Alvise Sommariva

Course: Introduction to multivariate polynomial approximation  

General Info

• Doctoral Program in Mathematical Sciences.

• PhD course, a.a. 2026-2027, Numerical cubature and its applications.

  Scheduling

1. Nov 9, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

2. Nov 10, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

3. Nov 11, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

4. Nov 12, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

5. Nov 16, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

6. Nov 17, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

7. Nov 18, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

8. Nov 19, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

9. Nov 23, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

10. Nov 24, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

11. Nov 25, 2026 14:00-16:00, Torre Archimede, Room 2BC30:

12. Nov 26, 2026 16:30-18:30, Torre Archimede, Room 2BC30.

Introduction and background requirements

Introduction and background requirements: the course requires 

• the knowledge of the basics of measure theory and multivariate calculus,

• some concepts of numerical analysis as interpolation, least-squares and one dimensional numerical integration; where necessary, 

The lecturer will introduce elementary notions, so to broad the course to a wider audience.

Aim

• The goal is to provide some ideas on multivariate polynomial approximation on classical domains as simplex, sphere, cubes, but also on more complicated ones that arise in Finite or Virtual Elements Methods. 

• In general these topics are not treated in university courses, though they are of fondamental interest in applied sciences. 

• The outcomes are of practical and theoretical nature. 

• The student will learn how to approach numerically multivariate polynomial approximation, but also deeper mathematical details concerning error analysis. 

• Numerical experiments will be given to provide more insights on the topics.

Course contents

• univariate and multivariate interpolation (on well-choosen sets) and Lebesgue constants, 

• hyperinterpolation and variants for data denoising, 

• numerical cubature on multivariate domains: application to hyperinterpolation, FEM and VEM, 

• (weakly) admissible meshes and good interpolation/least squares pointsets (time permitting).

References

1. P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, Dover Publications, 2007. 

2. N. Levenberg, J-P. Calvi, Uniform approximation by discrete least squares polynomials, Journal of Approximation Theory, 152, Issue 1 (2008), pp.82–100. 

3. A. Sommariva and M. Vianello, Computing approximate Fekete points by QR factorisations of Vandermonde matrices, Computers Mathematics with Application, 57 (2009), pp.1324-1336. 

4. I.H. Sloan, Polynomial Interpolation and Hyperinterpolation over General Regions, Journal of Approximation Theory, 83, Issue 2 (1995), pp.238–254.

  Material

  Examination

• Oral exam on the material given during the course;

• the oral exam will be decided by the student and the teacher, depending on their duties.