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Faculty of Mathematics

 
  • 2016 - date: PhD student, DAMTP, CCA, University of Cambridge.¬†
  • 2015 - 2016: M.Math. (Part III), University of Cambridge.
  • 2012 - 2015: B.A. Maths, University of Cambridge.

Research

Matthew is a member of the Applied Functional and Harmonic Analysis group within the Cambridge Centre for Analysis at DAMTP. Details of his research can be found here. His current interests are:

  • Spectral theory, in particular the existence (or non-existence!) of algorithms to solve computational problems.
  • Deep learning, its mathematical foundations and the problem of stability.
  • Spectral/pseudospectral methods and linear algebra in infinite dimensions.
  • Scattering problems and other problems related to computational fluid mechanics.

Publications

A full list of Matthew's publications can be seen here. Below is a list updated September 2019.

Journal Articles

(*denotes first authors where applicable, others alphabetical)
(paper prizes shown in orange)

Computational Spectral Theory

Pseudoergodic operators and periodic boundary conditions
Matthew Colbrook*
Mathematics of Computation, 2019.
preprint journal link
How to compute spectra with error control
(Smith-Knight/Rayleigh-Knight Prize Class I 2018)
Matthew Colbrook*, Bogdan Roman, Anders Hansen
Physical Review Letters, 2019 (chosen as cover article)
preprint journal link
On the infinite-dimensional QR algorithm
Matthew Colbrook*, Anders Hansen
Numerische Mathematik, 2019.
preprint journal link

Spectral Methods

A hybrid analytical-numerical technique for elliptic PDEs
Matthew Colbrook*, Thanasis Fokas, Parham Hashemzadeh
SIAM Journal on Scientific Computing (SISC), 2019.
preprint journal link
Extending the unified transform: curvilinear polygons and variable coefficient PDEs
Matthew Colbrook*
IMA Journal of Numerical Analysis, 2018.
preprint journal link
On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains
Matthew Colbrook*, Natasha Flyer, Bengt Fornberg
Journal of Computational Physics, 2018.
preprint journal link

Scattering Problems and Computational Fluid Mechanics

A spectral collocation method for acoustic scattering by multiple elastic plates
Matthew Colbrook*, Lorna Ayton
Journal of Sound and Vibration, 2019.
preprint journal link
The unified transform for mixed boundary condition problems in unbounded domains
(IMA Lighthill-Thwaites 2nd Prize 2019)
Matthew Colbrook*, Lorna Ayton, Athanassios Fokas
Proceedings of the Royal Society A, 2019.
preprint journal link
Scaling laws of passive-scalar diffusion in the interstellar medium
Matthew Colbrook*, Xiangcheng Ma, Philip Hopkins, Jonathan Squire
Monthly Notices of the Royal Astronomical Society, 2017.
preprint journal link

Other

A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line
Felipe de Barros, Matthew Colbrook, Athanassios Fokas
International Journal of Heat and Mass Transfer, 2019.
preprint journal link
Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons
Matthew Colbrook*, Athanassios Fokas
Applied Numerical Mathematics, 2018.
preprint journal link

Refereed Conference Articles

The unified transform: a spectral collocation method for acoustic scattering
Lorna Ayton, Matthew Colbrook, Athanassios Fokas
AIAA/CEAS Aeroacoustics, 2019.
preprint journal link
On the existence of stable and accurate neural networks for image reconstruction
Matthew Colbrook*, Vegard Antun, Anders Hansen
Signal Processing with Adaptive Sparse Structured Representations (SPARS), 2019.
preprint

Submitted Articles/Preprints

Computing spectral measures and spectral types: new algorithms and classifications
Matthew Colbrook*
arXiv preprint
The foundations of spectral computations via the solvability complexity index hierarchy: Part I
Matthew Colbrook*, Anders Hansen
arXiv preprint
The foundations of spectral computations via the solvability complexity index hierarchy: Part II
Matthew Colbrook*
arXiv preprint
On the solvability complexity index hierarchy and towers of algorithms
Jonathan Ben-Artzi, Matthew Colbrook, Anders Hansen, Olavi Nevanlinna, Markus Seidel
arXiv preprint
Kernel density estimation with linked boundary conditions
Matthew Colbrook*, Zdravko Botev, Karsten Kuritz, Shev MacNamara
arXiv preprint
Fast and spectrally accurate numerical methods for perforated screens
(with applications to Robin boundary conditions)

Matthew Colbrook, Matthew Priddin
preprint
Stable and accurate neural networks for image reconstruction
Matthew Colbrook*
In prep.

Publications

On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains
MJ Colbrook, N Flyer, B Fornberg
– Journal of Computational Physics
(2018)
374,
996
Extending the unified transform: curvilinear polygons and variable coefficient PDEs
MJ Colbrook
– IMA Journal of Numerical Analysis
Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons
MJ Colbrook, AS Fokas
– Applied Numerical Mathematics
(2018)
126,
1
Scaling laws of passive-scalar diffusion in the interstellar medium
MJ Colbrook, X Ma, PF Hopkins, J Squire
– Monthly Notices of the Royal Astronomical Society
(2017)
467,
2421
Kernel Density Estimation with Linked Boundary Conditions
MJ Colbrook, ZI Botev, K Kuritz, S MacNamara

Research Groups

CCA[Applied]
Numerical Analysis

Room

F0.13

Telephone

01223 339877