# A Novel Use of Pseudospectra in Mathematical Biology: Understanding HPA Axis Sensitivity

We consider a nonlinear delay differential equation model and calculate pseudospectra of three different linearizations: a time-dependent Jacobian, linearization around the limit cycle, and dynamic mode decomposition (DMD) analysis of Koopman operators (global linearization).

# Limits and Powers of Koopman Learning

This paper addresses a fundamental open question: \textit{When can we robustly learn the spectral properties of Koopman operators from trajectory data of dynamical systems, and when can we not?}

# Multiplicative Dynamic Mode Decomposition

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.

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# Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators

We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators.

3

# On the Convergence of Hermitian Dynamic Mode Decomposition

We show that, under suitable conditions, the eigenvalues and eigenfunctions of HDMD converge to the spectral properties of the underlying Koopman operator.

# The Multiverse of Dynamic Mode Decomposition Algorithms

1 code implementation30 Nov 2023

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.

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# Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems

Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators.

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# Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods

However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates.

1

# The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems

1 code implementation6 Sep 2022

We introduce measure-preserving extended dynamic mode decomposition ($\texttt{mpEDMD}$), the first truncation method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems.

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# Residual Dynamic Mode Decomposition: Robust and verified Koopmanism

Challenges include spurious (unphysical) modes, and dealing with continuous spectra, both of which occur regularly in turbulent flows.

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# Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems

1 code implementation29 Nov 2021,

This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20, 046-dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number $>10^5$ that has a 295, 122-dimensional state space.

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# WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions

1 code implementation24 Oct 2021

We show how several quantities controlling recovery performance also provide explicit approximate sharpness constants.

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# Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem

We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities, however, there does not exist any algorithm, even randomised, that can train (or compute) such a NN.

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# On the computation of geometric features of spectra of linear operators on Hilbert spaces

no code implementations26 Aug 2019

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences.

Spectral Theory 65J10, 65L15, 65F99, 47A10, 46N40, 47A12, 47N50, 15A60, 28A12, 28A78

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