APPROXIMATION METHODS A
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
Programming, Comput. Math., Numerical Analysis, Linear Algebra, Calculus.
Oral exam (possibly accompanied by a seminar and intermediate exams)
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
Borrowed from
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