Inverse Problems (Ill-Posedness and Regularization): Theory, Techniques, and Engineering Applications

Discover the fundamentals of inverse problems, and focus on the most advanced inverse solution procedures

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.


  1. F. D. Moura Neto, A. J. da Silva Neto, “An Introduction to Inverse Problems with Applications”. Springer, 2013.
  2. A. Tarantola, “Inverse Problem Theory and Methods for Model Parameter Estimation”. SIAM, 2005.
  3. R. C. Aster, B. Borchers, and C. H. Thurber, “Parameter Estimation and Inverse Problems”. Elsevier, 2013.
  4. “Microwave Imaging and Diagnostics: Theory, Techniques, and Applications”, European School of Antennas (ESoA) and European Cooperation in Science and Technology (COST Actions TD1301/TU1208), Madonna di Campiglio, Italy, 24-28 March 2014.
  5. “Microwave Imaging and Diagnostics: Theory, Techniques, and Applications”, European School of Antennas (ESoA), Madonna di Campiglio, Italy, 19-23 March 2018.
  6. “Microwave Imaging and Diagnostics: Theory, Techniques, and Applications,”  European School of Antennas (ESoA), Napoli, Italy, 1-5 February 2021.