Andrew Horning
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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: A family of high-order accurate contour integral methods for strongly continuous semigroups
A family of high-order accurate contour integral methods for strongly continuous semigroups Open
Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time approxi…
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: ContHutch++: Stochastic trace estimation for implicit integral operators
ContHutch++: Stochastic trace estimation for implicit integral operators Open
Hutchinson's estimator is a randomized algorithm that computes an $ε$-approximation to the trace of any positive semidefinite matrix using $\mathcal{O}(1/ε^2)$ matrix-vector products. An improvement of Hutchinson's estimator, known as Hutc…
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: 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)…
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: quantised bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations…
View article: Computing Spectral Measures of Self-Adjoint Operators
Computing Spectral Measures of Self-Adjoint Operators Open
Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high-orders of convergence in terms of a smoothi…
View article: Twice is enough for dangerous eigenvalues
Twice is enough for dangerous eigenvalues Open
We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous e…
View article: FEAST for Differential Eigenvalue Problems
FEAST for Differential Eigenvalue Problems Open
An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane. Unbounded search regions are handled with a novel ratio…
View article: The Harvard method of Tau calculation is incorrect.
The Harvard method of Tau calculation is incorrect. Open