Low Rank Approximation of Functions in Polar and Spherical Geometries

A collection of algorithms for computing with functions defined on the unit disk or the surface of the unit two-sphere is presented. Central to these algorithms is a new scheme for approximating functions to essentially machine precision that combines a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. The scheme produces low rank approximations of functions on the disk and sphere, ameliorates oversampling issues near the origin of the disk and poles of the sphere, converges geometrically for sufficiently analytic functions, and allows for stable differentiation. The low rank representation makes operations such as function evaluation, differentiation, and integration particularly efficient. A demonstration of the algorithms using the new Diskfun and Spherefun features of Chebfun will also be given. This is joint work with Prof. Alex Townsend and Heather Wilber (both at Cornell University).