Andrew Horning

horninga@mit.edu

Welcome!

I’m an Applied Math Instructor at MIT Mathematics. My research interests are in numerical analysis and scientific computing with an emphasis on fast numerical linear algebra, approximation theory, and all things eigenvalue-related!

Education

I completed my Ph.D. in applied mathematics at Cornell’s Center for Applied Math, where I worked closely with my supervisor, Alex Townsend in the math department. My research committee included David Bindel and Anil Damle from the computer science department.

Before that, I recieved a dual B.Sc in physics and mathematics from Rensselaer Polytechnic Institute (RPI), where I conducted research in cellular motors (math modeling), nanomaterials (computational physics), and ultrathin films (experimental physics). My interests in math and physics accelerated after high school while working as an R&D technician at Praxis Technology, a small biomedical firm in Glens Falls, NY.

Research

The spectral properties of infinite-dimensional operators often provide visual and intuitive insights into the behavior of complex interactions. For instance, the stability of a fluid flow, the scattering cross-section of a nucleus, and the influence of decentralized hubs in large networks are intimately connected to the spectral properties of such operators. I design algorithms that accurately and efficiently extract eigenvalues, eigenfunctions, and spectral measures of differential and integral operators. The algorithms use a mix of functional analysis and approximation theory to preserve and exploit the structure of the continuous operator, regardless of the underlying discretization used for numerical computation. In practice, this means

Software implementations exploiting fast linear algebra are available at https://github.com/SpecSolve.

Papers

  1. J. Zvonek, A. Horning, and A. Townsend, “ContHutch++: Stochastic trace estimation for implicit integral operators.” Submitted 2023.
  2. M.J. Colbrook, A. Horning, K. Thicke, and A.B. Watson, “Computing spectral properties of topological insulators without artificial truncation or supercell approximation.” IMA Journal of Applied Mathematics 88.1 (2023): 1-42.
  3. A. Horning and Y. Nakatsukasa, “Twice is enough for dangerous eigenvalues.” SIAM Journal on Matrix Analysis and Applications 43.1 (2022): 68-93.
  4. M.J. Colbrook, A. Horning, and A. Townsend, “Computing spectral measures of self-adjoint operators.” SIAM Review 63.3 (2021): 489-524.
  5. A. Horning, and A. Townsend, “FEAST for differential eigenvalue problems.” SIAM Journal on Numerical Analysis 58.2 (2020): 1239-1262.
  6. Y.P. Timalsina, et al. “Effects of nanoscale surface roughness on the resistivity of ultrathin epitaxial copper films.” Nanotechnology 26.7 (2015): 075704.
  7. C. Daniels, et al. “Elastic, plastic, and fracture mechanisms in graphene materials.” Journal of Physics: Condensed Matter 27.37 (2015): 373002.

Conference Proceedings

  1. M.J. Colbrook and A. Horning, “SpecSolve: Spectral methods for spectral measures” ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, Springer vol. 137 (2023).
  2. A. Horning, R. Morgan, and E. Nielson, “Minimum number of observations for exoplanet orbit determination.” Techniques and Instrumentation for Detection of Exoplanets IX. 4. Vol. 11117. SPIE (2018).

Invited and contributed talks