Bio Sketch

I’m an Applied Math Instructor at MIT Mathematics. I work in numerical analysis and scientific computing, with current interests in

infinite-dimensional spectral problems
randomized numerical linear algebra
data-driven modeling and analysis

Before coming to MIT, I obtained a Ph.D. in Applied Mathematics at Cornell University and a B.Sc. in Physics and Mathematics at Rensselaer Polytechnic Institute (RPI).

Research

Operators that act on an infinite-dimensional space of functions or vectors may exhibit a range of spectral phenomena that have no analog in the finite-dimensional world of matrices. What role do these play in practical problems and how can we compute them? As a computational scientist and numerical analyst, I am fascinated by three broad themes:

Continuous spectrum. Operators with continuous spectrum play a key role in resonance phenomena and wave-propagation in electromagnetics, acoustics, quantum mechanics, and various regimes of fluid flow. How does one capture the continuous spectrum on a computer?

Robust eigensolvers. Discretizations of infinite-dimensional operators may miss eigenvalues, converge to false eigenvalues, and amplify the sensitivity of the spectrum to small perturbations. Can one avoid the pitfalls of discretization when computing eigenvalues in infinite dimensions?

Data-driven design. Interactions between mathematical models and real data are a key ingredient in engineering analysis and design. Modal decompositions play a key role in constructing models and analyzing data from complex and nonlinear systems, but infinite-dimensional challenges abound. Can rigorous mode decompositions provide new bridges between models and data?

Software implementations of efficient and robust algorithms for infinite-dimensional spectral computations are available at https://github.com/SpecSolve.

Papers

J. Zvonek, A. Horning, and A. Townsend, “ContHutch++: Stochastic trace estimation for implicit integral operators.” Submitted 2023.
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.
A. Horning and Y. Nakatsukasa, “Twice is enough for dangerous eigenvalues.” SIAM Journal on Matrix Analysis and Applications 43.1 (2022): 68-93.
M.J. Colbrook, A. Horning, and A. Townsend, “Computing spectral measures of self-adjoint operators.” SIAM Review 63.3 (2021): 489-524.
A. Horning, and A. Townsend, “FEAST for differential eigenvalue problems.” SIAM Journal on Numerical Analysis 58.2 (2020): 1239-1262.
Y.P. Timalsina, et al. “Effects of nanoscale surface roughness on the resistivity of ultrathin epitaxial copper films.” Nanotechnology 26.7 (2015): 075704.
C. Daniels, et al. “Elastic, plastic, and fracture mechanisms in graphene materials.” Journal of Physics: Condensed Matter 27.37 (2015): 373002.

Conference Proceedings

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).
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

December 2022, “Differentiating functionals of data-sparse matrix factors,” SCAN Seminar, Cornell University, Ithaca, NY.
July 2022, “Computing spectral properties of topological insulators with disorder,” WAVES 2022, Paris, France.
July 2022, “Computing spectral properties of topological insulators with disorder,” SIAM AN22, Pittsburgh, PA.
November 2021, “Computing spectra of infinite-dimensional operators,” NMPDE Seminar, MIT, Cambridge, MA.
January 2021, “Computing spectral properties of infinite-dimensional operators,” Applied Math Seminar, UC Berkeley / Lawrence Berkeley Laboratory, Berkeley, CA.
December 2020, “Twice is enough for dangerous eigenvalues,” CMS Winter Meeting, Montreal, CA (virtual conference platform).
November 2020, “Twice is enough for dangerous eigenvalues,” NLA Seminar, University of Manchester, Manchester, UK.
Jul. 2020, “How to diagonalize differential and integral operators with continuous spectrum,” SIAM Annual Meeting, Toronto, CA (virtual conference platform).
Dec. 2019, “Computing spectral measures of differential and integral operators,” Complex Analysis Workshop, Isaac Newton Institute, Cambridge, UK.
Jul. 2019, “Computing the spectrum of a differential operator: a resolvent-based approach,” CAKE, University of Cambridge, Cambridge, UK.
Jun. 2019, “Computing the spectrum of a differential operator: a resolvent-based approach,” The 28th BNAC, University of Strathclyde, Glasgow, Scotland.
Sept. 2018, “A continuous analogue of FEAST for differential eigenvalue problems,” SCAN Seminar, Cornell University, Ithaca, NY.
Jul. 2018, “A continuous analogue of FEAST for differential eigenvalue problems,” ICOSAHOM, Imperial College, London, UK.
April 2018, “A continuous analogue of FEAST for differential eigenvalue problems,” Applied Math Days, RPI, Troy, NY.
Aug. 2015, “Mathematical model for steric disinhibition,” Mathematical Biosciences Institute Capstone Event, Ohio State, Columbus OH.
Oct. 2014, “Electrical resistivity in thin copper films,” American Vacuum Society, RPI, Troy, NY.