Conquering the pre-computation in two-dimensional harmonic polynomial transforms
This work addresses the pre-computation bottleneck in harmonic polynomial transforms for computational scientists, offering a significant reduction in memory and setup time.
The paper reduces storage and pre-computation for two-dimensional harmonic polynomial transforms to superoptimal complexities, with only an O(log n) increase in execution time. It derives a banded generalized eigenvalue problem for accelerating spherical harmonic transforms and extends the method to various geometries.
We describe a skeletonization of the spherical harmonic connection problem that reduces the storage and pre-computation to superoptimal complexities at the cost of increasing the execution time by the modest multiplicative factor of $\mathcal{O}(\log n)$. One advantage of accelerating the spherical harmonic connection problem over accelerating synthesis and analysis is that neighbouring layers (in steps of two) may be expanded in eachother's bases. The proposed skeletonization maximizes this interconnectivity by overlaying a dyadic partitioning on the connection problem. We derive the symmetric-definite banded generalized eigenvalue problem required to accelerate spherical harmonic transforms. We also include a full analysis of the weighted normalized Jacobi connection problem with applications to fast harmonic polynomial transforms on the disk, triangle, rectangle, deltoid, wedge, and any other geometry with a bivariate analogue of Jacobi polynomials.