Matthew J. Colbrook
YOU?
Author Swipe
View article: Computation and Verification of Spectra for Non-Hermitian Systems
Computation and Verification of Spectra for Non-Hermitian Systems Open
We establish a connection between quantum mechanics and computation, revealing fundamental limitations for algorithms computing spectra, especially in non-Hermitian settings. Introducing the concept of locally trivial pseudospectra, we sho…
View article: Avoiding spectral pollution for transfer operators using residuals
Avoiding spectral pollution for transfer operators using residuals Open
Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) opera…
View article: Noisy PDE Training Requires Bigger PINNs
Noisy PDE Training Requires Bigger PINNs Open
Physics-Informed Neural Networks (PINNs) are increasingly used to approximate solutions of partial differential equations (PDEs), especially in high dimensions. In real-world applications, data samples are noisy, so it is important to know…
View article: Deep greedy unfolding: Sorting out argsorting in greedy sparse recovery algorithms
Deep greedy unfolding: Sorting out argsorting in greedy sparse recovery algorithms Open
Gradient-based learning imposes (deep) neural networks to be differentiable at all steps. This includes model-based architectures constructed by unrolling iterations of an iterative algorithm onto layers of a neural network, known as algor…
View article: Restarts Subject to Approximate Sharpness: A Parameter-Free and Optimal Scheme For First-Order Methods
Restarts Subject to Approximate Sharpness: A Parameter-Free and Optimal Scheme For First-Order Methods Open
Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts . However, sharpness inv…
View article: On the convergence of Hermitian Dynamic Mode Decomposition
On the convergence of Hermitian Dynamic Mode Decomposition Open
View article: Computing Generalized Eigenfunctions in Rigged Hilbert Spaces
Computing Generalized Eigenfunctions in Rigged Hilbert Spaces Open
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 does not require prior knowledge about the eigenfunctions,…
View article: Another look at residual dynamic mode decomposition in the regime of fewer snapshots than dictionary size
Another look at residual dynamic mode decomposition in the regime of fewer snapshots than dictionary size Open
View article: Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals
Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals Open
We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and ot…
View article: Multiplicative Dynamic Mode Decomposition
Multiplicative Dynamic Mode Decomposition Open
Koopman operators are infinite-dimensional operators that linearize nonlinear dynamical systems, facilitating the study of their spectral properties and enabling the prediction of the time evolution of observable quantities. Recent methods…
View article: Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators
Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators Open
We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. By considering the evolution of observables, Koopman operators transform complex nonli…
View article: Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size
Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size Open
Residual Dynamic Mode Decomposition (ResDMD) offers a method for accurately computing the spectral properties of Koopman operators. It achieves this by calculating an infinite-dimensional residual from snapshot data, thus overcoming issues…
View article: On the Convergence of Hermitian Dynamic Mode Decomposition
On the Convergence of Hermitian Dynamic Mode Decomposition Open
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 approximates the Koopman operator associated with an unknown no…
View article: Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems
Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems Open
Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands o…
View article: The Multiverse of Dynamic Mode Decomposition Algorithms
The Multiverse of Dynamic Mode Decomposition Algorithms Open
Dynamic Mode Decomposition (DMD) is a popular data-driven analysis technique used to decompose complex, nonlinear systems into a set of modes, revealing underlying patterns and dynamics through spectral analysis. This review presents a com…
View article: Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems
Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems Open
Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and Dynamic Mode Decomposition (DMD) stands o…
View article: Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems
Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems Open
Koopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and inf…
View article: Avoiding discretization issues for nonlinear eigenvalue problems
Avoiding discretization issues for nonlinear eigenvalue problems Open
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 problems. Using examples, we show that discretization can: …
View article: Computing spectral properties of topological insulators without artificial truncation or supercell approximation
Computing spectral properties of topological insulators without artificial truncation or supercell approximation Open
Topological insulators (TIs) are renowned for their remarkable electronic properties: quantized bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations…
View article: Residual dynamic mode decomposition: robust and verified Koopmanism
Residual dynamic mode decomposition: robust and verified Koopmanism Open
Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman oper…
View article: Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods
Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods Open
Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness invo…
View article: On the Computation of Geometric Features of Spectra of Linear Operators on Hilbert Spaces
On the Computation of Geometric Features of Spectra of Linear Operators on Hilbert Spaces Open
Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to deter…
View article: The foundations of spectral computations via the Solvability Complexity Index hierarchy
The foundations of spectral computations via the Solvability Complexity Index hierarchy Open
The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We est…
View article: The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems
The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems Open
Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite-dimensional, and …
View article: Bulk localized transport states in infinite and finite quasicrystals via magnetic aperiodicity
Bulk localized transport states in infinite and finite quasicrystals via magnetic aperiodicity Open
Robust edge transport can occur when charged particles in crystalline lattices interact with an applied external magnetic field. Such systems have a spectrum composed of bands of bulk states and in-gap edge states. For quasicrystalline sys…
View article: Residual Dynamic Mode Decomposition: Robust and verified Koopmanism
Residual Dynamic Mode Decomposition: Robust and verified Koopmanism Open
Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman oper…
View article: The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problem
The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problem Open
Significance Instability is the Achilles’ heel of modern artificial intelligence (AI) and a paradox, with training algorithms finding unstable neural networks (NNs) despite the existence of stable ones. This foundational issue relates to S…
View article: Computing Semigroups with Error Control
Computing Semigroups with Error Control Open
We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t>0$, an arbitrary initial vector $u_0$ and an error tolerance $ε>0$, t…
View article: A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations
A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations Open
We develop a rapid and accurate contour method for the solution of\ntime-fractional PDEs. The method inverts the Laplace transform via an optimised\nstable quadrature rule, suitable for infinite-dimensional operators, whose\nerror decrease…
View article: SpecSolve: Spectral methods for spectral measures
SpecSolve: Spectral methods for spectral measures Open
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 spectral measures. The SpecSolve package [SIAM Rev., 63(3)…