INVERSE PROBLEMS:METHODS AND APPLICATIONS
Academic year and teacher
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- Versione italiana
- Academic year
- 2021/2022
- Teacher
- GAETANO ZANGHIRATI
- Credits
- 6
- Curriculum
- APPLICATIVO
- Didactic period
- Primo Semestre
- SSD
- MAT/08
Training objectives
- The course goal is to introduce the students to the basic theoretical aspects of linear inverse problems and some of the main numerical solution methods and algorithms. The class of inverse problems mathematically describes those scientific problems where we want to determine the unknown "cause" that has produced certain observable and measurable "effects". Many relevant real-world applications require to solve inverse problems, such as medical imaging or machine learning. For these reasons, the course plans to provide the following knowledge:
- knowledge of theoretical mathematical tools to describe the class of linear inverse problems between Hilbert spaces;
- knowledge of the theoretical bases of some of the main regularization methods, of both Tikhonov and iterative types, to solve finite-dimensional inverse problems;
- knowledge of the probabilistic interpretation of the regularization of inverse problems;
- knowledge of some software tools useful for applications.
At the end of the classes, students are expected to have acquired the following skills (meant as the ability to apply knowledge):
- be able to identify some of the main mathematical models associated with ill-posed problems;
- be able to choose the correct formulation to model simple inverse linear problems;
- be able to properly use some Tikhonov-type regularization methods and iterative regularization methods, as well as some methods of choosing the regularization parameter;
- be able to numerically solve simple problems of image deconvolution and supervised machine learning,
- be able to implement Matlab code for the solution of the aforementioned inverse problems. Prerequisites
- Elements of numerical analysis: direct and iterative numerical methods for linear and nonlinear systems, Euclidean polynomial approximation methods, integration methods.
Elements of functional analysis: Hilbert spaces, bounded operators.
Basic programming in Matlab language. Course programme
- The covered topics will be as follows (the scheduled hours are reported in brackets):
- linear inverse problems between finite-dimensional Hilbert spaces: operators, ill-position, ill-conditioning, spectral resolution, pseudo-solution, generalized solution, generalized inverse operator, singular cases; (6)
- Tikhonov-like regularization algorithms: TSVD, Tikhonov. (10)
- iterative regularizing methods: Landweber and conjugate gradient (CG) methods; (10)
- choice of the regularization parameter: Morozov discrepancy principle, generalized cross-validation (GCV), L-curve; (6)
- statistical approach to inverse problems: maximum likelihood and maximum a posteriori; (8)
- application examples: image reconstruction in computed tomography (CT), machine learning in the supervised case. (8)
This is a newly proposed course, so the hourly plan is hypothetical and could undergo some variations. Didactic methods
- The teaching activity will consist of about 40 hours of frontal lectures, dedicated to the theoretical aspects, and about 8 hours of lab activities in Matlab. The latter will be dedicated to some of the most relevant applications of the theory. Lectures will be given in classroom, unless different settings will be required by the University.
Learning assessment procedures
- The exam goal is to verify a suitable level of achievement of the training objectives of the course, with respect to both knowledge and skills, including the laboratory part in Matlab.
The exam will consist of an oral test, divided into two parts:
- the first part with questions ranging on all the topics covered in the classes;
- the second part with a discussion on lab exercises, including both those carried out in the classes and those assigned.
The exercises sources have to be delivered to the teacher before the exam.
The final score is given by the sum of the evaluation of both parts. In general, the evaluation of the first part cannot exceed 28/30. The maximum score on the second part can range between 2/30 and 10/30, depending on the amount of time actually dedicated to the lab. Therefore, the latter cannot be known in advance. However, if no Matlab exercises are delivered, the exam is compromised. Reference texts
- - Teacher's notes
- M. Bertero, P. Boccacci, "Introduction to Inverse Problems in Imaging", IOP, Bristol, 1996.
- H.W. Engl, M. Hanke, A. Neubauer, "Regularization of Inverse Problems", Kluwer Academic Publishers, 1996.
- C. Vogel, "Computational methods for inverse problems", SIAM, 2002.
- J. Mueller, S. Siltanen, “Linear and Nonlinear Inverse Problems with Practical Applications”, SIAM, 2012.
