TY - JOUR
T1 - Computing with Functions in Spherical and Polar Geometries I. The Sphere
AU - Townsend, Alex
AU - Wilber, Heather
AU - Wright, Grady B.
N1 - Publisher Copyright:
© 2016 Society for Industrial and Applied Mathematics.
PY - 2016
Y1 - 2016
N2 - A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in 1 minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.
AB - A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in 1 minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.
KW - Approximation theory
KW - Functions
KW - Gaussian elimination
KW - Low rank approximation
UR - http://www.scopus.com/inward/record.url?scp=84990985713&partnerID=8YFLogxK
UR - https://scholarworks.boisestate.edu/math_facpubs/182
U2 - 10.1137/15M1045855
DO - 10.1137/15M1045855
M3 - Article
SN - 1064-8275
VL - 38
SP - C403-C425
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 4
ER -