SYMPIX: A SPHERICAL GRID FOR EFFICIENT SAMPLING OF ROTATIONALLY INVARIANT OPERATORS

Abstract

ABSTRACT We present SymPix, a special-purpose spherical grid optimized for efficiently sampling rotationally invariant linear operators. This grid is conceptually similar to the Gauss–Legendre (GL) grid, aligning sample points with iso-latitude rings located on Legendre polynomial zeros. Unlike the GL grid, however, the number of grid points per ring varies as a function of latitude, avoiding expensive oversampling near the poles and ensuring nearly equal sky area per grid point. The ratio between the number of grid points in two neighboring rings is required to be a low-order rational number (3, 2, 1, 4/3, 5/4, or 6/5) to maintain a high degree of symmetries. Our main motivation for this grid is to solve linear systems using multi-grid methods, and to construct efficient preconditioners through pixel-space sampling of the linear operator in question. As a benchmark and representative example, we compute a preconditioner for a linear system that involves the operator D ^ + B ^ T N − 1 B ^ ?> , where B ^ ?> and D ^ ?> may be described as both local and rotationally invariant operators, and N ?> is diagonal in the pixel domain. For a bandwidth limit of ℓ max ?> = 3000, we find that our new SymPix implementation yields average speed-ups of 360 and 23 for B ^ T N − 1 B ^ ?> and D ^ ?> , respectively, compared with the previous state-of-the-art implementation.