# APPROXIMATION METHODS B

Degree course:
Corso di Second cycle degree in PHYSICS
Academic year when starting the degree:
2018/2019
Year:
1
Academic year in which the course will be held:
2018/2019
Course type:
Supplementary compulsory subjects
Language:
English
Credits:
8
Period:
First Semester
Standard lectures hours:
64
Detail of lecture’s hours:
Lesson (64 hours)
Requirements:

Programming, Computational Mathematics, Numerical Analysis, Linear Algebra, Calculus

Final Examination:
Orale

Oral exam (possibly accompanied by a seminar and intermediate exams)

Assessment:
Voto Finale

Understanding the complexity of a problem; ability in decomposing in into smaller and easier subproblems, by exploiting interdisciplinary tools, deriving from Numerical Analysis, Matrix Theory, Linear Algebra, and Approximation techniques in Analysis and Numerical Analysis

Definition of Structured Matrices
Examples of Structured Matrices (Vandermode, Toeplitz, Hankel, Circulants etc)
Vandermonde matrix, the interpolation problem, necessary and sufficient conditions for invertibility
Vandermonde matrix and its (asymptotic) conditioning as a function of the distribution of points
Generalized Vandermonde matrices, the special case of the Fourier Matrix
Fourier Matrix and quadrature formulae for the Fourier coefficients
Discrete Fourier Transform and the computational challenge of a fast algorithm
Algebraic properties of the Fourier Matrix, basics of the tensor calculus
Fast Fourier Transform in special dimensions: the recursive algorithm and its computational cost
Fast Fourier Transform and the direct tensor decomposition: the direct algorithm and its computational cost
Circulant matrices, algebra of matrices via the Cayley Hamilton Theorem
Circulant matrices and Fast Fourier Transform
Fast matrix vector product with Toeplitz, Hankek, g-Toeplitz, g-Hankel
Fast Fourier Transform for every matrix size
Spectral Analysis of Circulants, Toeplitz
Approximation of elliptic differential operators via Finite Differences
Approximation of elliptic differential operators via Finite Elements
Spectral analysis of Locally Toeplitz Sequences
Spectral analysis of Generalized Locally Toeplitz Sequences
Applications of approximation of differential and integral operators

Garoni, Carlo; Serra-Capizzano, Stefano
Generalized locally Toeplitz sequences: theory and applications. Vol. I. Springer, Cham, 2017.

Serra-Capizzano, Stefano: notes the Fast Fourier Transform

Serra-Capizzano, Stefano: notes on preconditioning for Toeplitz-like structures

Classroom teaching; practical exercises (on blackboard)

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Degree course in: MATHEMATICS