Chapter 2: Ill-Posed Problems and Regularization Theory

Learning Objectives

• State Hadamard's three conditions for well-posedness and identify which conditions fail for each class of imaging inverse problem
• Compute the Moore-Penrose pseudoinverse via the SVD and explain why it is unbounded for compact operators
• Analyse a regularization scheme via its spectral filter function and compute bias-variance decompositions
• Apply truncated SVD, Tikhonov, and Landweber regularization to a discrete inverse problem
• Select the regularization parameter using Morozov's discrepancy principle, the L-curve, GCV, and SURE
• Formulate LASSO and TV as variational regularization problems and interpret them as MAP estimates under specific priors
• Linearize a nonlinear forward operator and apply the iteratively regularized Gauss-Newton method