Alvise Sommariva

Sets

Last update: July 9, 2023.

In this homepage we have stored several pointsets suitable for interpolation or cubature on intervals, simplex (triangle), square, disk and polygons.

We point out that concerning cubature rules in Phyton, a comprehensive list can be found at https://awesomeopensource.com/project/nschloe/quadpy.

INTERVAL:

» Interpolation

1. General set with low Lebesgue constant: [.m]
2. Extended Chebyshev: [.m]
3. Gauss-Legendre-Lobatto (Fekete points in the interval): [.m]

SIMPLEX (TRIANGLE):

» Interpolation/Least squares sets

• General set with low Lebesgue constant: [.m] (last update: Jan 08, 2017, old version: [.m]).
• General set with low Lebesgue constant and Gauss-Legendre-Lobatto distribution on the side: [.m]
• General set with high absolute value of the Vandermonde matrix, i.e. (quasi-) Fekete points: [.m]
• Symmetric set with low Lebesgue constant: [.m]
• Weakly Admissible Mesh: [.m]
• Approximate Fekete points (with degree from 1 to 70): [.m]

» Cubature


    → Note: The reference simplex has vertices (0,0), (1,0), (0,1).

• Best cubature sets on the triangle (up to degree 50): [.m]
• Slobodkins-Tausch cubature sets on the triangle (up to degree 50): [.m]

» 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 [.m] as orthonormal basis;
• Proriol-Dubiner [.m] 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).

SQUARE:

» Interpolation

• General set with low Lebesgue constant: [.m] (last update: Jan 08, 2017, old version: [.m]).
• General set with high absolute value of the Vandermonde matrix, i.e. (quasi-) Fekete points: [.m]
• Padua-Jacobi points with low Lebesgue constant: [.m]
• Padua-Jacobi points with high absolute value of the Vandermonde matrix: [.m]
• Padua points: [.m]

» Cubature


    → Note: The reference square is [-1,1] x [-1,1].

• Sommariva-Festa rules (Legendre weight): [.m]
• Slobodkins-Tausch rules (Legendre weight): [.m]
• Almost minimal rules (Legendre weight, lowest known cardinality): [.m]

DISK:

» Interpolation/Least squares sets

• Good interpolation sets on the unit disk [.html]
• Weakly Admissible Mesh: [.m]

» Cubature

• Cubature rules (Legendre weight): [.m]

SPHERE:

» 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: [.m].
• Heo-Xu rules: [.m].
• Lebedev rules: [.m] (see also [.html]).
• McLaren rules: [.m].

TETRAHEDRON:

» 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.

• almost minimal rules (best known rules up to degree 25): [.m].
• Slobodkins-Tausch: [.m].
• some low cardinality rules (up to degree 20): [.m].