Speaker: Mikael Richard Slevinsky (Manitoba)

Title: Fast and stable computations with spherical harmonics

Abstract: I will present rapid transformations between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all orders are converted to those of order zero and one; then, these intermediate expressions are re-expanded in trigonometric form. Total pre-computation requires at best O(n³ log n) flops; and, asymptotically optimal execution time of O(n² log² n) is rigorously proved via connection to Fourier integral operators. I have released free and open source software in C and Julia that implements the algorithms and will demonstrate it live on nonlocal reaction-diffusion equations.