APPROXIMATION METHODS A

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2019/2020
Year: 
1
Academic year in which the course will be held: 
2019/2020
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Programming, Comput. Math., 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.

Lineari positive operators (LPO). General properties and concrete examples

The Korovkin Meta-Theorems I, II, III, IV

Bernstein polynomials in dimension d as LPOs and the Weierstrass Theorems (algebraic version)

Toeplitz matrices as LPOs. Spectral analysis of Toeplitz matrices generated by a symbol

Cesaro sums as LPOs and the Weierstrass Theorems (trigonometric version)

Acceleration of the convergence via extrapolation; Jackson Theorems and optimal approximation

Estrapolation techniques in the LPO setting (Bernstein and Cesaro)

Singular values and SVD (singular value decomposition)

Fronenius optimal approximation as LPO: the Toeplitz case and the trigonometric algebras (FFT and related transforms)

Korovkin Theorem in the Toeplitz-Frobenius setting
Conjugate gradient method and Preconditioning

The Frobenius optimal preconditioning

LPOs and approximation of differential operators and PDEs

The specific cases of Finite Differences and Finite Elements

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

Serra-Capizzano, Stefano: notes on the Korovkin theory

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

Classroom teaching; practical exercises (on blackboard)

for discussing with the professors, please use email: stefano.serrac@uninsubria.it