Alvise Sommariva

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

We discuss and compare two greedy algorithms that compute discrete versions of Fekete-like points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the so-called approximate Fekete points by QR factorization with column pivoting of Vandermonde-like matrices. The second computes discrete Leja points by LU factorization with row pivoting. Moreover, we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes.

Computing approximate Fekete points by QR factorizations of Vandermonde matrices

We propose a numerical method (implemented in Matlab) for computing approximate Fekete points on compact multivariate domains. It relies on the search of maximum volume submatrices of Vandermonde matrices computed on suitable discretization meshes, and uses a simple greedy algorithm based on QR factorization with column pivoting. The method gives also automatically an algebraic cubature formula, provided that the moments of the underlying polynomial basis are known. Numerical tests are presented for the interval and the square, which show that approximate Fekete points are well suited for polynomial interpolation and cubature.

Gauss-Green cubature and moment computation over arbitrary geometries

We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2n-1 using N~cmn^2 nodes, 1@?c@?p, m being the total number of points given on the boundary. It does not need any decomposition of the domain, but relies directly on univariate Gauss-Legendre quadrature via Greens integral formula. Several numerical tests are presented, including computation of standard as well as orthogonal moments over a nonstandard planar region.

Weakly Admissible Meshes and Discrete Extremal Sets

We present a brief survey on (Weakly) Admissible Meshes and corresponding Discrete Extremal Sets, namely Approximate Fekete Points and Discrete Leja Points. These provide new computational tools for polynomial least squares and interpolation on multidimensional compact sets, with different applications such as numerical cubature, digital filtering, spectral and high-order methods for PDEs.

Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the so-called Padua points on rectangles, and the corresponding versions for algebraic cubature.

Computing Fekete and Lebesgue points: Simplex, square, disk

We have computed point sets with maximal absolute value of the Vandermonde determinant (Fekete points) or minimal Lebesgue constant (Lebesgue points) on three basic bidimensional compact sets: the simplex, the square, and the disk. Using routines of the Matlab Optimization Toolbox, we have obtained some of the best bivariate interpolation sets known so far.

Least-squares polynomial approximation on weakly admissible meshes: Disk and triangle

We construct symmetric polar WAMs (weakly admissible meshes) with low cardinality for least-squares polynomial approximation on the disk. These are then mapped to an arbitrary triangle. Numerical tests show that the growth of the least-squares projection uniform norm is much slower than the theoretical bound, and even slower than that of the Lebesgue constant of the best known interpolation points for the triangle. As opposed to good interpolation points, such meshes are straightforward to compute for any degree. The construction can be extended to polygons by triangulation.

Approximation on the sphere using radial basis functions plus polynomials

In this paper we analyse a hybrid approximation of functions on the sphere

Numerical Cubature on Scattered Data by Radial Basis Functions

{\mathbb S}^2 \subset {\mathbb R}^3

Adaptive Bivariate Chebyshev Approximation

by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation.