Event Details
This event is running from 14 November 2024 until 12 December 2024. It is next occurring on 14 November 2024 13:00.
Mini-course:
An accurate polynomial approximation on equispaced nodes via an interpolation-regression approach with applications
Lecturer: Federico Nudo, Università della Calabria (Italy) & Université de Pau et des Pays de l'Adour (France).
ABSTRACT
A very common problem in computational science is determining an approximation, within a fixed interval, of a function evaluated only at a finite set of points. A popular approach to solving this problem relies on polynomial interpolation, which involves determining a polynomial that corresponds to the function at the specified points. A case of particular practical interest is when these points are equispaced within the interval. Under these conditions, a known issue with polynomial interpolation is the Runge phenomenon, where interpolation errors increase near the ends of the interval. In 2009, J. Boyd and F. Xu demonstrated that the Runge phenomenon could be mitigated by interpolating the function over a specific subset of points close to the Chebyshev-Lobatto nodes, known as mock–Chebyshev nodes. However, this strategy entails discarding most of the available data. To enhance the accuracy of Boyd and Xu’s method while fully using all data points, S. De Marchi, F. Dell’Accio, and M. Mazza introduced a technique called the constrained mock-Chebyshev least squares approximation. In this method, the nodal polynomial plays a critical role in ensuring interpolation at mock-Chebyshev nodes. Extending this approach to the bivariate case, however, requires alternative techniques. A recent development by F. Dell’Accio, F. Di Tommaso, and F. Nudo uses the Lagrange multipliers method to define the constrained mock-Chebyshev least squares approximation on a uniform grid. This innovative technique, analytically equivalent to the earlier univariate method, also proves to be more accurate in numerical terms. This mini-course will explore the main properties of this approximation operator and its applications to numerical quadrature, differentiation problems, and the histopolation method.
