## 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 of the main regularization methods.

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 50 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. Discrete Fourier transform.

• Statistical approach to inverse problems: Maximum Likelihood and Maximum A-Posteriori.

• Methods for the choice of the regularization parameter:

- redictive risk
- Generalized Cross Validation
- L-curve.

• Extension of the Tikhonov regularization to the dynamic case: the Kalman filter.

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

**Office hours:** By appointment via email.

**Office hours:** By appointment

Exam Board

CLAUDIO ESTATICO (President)

ALBERTO SORRENTINO

FEDERICO BENVENUTO (President Substitute)

## LESSONS

TEACHING METHODS

The teaching activity is in presence and consists of:

• traditional lectures, for a total of 50 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.

LESSONS START

The class will start according to the academic calendar.

Class schedule

## EXAMS

EXAM DESCRIPTION

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

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.