THEORY OF APPROXIMATION AND APPLICATIONS

Second cycle degree in MATHEMATICS

Campus: PADOVA

Language: English

Teaching period: First Semester

Lecturer: STEFANO DE MARCHI

Number of ECTS credits allocated: 7


Syllabus
Prerequisites: The course requires the basic courses of Numerical Calculus and Numerical Analysis. It is also useful to have attended a course of Functional Analysis. It is assumed that the students know the programming language Matlab.
Examination methods: The final exam is a written test on the topics of the course. There will be also an oral part in which the student will discuss the lab exercises given during the course.
Course unit contents: The course can be subdvided in 2 theoretical parts of 24h each, in total 48h corresponding to 6CFU. Moreover there will be 16h of lab exercises.

PART I (24h+2h): from univariate to multivariate polynomial approximation
- best approximation approximation
- modulus of continuity and Lebesgue constant
- nearly optimal distribution of points in 1-dimension
- Padua points for interpolation and cubature
- (Weakly) admissible meshes
- applications and lab (6h)

PART II (24h+6h): Radial Basis Functions (RBF)
- learning from splines
- positive and conditionally definite functions
- native spaces, power function and error estimates
- application to the solution of elliptic PDEs
- applications and lab (10h)