LEARNING OUTCOMES

The course allows students to acquire basic knowledge, both theoretical and applicative, relating to the study of inverse problems. The student will in fact be able to theoretically model the class of ill-posed problems deriving from the inversion of linear operators and to apply the main numerical regularization methods, both analytical and stochastic, to these problems. For this purpose, together with lectures on the theory, computational laboratory activities are planned.

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).