66736 - Numerical Methods

Learning outcomes

At the end of the course, the student has the theoretical and practical advanced numerical instruments for the solution of some applicative problems and is able to analyze the obtained results.

Course contents

The course consists of two parts (modules), Part I and Part II, carried out sequentially and concerning two important applicative fields of advanced numerical analysis techniques. In the following, the contents of the two modules are reported.

Part I – Numerical Solution of (Linear) Inverse Ill-Posed Problems: Applications to the Restoration of Signals/Images.

• Definition of Linear Inverse Ill-Posed Problems
• Numerical Techniques for the solution, among which:
  • Truncated SVD (TSVD)
  • Tikhonov Regularization
  • Total Variation (TV) Regularization
• Criteria for the (automatic) selection of the regularization parameter
• Applications to the restoration of signals/images corrupted by blur and noise

Part II - Theoretical and practical analysis of numerical methods for unconstrained nonlinear optimization.

• Definition of optimization problem and optimality conditions.
• The steepest-descent method for quadratic problems.
• The gradient method, the Newton's method, quasi-Newton methods for nonlinar optimization methods.
• Globalization techniques: line-search and trust-region methods.
• Application to nonlinear systems and to nonlinear least-squares.
  

 

Readings/Bibliography

Part I:

• P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical aspects of Linear Inversion, SIAM, 1998.
• P. C. Hansen, J. G. Nagy, D. P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM, 2006.
• P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, 2010.

Part II:

• Nocedal, J., & Wright, S. (2006). Numerical optimization. Springer Science & Business Media.
  
• Dennis Jr, J. E., & Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations (Vol. 16). SIAM.

Teaching methods

Theoretical lessons and exercises in the computer laboratory using the Matlab software. The laboratory exercises will be partly guided by the teacher, partly solved by the students (either individually or in groups). The results of the exercises will be analyzed during the lessons and discussed by the students during the oral examination.

Assessment methods

The examination consists of the oral discussion of a project assignment carried out by the students; the project consists of two sub-projects, one for the first and one for the second part (module) of the course.

The oral examination is aimed at evaluating the ability of the student to solve numerical problems on the computer and to critically analyze the obtained results.

Teaching tools

Slides and notes from the teachers, and other electronic material (Matlab source codes, etc.)

Office hours

See the website of Alessandro Lanza

See the website of Margherita Porcelli