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.
Course Prerequisites
Programming, Comput. Math., Numerical Analysis, Linear Algebra, Calculus.
Teaching Methods
Classroom teaching; practical exercises (on blackboard)
Assessment Methods
Oral exam (possibly accompanied by a seminar and intermediate exams)
Contents
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
More information
for discussing with the professors, please use email: stefano.serrac@uninsubria.it
