Heather Wilber, Oden Institute, University of Texas at Austin
Rational functions in computational mathematics
From dynamical systems and signal processing theory to core algorithms in numerical linear algebra, rational approximation theory has always shaped the way we think about computational mathematics. Even so, outside of a few very active areas, it is sometimes seen as a specialized technique that isn’t readily accessible to practitioners. Work from the last few decades has begun to change this, and we are currently living through a bit of a renaissance in computational rational approximation methods. In this talk, we give a whirlwind overview of some of these exciting developments. Joint work between myself, Per-Gunnar Martinsson, and Ke Chen is highlighted: we extend a classical technique for approximating the square root function to derive a new class of rational approximants, which we use to solve the spectral fractional Poisson equation. Our approach combines rational approximation with powerful, high-accuracy direct solvers (such as the hierarchical Poincaré Steklov method). The result is a fast, spectrally accurate method that can handle complicated geometries.