Heather Wilber

Current projects

Parametrized PDEs

Many applications require solutions to families of PDEs dependent on continuous parameters. Such families often have interesting shared structures: if these can be thoroughly explained, fast numerical methods that take advantage of them can be developed. Another exciting angle of this work is the development of new numerical methods based on parametrization. As a simple example, multi-dimensional domains can be re-imagined as volume integrals over lower-dimensional slices dependent on parameters that vary continously under the integral. Exploiting shared structures in the slices gives us new discretization schemes for solving challenging PDEs. This work fuses together fast direct solvers, hierarchical low rank structures, and boundary integral equation methods. The picture on the right is a computation of the magnitude of an acoustic wave trapped in a keyhole region.

• Paper on acoustic trapping
• Acoustic wave simulations

Fast methods for structured matrices and operators

Hierarchical low rank structures arise in matrices related to interactions between particles, planets, chemical species, and various other creatures. They also show up in data science in numerous ways (e.g., covariance matrices). While decades of work have been devoted to exploiting these structures in the context of PDEs and square systems, far less work has focused on how they can be used in rectangular systems and in the design of fast methods for solving standard optimization problems. From least-squares to LASSO, we are investigating the ways that hierarchical numerical methods can be adapted to optimization algorithms. Closely related to this work is our continued investigation into highly structured matrices in computational mathematics, including Cauchy, Toeplitz, Hankel, and Vandermonde matrices, as well as block-variants of these. The picture on the right shows an example of the structure found in a rectangular hierarchical matrix.

• Paper on hierarchical least squares for inverting nonuniform discrete Fourier transform matrices
• Slides from talk on why Cauchy matrices explain a lot in computational math (PDF)

Rational approximation

Rational approximation methods are a cornerstone of applied and computational mathematics. Any iterative method involving the application of matrix inverses probably has a rational function at its core: studying how and why rational functions work well as approximants of more complicated functions helps us explain these methods and design new methods. Rational functions are also useful in signal processing and data science, serving as an approximation class that is especially good for tasks requiring the capture or identification of sharp features and transitions, extrapolation, or slow frequency decay. Fully explaining these properties in a rigorous sense, as well as designing new methods for making use of them in signal processing, is the major goal of our current work. The image on the left is a rational approximation to the sign function defined over the white lines, shown in the complex plane. The poles of the function (white dots) are useful in understanding key properties of function approximation in the domain.

• Paper on computing Zolotarev rational functions
• REfit software for computing in MATLAB with exponential sums and rational functions

Working with me

Papers and Presentations

papers

• M.A. Gilles, H. Wilber — Low-rank approximation by randomly pivoted LUsubmitted · SIMAX

  H. Wilber, W. Vaes, A. Gopal, P.G. Martinsson — A time-frequency method for acoustic scattering with trappingsubmitted · J. Comp. Phys.

  C. Ballew, T. Trogdon, H. Wilber — The Akhiezer iteration and an inverse-free solver for Sylvester matrix equationssubmitted · IMA J. Num. Analysis

  B. Beckermann, D. Kressner, H. Wilber — Compression properties for large Toeplitz-like matrices

  Num. Algorithms 100.4 (2025), 2041–2067

  L.N. Trefethen, H. Wilber — Computation of Zolotarev rational functions

  SISC 47.4 (2025), A2205–A2220

  H. Wilber, E.N. Epperly, A.H. Barnett — A superfast direct inversion method for the nonuniform discrete Fourier transform

  SISC 47.3 (2025), A1702–A1732

  H. Wilber, A. Damle, A. Townsend — Data-driven algorithms for signal processing with trigonometric rational functions

  SISC 44-3 (2022), C185–C209

  D. Rubin, A. Townsend, H. Wilber — Bounding Zolotarev numbers using Faber rational functions

  Const. Approx. 56 (2022), 1–26

  Quinn, Wilber, Townsend, Sethna — Chebyshev approximation and the global geometry of model predictions

  Phys. Rev. Lett. 122 (2019), 158302

  A. Townsend, H. Wilber — On the singular values of matrices with high displacement rank

  LAA 548 (2018), 19–41

  H. Wilber, A. Townsend, G.B. Wright — Computing with functions in spherical and polar geometries II: the disk

  SISC 39 (2017), C238–C262

  A. Townsend, H. Wilber, G.B. Wright — Computing with functions in spherical and polar geometries I: the sphere

  SISC 38 (2016), C403–C425

theses

• M.S. Thesis
  PDF of M.S. Thesis
• PhD Thesis
  PDF of PhD Thesis

upcoming conferences & visits

• May 2026 27th Conference of the ILAS — Blacksburg, VA
• Mar 2026 Berkeley Applied Mathematics Seminar — Berkeley, CA
• Mar 2026 BIRS Workshop: Women in computational methods for PDEs — Banff, Canada
• Sept 2025 Mittag-Leffler Program: Interfaces and Unfitted Discretizations — Djursholm, Sweden

selected presentations

• Acoustic waves in unfriendly domains slides (PDF)
• Zolotarev numbers and the nonuniform discrete Fourier transform slides (PDF)
• REfit: data-driven computing with trigonometric rationals poster (PDF)
• Low rank numerical methods via rational approximation slides (PDF)
• Computing with functions on the sphere and disk poster (PDF)

software

Select Awards & Fellowships

• Householder Prize2022
• AWM Dissertation Prize2022
• NSF Mathematical Sciences Postdoctoral Research Fellowship2021–2023
• SIAM UKIE Prize: best student presentation, 27th Biennial Numerical Analysis Conference2019
• NSF Graduate Research Fellowship2016–2020
• Boise State University Distinguished M.S. Thesis in STEM2017
• NASA ISGC Fellowship2015–2016

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