Low Rank Function Approximation on the Sphere and Disk

Alex Townsend, Grady Wright, Heather Wilber (Boise State University)

Advances in fast and accurate computations with functions on the sphere are crucial in a host of domain sciences, including climate science, astrophysics, heliophysics, and the geosciences. A common strategy for numerically representing these functions involves mapping the sphere to a rectangular latitude-longitude coordinate grid and thereby enabling fast algorithms. However, this fails to take advantage of the natural periodicity of the sphere which occurs in the latitude (polar) direction. It also induces unneeded oversampling near the poles, and creates artificial singularities at the poles. Using the Double Fourier Sphere method originally proposed by Merilees [1], we apply a mapping which is 2π-periodic in both directions, and additionally introduces a useful symmetry structure to the problem. We develop an iterative, structure-preserving variant of Gaussian Elimination to form a low rank approximation method that geometrically converges for sufficiently analytic functions. Our method alleviates the issue of oversampling near the poles, and results in low rank approximations that are smooth at every level of approximation. This technique can also be used for functions on the unit disk.

We introduce a suite of algorithms related to this technique, including the ability to integrate, differentiate, evaluate and plot functions on the sphere and disk with speed and ease. Our algorithms will be publicly available in the open-source software package Chebfun as libraries spherefun and diskfun.

References

[1] P. E. Merilees, The pseudospectral approximation applied to the shallow water equations on a sphere, Atmosphere, 11 (1973), pp. 13–20.