Discretization of elliptic problems. Introduction to the reconstruction of blurred images. Spectral factorizations of structured matrices (Toeplitz, Hankel, circulant). Direct methods for sparse linear systems (basics). Iterative methods for linear system: relaxed Richardson iteration; gradient method. Krylov methods: conjugate gradient iteration; GMRES iteration; Arnoldi process; preconditioning.
Hansen, Nagy, O’Leary, “Deblurring Images. Matrices, Spectra and Filtering”, Fundamentals of Algorithms, SIAM, Philadelphia, 2006.
Kelley, "Iterative Methods for Linear and Nonlinear Equations", Frontiers in Applied Mathematics, v. 16, SIAM, Philadelphia, 1995.
Saad, “Iterative Methods for Sparse Linear Systems”, II ed., SIAM, Philadelphia, 2003.
Learning Objectives
Knowledge acquired:
In applied mathematics, the need to solve linear systems of equations involving a very large number of unknowns occurs frequently. Aim of the course is to present numerical methods that can be used to solve such problems efficiently.
Purpose of the course is also to discuss implementation issues for the numerical methods under study, and to show how the algorithms perform, via a limited number of computational examples developed in Matlab. These examples arise from the discretization of some elliptic problems and in the field of image deblurring.
Competence acquired:
Understanding of the linear algebra behind many applied problems, like the reconstruction of blurred images, and of the numerical methods for solving large-scale linear systems.
Skills acquired (at the end of the course):
Ability to develop simple programs and to use software available in Matlab in order to solve some large problems arising from the discretization of elliptic equations and in the field of image deblurring. Understanding of the obtained numerical results.
Prerequisites
Courses recommended: first level courses of numerical analysis or numerical linear algebra, and Matlab.
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 150
Hours reserved to private study and other indivual formative activities: 102
Contact hours for: Lectures (hours): 30
Contact hours for: Laboratory (hours): 18
Further information
Frequency of lectures, practice and lab: Recommended
Teaching Tools
UniFi E-Learning: http://e-l.unifi.it
Office Hours:
Wednesdays, 14.00-16.00 or by appointment.
Dipartimento di Energetica "Sergio Stecco".
Viale Morgagni, 40/44 - 50134 Firenze
Tel. 055 4796716
Fax. 055 4796744
alessandra.papini@unifi.it
Type of Assessment
Oral
Course program
Finite difference discretization of elliptic problems (basics). Introduction to the reconstruction of blurred images: representation and storage of an image; the blurring function; Toeplitz, Hankel, circulant matrices and spectral factorizations; regularization by spectral filtering. Direct methods for large-scale sparse linear systems (basics). Iterative methods for large-scale linear system: basics on stationary and nonstationary methods; relaxed Richardson iteration; gradient method. Introduction to Krylov methods: minimization properties and finite termination; conjugate gradient iteration; GMRES iteration; Arnoldi process; implementation details and practical behaviour; preconditioning techniques. Computational examples using Matlab: simple elliptic problems; reconstruction of blurred images.