Richard Mikael Slevinsky
University of Manitoba
Title: "FastTransforms and A Nonlocal Vector Calculus on the Sphere"
Abstract: I will present FastTransforms, a software package that provides computational routines for the associated connection problems and synthesis and analysis of harmonic functions on the disk, sphere, and triangle using the appropriate bivariate orthogonal polynomials. The algorithms are backward stable with a runtime complexity of O(n^3), where n is the polynomial degree, they are vectorized by Intel SSE, AVX, and AVX-512 intrinsics, and are parallelized by OpenMP.
Next, I will discuss a nonlocal vector calculus on the sphere. For particular classes of integral kernels, the nonlocal analogues of the local vector calculus operators (div, grad, curl, Laplacian) are shown to act as diagonal operators when scalar and vector spherical harmonics are used as bases for the domain and range, though the diagonal scalings must be computed numerically. Together with the FastTransforms software package, the first simulation of nonlocal reaction--diffusion equations on the sphere is straightforward, exhibiting fascinating differences with the local analogues.
