OVERVIEW

“Inverse problems" indicates a large class of problems in which the measurement of some effects allows to calculate their causes. The course deals with the mathematical theory of regularization methods for the solution of inverse problems, which are modelled by linear operators between Hilbert spaces, representative of the "cause-effect" maps. The solution of these problems is important in real applications, such as signal processing and learning from examples.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to define the ill-posed problems resulting from the inversion of linear operators and to give an overview, both theoretical and applied, of the main regularization methods. The student will be able to identify the class of inverse problems associated with the inversion of linear operators and to apply to these problems the main numerical regularization methods, both analytical and stochastic. Together with lectures, computational laboratory activities are planned. Important examples of inverse problems in the application area are biomedical imaging methods (CT, Computerized Axial Tomography), satellite remote sensing in climatology, oceanographic acoustic tomography and non-destructive analysis in civil engineering, image reconstruction, automatic learning from examples.

AIMS AND LEARNING OUTCOMES

The course allows students to understand the basic mathematical tools for the solution of linear inverse problems. To this end, along with lectures on the regularization theory, computer lab activities are planned.

At the end of the course the student will have acquired sufficient theoretical knowledge:

• to identify the main mathematical models associated with ill-posed problems;

• to manage functional analysis tools, such as the theory of linear operators in infinite-dimensional spaces, to solve inverse problems;

• to understand and classify Tikhonov regularization methods and iterative regularization methods;

• to understand the probabilistic approach to the regularization of inverse problems;

• to understand the optimality criteria for the best approximation;

• to use the techniques for estimating the optimal approximation, both deterministic and statistical;

• to solve linear inverse problems with the use of spectral regularization coupled with the optimal choice of the regularization parameter;

• to apply numerical methods to problems of image deconvolution and dynamic inverse problems, in which the unknown varies over time.

PREREQUISITES

All the mathematical tools to understand the arguments are given in the course. For an in-depth understanding it can be useful to have some basis of:

theory of linear operators in Hilbert spaces;

iterative methods for linear systems;

probability theory and statistic;

TEACHING METHODS

The teaching activity is in presence and consists of:

• traditional lectures, for a total of 42 hours, in which the subjects are introduced and explained in their classical theoretical setting;

• an additional 10 hours of computational laboratory activity, in which the theoretical tools are applied to the resolution of some applicative inverse problems.

Although attendance is optional, it is strongly recommended.

SYLLABUS/CONTENT

The program focuses on the following main topics:

• Linear operators on Hilbert spaces: operators with closed and non-closed ranges, compact operators and spectral resolution of self-adjoint operators.
• Ill-posed problems, generalized inverse. Case of compact operators. Singular system.
• Regularization methods: regularization algorithms in the sense of Tikhonov, theoretical study by spectral resolution.
• Regularization iterative methods: Landweber-Fridman method and conjugate gradient.
• Problems of image reconstruction and deconvolution. The previously introduced regularization methods are analyzed in the context of Fourier analysis. 
• Statistical approach to inverse problems and choice of the regularization parameter.
• Bayesian approach and Maximum Likelihood estimation.
• Predictive risk, Generalized Cross Validation, L-curve for Gaussian noise.
• Karush Khun Tucker optimality conditions and Expectation-Maximization method for Poisson noise.

Laboratory numerical computation with MatLab language will be done during the course.

RECOMMENDED READING/BIBLIOGRAPHY

In general, the notes taken during class lessons and some downloadable materials from the course web page are sufficient. In addition, the following texts may be useful:

• M.Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, Bristol, 1996)
• C.W.Groetsch, Generalized Inverses of Linear Operators (New York and Basel: Marcel Dekker Inc., USA, 1997)
• H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer academic Publishers, 1996)
• C. Vogel, Computational methods for inverse problems (SIAM, 2002).

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar, please see

https://corsi.unige.it/corsi/11907/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam is oral. In some cases, a computational laboratory activity may be discussed.

Students with DSA certification ("Specific Learning Disorders"), see section FURTHER INFORMATION.

ASSESSMENT METHODS

The oral exam focuses on the theory, and aims to ascertain its understanding, also through the discussion of the analytical concepts and the examples. In some cases it will also be possible to evaluate a written laboratory report.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it

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Ask professors for other information not included in the teaching schedule.