Interpolation and Cubature on the Sphere

MD50 points, Delaunay triagulation

Rob Womersley

MD50 points, Lagrangian 1-norm MD50 points, Lagrangian 2-norm MD50 points, Lagrangian infinity-norm

What is the best choice of points on the sphere for
polynomial interpolation and interpolatory cubature?

Contents

  1. Notation
  2. Point Sets
  3. Mesh Norms
  4. Interpolation Norms
  5. Condition numbers and eigenvalues
  6. Uniform errors in approximations
  7. Cubature weights
  8. Related publications
  9. Related web sites

Notation

Point Sets

Click on the following links to obtain images or tables of different point sets. Point sets obtained by the four criteria below have very different characteristics, the differences being generally not understood theoretically.

Caveat: All point sets are only approximate local optimizers of their respective criteria.

Recent point sets

For these point sets attempts to find global optima have been made, but with decreasing reliability as the degree n of the polynomials increases.

Point Sets from the "How Good ... " paper

Code Points, Data and Images Criterion
ME Minimum Energy points Minimize the potential energy (from Fliege and Maier)
MD Maximum Determinant points Maximizime det(G)
EV Eigenvalue points Maximize lmin, the smallest eigenvalue of G
MN Minimum Norm points Minimize ||Ln||, the interpolation operator norm

Mesh norms

The mesh norm gives a measure of how geometrically well distirbuted the points on the sphere are.

  Interpolation norms

The norm of the ploynomial interpolation operator, as a map from C(S2) to C(S2), gives a bound on the quality of the polynomial interpolant.

Points with small mesh norm do not necessarily correspond to points with low interpolation norm.

Condition number and smallest eigenvalues

Some point sets, like the minimal energy points sets of Fliege and Maier and the generlaized spiral points, have interpolation matrices which are very close to singular, making it numerically difficult to accuratley solve the linear system for the interpolation weights. A very small eigenvalue also makes the Reimers bound R on the Lebesgue constant is large. Plot of the condition number, largest and smallest eigenvalues for the ME, EV, MD and MN points

Uniform error of approximation

To illustrate the effect that a poor choice of interpolation points can have, the interpolants and the errors in the interpolation are plotted for the minimum energy (ME) and minimum norm (MN) points for the cosine cap.

Approximation of cosine cap

Cubature weights

Cubature weights w so that the rule Q(f) is exact for all polynomials of degree at most n are given by G*w = e. These weights may not be positive for any given point set. However the maximum determinant point set numerically has positive weights. As the weights sum to give the surface area of the sphere, the plots show the minimum and maximum of the scaled weights wj / wavg.

Scaled cubature weights

Cubature weights for maximum determinant points

Related publications by Ian Sloan and Robert Womersley

Related web sites