SymPix: A spherical grid for efficient sampling of rotationally invariant operators

We present SymPix, a special-purpose spherical grid optimized for efficient sampling of 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 over-sampling near the poles and ensuring nearly equal sky area per grid point. The ratio between the number of grid points in two neighbouring 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. The GL grid is not suitable for these purposes due to its massive over-sampling near the poles, leading to nearly degenerate linear systems, while HEALPix, another commonly used spherical grid, exhibits few symmetries, and is therefore computationally inefficient for these purposes. As a benchmark and representative example, we compute a preconditioner for a linear system with both HEALPix and SymPix 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 pixel domain. For a bandwidth limit of $\ell_\text{max}=3000$, we find that SymPix, due to its higher number of internal symmetries, yields average speed-ups of 360 and 23 for $B^T N^{-1} B$ and $D$, respectively, relative to HEALPix.