Bernstein polynomials. Totally positive systems of functions. Classical and generalized splines: the B-spline basis, theorem of Curry-Schoenberg. Tensor-product bivariate functional spaces.
The finite element method for elliptic problems. Elements of Numerical Grid Generation. An Introduction to Isogeometric Analysis.
Basics on image deblurring. SVD and low rank approximation of linear operators. Structured matrices and spectral decompositions. Spectral filtering, Tikhonov regularization
De Boor, C. “ A Practical Guide to Splines” II ed., Springer, Berlin, 2001.
A. Quarteroni, “Modellistica Numerica per problemi differenziali”, IV edizione, Springer-Verlag, Milano, 2008.
Hansen, Nagy, O’Leary, “Deblurring Images. Matrices, Spectra and Filtering”, Fundamentals of Algorithms, SIAM, Philadelphia, 2006.
Learning Objectives
Knowledge acquired: deepened knowledge on advanced methodologies and related algorithmic issues of current interest in the Numerical Analysis field, with emphasis on their use in practical applications. 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.
Competence acquired: advanced competence in numerical linear algebra and approximation, as well as in numerical modelling of elliptic differential problems.
Skills acquired: being able to select, use and compare numerical methods suited to solve the considered problems. Being also able to evaluate the numerical results.
Prerequisites
Courses to be used as requirements (required and/or recommended)
Courses required:
Courses recommended: first level courses of numerical analysis, and Matlab.
Teaching Methods
CFU: 9
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 153
Hours for lectures: 42
Hours for laboratory: 30
Hours for laboratory-field/practice: 0
Seminars (hours): 0
Stages (hours): 0
Intermediate examinations (hours): 0
Further information
Attendance of lectures, practice and lab:
Recommended
Teaching tools: textbooks, UniFi E-Learning: http://e-l.unifi.it
Office hours:
Prof. Papini
Monday, from 3:00 to 5:00 p.m. or by appointment. Dipatimento di Ingegneria Industriale, viale Morgagni 40, 50134 – Firenze
E-mail: alessandra.papini@unifi.it
Tel. 055 4796716 Fax 055 4796744
Prof. Brugnano
Tuesday, from 10:30 to 12:30 p.m. or by appointment. Dipartimento di Matematica e Informatica “U. Dini”, viale Morgagni 67/a, 50134 – Firenze E-mail: luigi.brugnano@unifi.it
Tel. 055 2751421 Fax 055 2751452
Type of Assessment
Oral exam
Course program
Bernstein polynomials: definition, recurrence, properties. Bezie’r curves and tensor-product patches. Related algebraic-geometric algorithms. Totally positive systems of functions and their graphical use. Classical and generalized splines. The truncated power basis and the B-spline basis, Graphical applications. Interpolation and least—squares approximation with splines: the Witney—Schoenberg theorem.
An outline of Sobolev spaces. Variational formulation of an elliptic differential problem. Galerkin approximation. Lagrange formulation of finite element methods in 1 and more dimensions. Conditioning of the stiffness matrix. A posteriori error-estimates. Methods for numerical grid generation: structured 2D meshes and triangulations. An outline of isogeometric analysis and and application to a 1D structural vibration problem.
Introduction to the reconstruction of blurred images: representation and storage of an image; linear blurring operators; Toeplitz, Hankel, circulant matrices and spectral factorizations; singular value decomposition and low rank approximation of linear operators; regularization by spectral filtering; Tikhonov regularization.