Alvise Sommariva
Alvise Sommariva
Punti per interpolazione, mesh polinomiali e formule di cubatura
Interpolation pointsets, polynomial meshes and cubature rules
In this homepage we have stored several pointsets suitable for interpolation or cubature on intervals, simplex (triangle), square, disk, sphere and tetrahedron.
A comprehensive list of cubature rules in Phyton can be found at
» Interpolation/Least squares sets
General set with low Lebesgue constant: [matlab] (last update: Jan 08, 2017, old version: [matlab]).
General set with low Lebesgue constant and Gauss-Legendre-Lobatto distribution on the side: [matlab]
General set with high absolute value of the Vandermonde matrix, i.e. (quasi-) Fekete points: [matlab]
Symmetric set with low Lebesgue constant: [matlab]
Weakly Admissible Mesh: [matlab]
Approximate Fekete points (with degree from 1 to 70): [matlab]
» Cubature
→ Note: The reference simplex has vertices (0,0), (1,0), (0,1).
Best cubature sets on the triangle (up to degree 50): [matlab]
Slobodkins-Tausch cubature sets on the triangle (up to degree 50): [matlab]
» Comparisons
For a comparison on several interpolation sets see also: [html].
For a comparison on several cubature sets see also: [html].
» Other contributions
You may find useful the M-file for the evaluation of the Vandermonde matrix w.r.t.Dubiner Legendre basis [matlab] as orthonormal basis; see also: [zip] where we test orthogonality of the basis.
Proriol-Dubiner [matlab] as orthogonal basis with its derivatives (in a form suitable for cubature, i.e. it is the transpose of the Vandermonde matrix for interpolation purposes).
» Interpolation
General set with low Lebesgue constant: [.m] (last update: Jan 08, 2017, old version: [matlab]).
General set with high absolute value of the Vandermonde matrix, i.e. (quasi-) Fekete points: [matlab]
Padua-Jacobi points with low Lebesgue constant: [matlab]
Padua-Jacobi points with high absolute value of the Vandermonde matrix: [matlab]
Padua points: [matlab]
» Cubature
→ Note: The reference square is [-1,1] x [-1,1].
» Cubature
→ Note: The reference tetrahedron has vertices is [0 0 0], [1 0 0], [0 1 0], [0 0 1]. In case of need we propose also the version with barycentrical coordinates, in which the sum of the weights is normalised to 1.
» Interpolation
Maximum Determinant (Fekete, Extremal) points on the sphere S2 (R. Womersley homepage).
Minimum Energy points on the sphere S2 (R. Womersley homepage).
Recursive Zonal Equal Area (EQ) Sphere Partitioning (P. Leopardi homepage).
» Cubature
Spherical designs: Efficient Spherical Designs with Good Geometric Properties (R. Womersley homepage).
Spherical designs: Quadrature Rules on Manifolds: Putatively Optimal Quadrature Rules on the Sphere S2 (M. Graf homepage).
Spherical designs: Spherical Designs (R. H. Hardin and N. J. A. Sloane homepage). Each file stores a unique vector v, so that x=v(1:3:end), y=v(2:3:end), z=v(3:3:end) and weights are equal.
Albrecht-Collatz rules: [matlab].
Heo-Xu rules: [matlab].
McLaren rules: [matlab].