05pm
Lind 305
Wilkins Aquino (Duke University)
1. Introduction and Motivation
-What are Inverse problems?
-Examples: detection of contaminant sources, image and voice recognition, medical imaging, subsurface imaging, materials identification

2. Theoretical aspects of (discrete) inverse problems
-Why are inverse problems (oftentimes) difficult to solve?
-Well-posed and ill-posed problems: existence, uniqueness, and stability of solutions
-Linear vs nonlinear inverse problems
-Singular Value Decomposition: a path to understanding inverse problems
i. What is the main idea?
ii. Truncated SVD
iii. Tikhonov regularization
-Selection of regularization parameters
i. Discrepancy principle
ii. L-Curve approach
iii. Cross-validation

3. Optimization framework for inverse problems
-Optimality conditions
-Ill-posedness in the context of optimization
-Regularization in the context of optimization

4. Infinite Dimensional Inverse Problems
-A brief description of PDE-constrained optimization
-Discretization aspects

5. Hands on Matlab Turorial: Acoustic Source Identification
-Problem description
-What we will learn:
i. How ill-posedness manifests
ii. Tikhonov Regularization
iii. Truncated SVD
iv. Truncated conjugate gradient
v. Selection of regularization parameters
MSC Code: