130 citations · 345 across the 38 of their papers we have counts for
21 papers · 1 filter
Convex Bernstein limits and sharp recovery on expanding intervals
Matthew J. Colbrook, Catherine Drysdale
We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most…
Computing Spectral Properties of Differential Operators on Surfaces
Gustav Conradie, Matthew J. Colbrook, Daniel Fortunato
Computing spectra of differential operators on surfaces is crucial in many applications, ranging from medical image analysis to nanoscale quantum structures. However, discretizatio…
A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients
Matthew J. Colbrook, George Stepaniants, Alex Townsend
Forsythe's conjecture, published in 1968, asserts that for each restart length , every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive defin…
A Proof of the Forsythe Conjecture for the Two-Step Restarted Conjugate Gradient Method
Matthew J. Colbrook, George Stepaniants, Alex Townsend
Forsythe conjectured in 1968 that the normalized residual directions of restarted conjugate gradients approach a two-cycle. We prove the conjecture for restart length two: unless t…
No Spectral Invisibility for Dissipative Barrier Truncations in Any Dimension
Matthew J. Colbrook, Marco Marletta
The dissipative barrier method suppresses spectral pollution, but whether it can itself conceal genuine spectral points has remained open. Known as the graveyard problem in computa…
Nyström Error Beyond -Matrices: A Minimal Diagonally Dominant Obstruction
Matthew J. Colbrook
We study the nuclear-norm error of a column-selected Nyström approximation to , where is symmetric diagonally dominant and . Our central question is whether…
Ten Digits on a Train: AI-Assisted Verification of Two Eigenvalue Problems
Matthew J. Colbrook
Accurate numerical eigenvalues are often difficult to certify, especially in singular or non-normal settings. This article reports a human--AI collaboration on two such computation…
An Introductory Guide to Koopman Learning
Matthew J. Colbrook, Zlatko Drmač, Andrew Horning
Koopman operators provide a linear framework for data-driven analyses of nonlinear dynamical systems, but their infinite-dimensional nature presents major computational challenges.…
Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces
Nicolas Boullé, Matthew J. Colbrook, Gustav Conradie
Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface t…
Universal Methods for Nonlinear Spectral Problems
Matthew J. Colbrook, Catherine Drysdale
Nonlinear spectral problems arise across a range of fields, including mechanical vibrations, fluid-solid interactions, and photonic crystals. Discretizing infinite-dimensional nonl…
Computing Generalized Eigenfunctions in Rigged Hilbert Spaces
Matthew J. Colbrook, Andrew Horning, Tianyiwa Xie
We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach…
On the Convergence of Hermitian Dynamic Mode Decomposition
Nicolas Boullé, Matthew J. Colbrook
We study the convergence of Hermitian Dynamic Mode Decomposition (DMD) to the spectral properties of self-adjoint Koopman operators. Hermitian DMD is a data-driven method that appr…
Avoiding discretization issues for nonlinear eigenvalue problems
Matthew J. Colbrook, Alex Townsend
The first step when solving an infinite-dimensional eigenvalue problem is often to discretize it. We show that one must be extremely careful when discretizing nonlinear eigenvalue…
The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems
Matthew J. Colbrook
Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems…
SpecSolve: Spectral methods for spectral measures
Matthew J. Colbrook, Andrew Horning
Self-adjoint operators on infinite-dimensional spaces with continuous spectra are abundant but do not possess a basis of eigenfunctions. Rather, diagonalization is achieved through…
Computing spectral properties of topological insulators without artificial truncation or supercell approximation
Matthew J. Colbrook, Andrew Horning, Kyle Thicke +1
Topological insulators (TIs) are renowned for their remarkable electronic properties: quantised bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in…
A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations
Matthew J. Colbrook, Lorna J. Ayton
We develop a rapid and accurate contour method for the solution of time-fractional PDEs. The method inverts the Laplace transform via an optimised stable quadrature rule, suitable…
Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems
Matthew J. Colbrook, Alex Townsend
Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. How…
WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions
Matthew J. Colbrook
Reconstruction of signals from undersampled and noisy measurements is a topic of considerable interest. Sharpness conditions directly control the recovery performance of restart sc…
Computing semigroups with error control
Matthew J. Colbrook
We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator , a time , an arb…
Computing spectral measures of self-adjoint operators
Matthew J. Colbrook, Andrew Horning, Alex Townsend
Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve a…