OVERVIEW

The course deals with the mathematical theory of regularizing methods, both deterministic character of stochastic, for solving ill-posed problems associated with inverse problems in real applications.

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 aims to mathematically define the class of ill-posed problems arising from the inversion of linear and non-linear operators, and to give an overview of the main numerical methods, analytical and Monte Carlo, for the resolution of these problems by regularization. Examples of inverse problems are the reconstruction of astronomical digital images, computed tomography for biomedical and civil applications, satellite remote sensing, geological prospecting.

TEACHING METHODS

Teaching style: In presence. Part of the course will be held in the computer laboratory.

SYLLABUS/CONTENT

Linear operators in Hilbert spaces:closed and non closed range operators. Ill-posed problems, generalized solution. Compact operators. Singular system and regularization methods: regularization algorithms in the sense of Tikhonov. 

Iterative methods: the Landweber method and the conjugate gradient method. Choice of the regularization parameter.

Problems of image reconstruction and of image deconvolution.Regularization methods are analyzed using the tools already exposed adapted its Fourier analysis

Statistical approach to inverse problems: Maximum Likelihood and Bayes Theorem.

Monte Carlo methods for non-linear inverse problems: importance sampling and Markov Chain Monte Carlo.

Methods for dynamic inverse problems: Kalman and particle filtering.

The course also includes numerical experiments with Matlab.

RECOMMENDED READING/BIBLIOGRAPHY

M.Bertero, P. Boccacci, 1998, An Introduction to Inverse Problems in Imaging (IOP, Bristol)

C.W.Groetsch, 1977, Generalized Inverses of Linear Operators (New York and Basel: Marcel Dekker Inc., USA)

Robert and Casella. Monte Carlo Statistical Methods. Springer, 2004.

TEACHERS AND EXAM BOARD

Exam Board

CLAUDIO ESTATICO (President)

ALBERTO SORRENTINO (President)

FEDERICO BENVENUTO

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

INVERSE PROBLEMS AND APPLICATIONS

EXAMS

EXAM DESCRIPTION

Oral exam.

ASSESSMENT METHODS

Oral examination, with previous evaluation of an exercise in computer laboratory.

FURTHER INFORMATION

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

operator theory in Hilbert spaces;
probability theory;
theory of Markov chains in discrete time.

Attendance: Optional.