Utilizing GPU Parallelism to Improve Fast Spherical Harmonic Transforms

Abstract

Spherical harmonics form an orthogonal basis for functions that live on the surface of a sphere and are useful for solving partial differential equations and for numerical integration. The complexity of transforming a set of function samples to their corresponding spherical harmonic coefficients is largely dominated by the computation of the associated Legendre transform. This associated Legendre transform requires the computation of $(L+1)$ dense matrix-vector products where $L$ is the order of the spherical harmonic expansion. Since the number of rows and columns of each of these matrices depends on $L$ , this step is essentially $O(L^{3})$ . In this paper, we explore the GPU parallelism available to improve the butterfly compression approach. We present some preliminary results showing performance increases for large problem sizes and eventually plan to release the MonarchSHT library for GPU spherical harmonic transforms.