Inverse problems (IPs) have been traditionally considered as mathematically challenging because they are intrinsically ill-posed. There are many practical IPs in a variety of engineering disciplines requiring suitable mathematical tools for their robust/stable solution, by recovering the well-posedness typical of forward/direct problems through suitable regularization and information-acquisition/exploitation techniques. Since industry requires fast and simple algorithms for the solution of a wide variety of IPs arising in several engineering fields, this implies a growing need for users that do not have a very high degree of mathematical education.

The course will review fundamentals and main issues of IPs, then focusing on classical/state-of-the-art and recently introduced inverse solution procedures and algorithms. Applicative examples including exercises will corroborate the theoretical concepts.

Course Topics

• Introduction and basics: motivations (methodological, applicative), synthesis and design problems in engineering as IPs;
• Formulation of IPs and numerical techniques for dealing with their resolution;
• Ill-posedness and non-linearity: on the role of information in IPs;
• Ill-posedness and the need for regularization;
• Non-linearity: physical meaning, degree of non linearity, the role of a-priori/available information;
• Solution of IPs as minimization/maximization of a cost-function/functional;
• Multi-resolution and information-acquisition strategies as an effective recipe to counteract ill-posedness and non-linearity;
• Applicative examples including exercises regarding specific engineering applications.