Past Event: Babuška Forum

Computing with Rational Approximations to the Square Root

Heather Wilber, Postdoctoral fellow, Oden Institute, UT Austin

Abstract

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 bit of a specialty 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 by focusing on a famous rational approximation problem that appears pervasively throughout computing applications, the approximation of the square root function. 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.  

Biography

Heather Wilber is an NSF postdoctoral fellow working with Prof. Gunnar Martinsson's research group in numerical analysis and computational mathematics. She graduated from Cornell University with her PhD in Applied Mathematics in May 2021, where she was supervised by Prof. Alex Townsend. Prior to that, she earned her M.S. in Applied Mathematics under the supervision of Prof. Grady Wright at Boise State University.

Heather's work involves problems at the interface of approximation theory, computational mathematics, and scientific computing. This includes the development and analysis of low rank approximation methods, direct solvers for partial differential equations, algorithms for solving linear matrix equations and evaluating functions of matrices, and robust algorithms for computing with rational approximations to functions.