Matthew Colbrook
You have reached the webpage of Matthew Colbrook, a Junior Research Fellow at Trinity College, Cambridge (UK). I completed my PhD, "The Foundations of Infinite-Dimensional Spectral Computations", in September 2020 at the University of Cambridge. For the acadmic year 2021-2022 I will be visiting École Normale Supérieure's Centre Sciences des Données under a Fondation Sciences Mathématiques de Paris Fellowship.
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Research Interests
My work concerns the development of algorithms related to spectral theory, solutions of PDEs, neural networks, compressed sensing and inverse problems/optimisation. More specifically:
- Spectral computational problems in infinite dimensions
Spectra appear in many applications such as quantum mechanics, scattering, condensed matter theory, vibrational analysis, fluid flow etc. and are fundamental to many branches of mathematics. However, infinite-dimensional spectral problems are notoriously difficult, with standard methods suffering from problems such as spectral pollution (false eigenvalues due to truncations/discretisation to a finite-dimensional problem) and the difficulty of dealing with continuous spectra. I develop algorithms that solve these issues and I also classify the difficulty of these problems (proving also that algorithms are optimal). I have developed a new family of resolvent-based algorithms which use a mixture of numerical analysis, functional analysis and approximation theory. Results include (see publications and research pages for more):
- General methods for computing spectral measures and spectral type (link to two papers).
Code: SpecSolve (written with Andrew Horning) for spectral measures of general discrete, integral and differential operators.
This work featured on the frontcover of SIAM Review. - Computing spectra with error control (link to main paper, link to pedagogical article).
This work featured on the frontcover of Physical Review Letters and received the 2021 Lighthill-Thwaites Prize. - Classifications and new algorithms in the SCI hierarchy for a zoo of different spectral problems.
- General methods for computing spectral measures and spectral type (link to two papers).
- Stability and accuracy in deep learning
Alarmingly, recent results show that deep learning methods tend to be unstable. This is not just a problem in image classification (e.g., adversarial attacks) but also in image denoising and reconstruction, and the issue is attracting an increasing amount of interest/awareness. I seek to fix this in the context of image reconstruction and other problems such as solving PDEs. Our recent paper (to appear in PNAS) explores these issues, providing fundamental barriers and describing conditions under which (stable) NNs with a given accuracy can be computed by an algorithm. We also introduce Fast Iterative REstarted NETworks (FIRENETs) which we prove are stable and also converge exponentially in the number of layers.
- The Solvability Complexity Index Hierarchy and foundations of computational mathematics
The SCI hierarchy provides tools for the classification of the difficulty of computational problems in scientific computing and the proof that algorithms are optimal. The hierarchy generalises the fundamental problems of S. Smale on the existence of algorithms. Part of my thesis concerned the extension of this hierarchy and I develop new techniques to study it. As well as its use in scientific computation, applied mathematics and the sciences, the SCI hierarchy provides tools for computer-assisted proofs (where 100% rigour is needed).
- Infinite-dimensional numerical linear algebra
Many problems such as computing spectral properties of operators, solving PDEs and computing semigroups require techniques that deal with infinite-dimensional objects (such as linear operators on infinite-dimensional spaces). I develop tools that allow one to perform numerical linear algebra in infinite dimensions to solve these problems directly rather than an underlying truncations/discretisation.
- Spectral and pseudospectral methods for PDEs
Spectral methods are a powerful way of solving many PDEs. However, they typically take a global approach and expand solutions in basis functions. This can be inefficient if the solution has different intrinsic length scales and/or the domain is infinite or has complicated geometry. I have developed various boundary-based spectral methods for PDEs. Advantages include dimension reductions, dealing with unbounded domains and rapid convergence.
- Scattering problems, acoustics, and other problems related to computational fluid mechanics
Main Academic Awards
Biannual international prize given by IMA to top early career applied mathematicians.
(see here for details).
Cecil King Travel Scholarship 2020
Annual prize of the London Mathematical Society given "to a young mathematician of outstanding promise".
(see here for details).
Smith-Knight/Rayleigh-Knight Prize 2018
Grade I (top grade). Best first-year graduate piece of work in Mathematics Department of University of Cambridge.
(see here for details).
Mayhew Prize 2016
Best masters degree in Department of Applied Mathematics and Theoretical Physics, University of Cambridge.
(see here for details).
Caltech SURF Scholarship 2015
(see here for details).