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

Lecturers: Andrea Massa (UNITN/DICAM), Paolo Rocca (UNITN/DICAM)

Timetable: July 2023 (to be confirmed)

Course Description:
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.

References:
[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.
 

Duration: 32 hours ( 4 credits)

Registration: in order to access the course please send an email to dicamphd [at] unitn.it