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 -Regularization: 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 -Formulation -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