The Commutation Problem

by Peter Alfeld

The purpose of this page is to organize results of some experiments I ran regarding the infamous commutation problem in multivariate splines, including some color pictures that are hard to handle or publish otherwise.

The underlying triangulation in all cases consists of four triangles all sharing one interior vertex.

The results are summarized in the table

The individual spaces are:

Parts of the notation, and the term commutes, are defined in a technical description of the commutation problem. The term buckles means that the diagram does not commute.


  1. Column 1 gives the reference number of the example.
  2. Column 2 gives the functional that is being minimized (subject to interpolation). The integration is over the domain Omega, which is the union of the triangles. The triangulation consists of one interior vertex shared by four triangles having vertices (1,1), (-1,1), (-1,-1), (1,-1). The interior vertex is allowed to vary in Omega. When it is at the origin we have a singular vertex where the dimension of the space of differentiable piecewise cubics or quadratics increases by 1 (from 15 to 16 for cubics and from 7 to 8 for quadratics).
  3. Column 3 gives the function space over which the functional is minimized. The space in example 6 is the space of continuous piecewise cubic functions that are differentiable at the vertices of the traingulation, but not in the interior of the edges shared by triangles.
  4. Column 4 gives the behavior of the interpolant for the cardinal function at V1, i.e., the function that's 1 at V1 and 0 at the other four vertices.
  5. Column 5 similarly describes the behavior for the cardinal function associated with the interior vertex.
  6. Column 6 refers to notes explained in this list.
  7. This is the most frequently considered situation. From a likelyhood point of view it is amazing that the diagram commutes.
  8. Remarkably, the solution for the interior vertex being in singular position turns out to be a single quadratic function, namely: As yet I am unable to compute the solution explicitly for a general non-singular vertex - instead the code solves an appropriate linear system that depends on the location of the interior vertex. It is remarkable that the solution at singularity is the same for all variation principles in examples 1, 2, and 3. The corresponding statement does not hold for non-singular vertices.
  9. You may see pictures of this example.