Course Details for A.Y. 2019/2020

The information on this page refer to the academic year 2019/2020. Therefore, several links are not available and data may be incomplete. To read the current information on the course, if it is still available, go to the university course catalogue . To read information relating other academic years, use the list at the bottom of this page.

A. Year:

2019/2020

Name:

Numerical Methods For Linear Algebra And Optimisation / Numerical Methods For Linear Algebra And Optimisation

Type:

Course

Basic information

Code:

DT0312

Sector:

MAT/08

Credits:

: Master Degree in Mathematical Engineering 6 CFU (c)

Term:

2^nd semester

Degree(s):

Master Degree in Mathematical Engineering 1^st anno curriculum Comune Elective

Language:

English

Teacher(s):

Raffaele D'Ambrosio

Course Objectives

This is a course on numerical linear algebra and numerical optimization. Together with the theoretical inspection on the numerical methods and their features, the course is also intended to provide the basic tools to develop a reliable software implementing the studied methods and analyze the obtained results on a selection of significant test problems.

Course Content

Relevant matrix factorizations: back to LU decomposition, Cholesky decomposition. Singular value decomposition and applications (image processing, recommender systems). QR decomposition and least squares. Householder triangularization. Conditioning and stability in the case of linear systems.
Eigenvalue problems. Reduction to Hessemberg form. Rayleigh quotient, inverse iteration. QR algorithm with and without shift. Jacobi method. Givens-Householder algorithm.
Iterative methods for linear systems. Overview of iterative methods. Arnold iterations, Krylov iterations. GMRES. Lanczos method. Conjugate gradient.
Introduction to numerical optimization. Continuous versus discrete optimization. Constrained and unconstrained optimization. Global and local optimization. Overview of optimization algorithms. Convexity.
Line search methods. Convergence of line search methods. Rate of convergence. Steepest descent, quasi-Newton methods. Step-length selection algorithms.
Trust region methods. Cauchy point and related algorithms. Dogleg method. Global convergence. Algorithms based on nearly exact solutions.
Conjugate gradient methods. Basic properties. Rate of convergence. Preconditioning. Nonlinear conjugate gradient methods: Fletcher-Reeves method, Polak-Ribiere method.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

acquire the knowledge of most meaningful numerical methods for the linear algebra and optimization, as well as of their implementation in an accurate and efficient mathematical software.

Prerequisites and Learning Activities

Basic Numerical Analysis and matrix theory.

Assessment Methods and Criteria

Oral examination and a final project consisting in applying the developed software on a selection of test problems. The final project will be discussed in the same day of the oral examination.

Textbooks

J. Stoer, R. Bulirsch, Introduction to numerical analysis , Springer. 2002.
J. Nocedal, S. J. Wright, Numerical optimization , Springer. 1999.

Course page updates

This course page is available (with possible updates) also for the following academic years:

To read the current information on this course, if it is still available, go to the university course catalogue .

Course information last updated on: 16 ottobre 2017, 18:44