130 citations · 345 across the 38 of their papers we have counts for
47 papers
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…
Crown graphs maximise the representation number of bipartite graphs
Matthew J. Colbrook, Catherine Drysdale
The representation number of a graph is the least positive integer for which its vertices can be arranged in a word, each occurring times, so that two distinct letters alte…
A counterexample to Kenig's conjecture for the Laplace double-layer operator
Matthew J. Colbrook, Siavash Sadeghi
We disprove Kenig's 1994 spectral-radius conjecture for the double-layer operator in every dimension . For every sufficiently close to , we construct a bounded…
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…
Finiteness and exponential growth of full graph -spectra
Matthew J. Colbrook
We prove that the full spectrum of the graph -Laplacian is finite for every finite graph and every real . This resolves the problem of finiteness of the full graph spectrum…
A sharp isoperimetric inequality for the Neumann--Poincaré operator in every dimension
Matthew J. Colbrook, Siavash Sadeghi
Let , , be a bounded connected domain with boundary of class , where . For the adjoint Neumann--Poincaré operator , no…
A computer-assisted counterexample to the planar Berenstein conjecture
Matthew J. Colbrook, Siavash Sadeghi, George Stepaniants
Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial,…
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…
A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures
Matthew J. Colbrook, George Stepaniants
The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier ze…
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…
Computation and Verification of Spectra for Non-Hermitian Systems
Catherine Drysdale, Matthew Colbrook, Michael T. M. Woodley
We establish a connection between quantum mechanics and computation, revealing fundamental limitations for algorithms computing spectra, especially in non-Hermitian settings. Intro…
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.…
Avoiding spectral pollution for transfer operators using residuals
April Herwig, Matthew J. Colbrook, Oliver Junge +2
Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional appro…
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…
Deep greedy unfolding: Sorting out argsorting in greedy sparse recovery algorithms
Sina Mohammad-Taheri, Matthew J. Colbrook, Simone Brugiapaglia
Gradient-based learning imposes (deep) neural networks to be differentiable at all steps. This includes model-based architectures constructed by unrolling iterations of an iterativ…
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…
Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response
Niccolò Zagli, Matthew Colbrook, Valerio Lucarini +2
The fluctuation-dissipation theorem is a cornerstone result in statistical mechanics that can be used to translate the statistics of the free natural variability of a system into i…
A Novel Use of Pseudospectra in Mathematical Biology: Understanding HPA Axis Sensitivity
Catherine Drysdale, Matthew J. Colbrook
The Hypothalamic-Pituitary-Adrenal (HPA) axis is a major neuroendocrine system, and its dysregulation is implicated in various diseases. This system also presents interesting mathe…
Computing Spectral Size: Rigorous Algorithms and the Limits of Computation
Matthew J. Colbrook, Mark Embree, Jake Fillman
Many structures in mathematical physics and dynamics exhibit intricate fractal geometry. Such behavior appears prominently in quantum mechanics and materials science through spectr…
Adversarial dynamical systems characterize when data-driven learning succeeds or fails
Matthew J. Colbrook, Igor Mezić, Alexei Stepanenko
Many systems resist analytical modeling, making data-driven inference of dynamics important. Yet data-driven methods can fail to converge or generalize, leaving open a central ques…
Multiplicative Dynamic Mode Decomposition
Nicolas Boullé, Matthew J. Colbrook
Koopman operators are infinite-dimensional operators that linearize nonlinear dynamical systems, facilitating the study of their spectral properties and enabling the prediction of…