Inverse problems

Know how to identify an inverse problem, understand the concept of ill-posed or ill-conditioned problems, understand the importance of regularization, master a few regularization methods, formulate regularization in a Bayesian context.

Course outline

·        Introductory example: deconvolution of sparse signals

◦    Direct modeling

◦    Naive inversion and least squares solutions

◦    Sparse regression (MP, OMP)

·        Characterization of inverse problems

◦    Ill-posed problems

◦    Conditioning

◦    SVD-based solutions

·        Regularization/penalization

◦    Penalized and constrained formulations

◦    Tikhonov regularizations

◦    Parcimonious regularizations

·        Probabilistic formulation

◦    Inversion and estimation

◦    Linear Gaussian case

◦    Bayesian regularization

·       (Digital) signal processing

·        Linear algebra

·        Probability

·        Statistics (estimation)

·        Advanced statistics (Bayesian estimation)