KNOWLEDGE AND UNDERSTANDING At the end of the course the student will know some modern numerical methodologies in the fields of the inverse problems. APPLYING KNOWLEDGE AND UNDERSTANDING At the end of the course the student will be able to use the studied numerical methodologies in order to solve problems coming from the applications. MAKING JUDGMENTS At the end of the course the student will be able to recognize the fundamental characteristics of the considered problems and will have the ability to appropriately choose the methods to solve them. COMMUNICATION SKILLS At the end of the course the student will be able to express himself appropriately in the description of the numerical methods studied, with mastery of language and asservativeness in the presentation. LEARNING SKILLS At the end of the course the student will be able to consult the literature in the field of the inverse problems to complete his/her knowledge.

Basic knowledge of the numerical linear algebra.

Ill posed and inverse problems. The singular value expansion. The Picard condition. The SVD and its generalizations. Discrete ill posed problems. Filter factors. Discrete Picard condition. Direct regularization. TSVD. Tikhonov regularization. Iterative regularization methods. Landweber iteration. Conjugate gradient method. CGLS. Bidiagonalization and LSQR iterationd. GMRES. Hybrid methods. Parameter choice methods. Discrepancy principle. Generalized cross validation. L-curve analysis.

1. P.C.Hansen, Rank deficient and discrete ill posed problems. 2. Lecture notes

Ill posed and inverse problems. The singular value expansion. The Picard condition. The SVD and its generalizations. Discrete ill posed problems. Filter factors. Discrete Picard condition. Direct regularization. TSVD. Tikhonov regularization. Iterative regularization methods. Landweber iteration. Conjugate gradient method. CGLS. Bidiagonalization and LSQR iterationd. GMRES. Hybrid methods. Parameter choice methods. Discrepancy principle. Generalized cross validation. L-curve analysis.

Lectures, both of theoretical character and aimed to the solution of exercises, also including the design and the execution of related computer programs.

At the end of each topic of the course, the teacher will provide a copy of his own notes.

Oral exam, where the students knowledge of the various numerical techniques tought in the course, as well as the ability to suitably apply them to the considered mathematical problems, is checked.