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We present SymPix, a special-purpose spherical grid optimized for efficientsampling of rotationally invariant linear operators. This grid is conceptuallysimilar to the Gauss-Legendre (GL) grid, aligning sample points withiso-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 skyarea per grid point. The ratio between the number of grid points in twoneighbouring 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 forthis grid is to solve linear systems using multi-grid methods, and to constructefficient preconditioners through pixel-space sampling of the linear operatorin question. The GL grid is not suitable for these purposes due to its massiveover-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 benchmarkand representative example, we compute a preconditioner for a linear systemwith 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 invariantoperators, 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 ofinternal symmetries, yields average speed-ups of 360 and 23 for $B^T N^{-1} B$and $D$, respectively, relative to HEALPix.

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