arXiv subjects

A. H. Barnett

Publications and source records attributed to A. H. Barnett.

10 recordsLinked to original sources

arXiv2020

Aliasing error of the exp$(β\sqrt{1-z^2})$ kernel in the nonuniform fast Fourier transform

The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel $ϕ$ to spread (or interpolate) between given nonuniform points and a uniform upsampled grid, combined with an FFT and diagonal scaling (deconvolution) in frequency space. The high performance of the recent FINUFFT library is in part due to its use of a new "exponential of semicircle" kernel $ϕ(z)=e^{β\sqrt{1-z^2}}$, for $z\in[-1,1]$, zero otherwise, whose Fourier transform $\hatϕ$ is unknown analytically. We place this kernel on a rigorous footing by proving an aliasing error estimate which bounds the error of the one-dimensional NUFFT of types 1 and 2 in exact arithmetic. Asymptotically in the kernel width measured in upsampled grid points, the error is shown to decrease with an exponential rate arbitrarily close to that of the popular Kaiser--Bessel kernel. This requires controlling a conditionally-convergent sum over the tails of $\hatϕ$, using steepest descent, other classical estimates on contour integrals, and a phased sinc sum. We also draw new connections between the above kernel, Kaiser--Bessel, and prolate spheroidal wavefunctions of order zero, which all appear to share an optimal exponential convergence rate.

math.NA↗

arXiv2012

High-order accurate Nystrom discretization of integral equations with weakly singular kernels on smooth curves in the plane

Boundary integral equations and Nystrom discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarithmically-singular kernels. Only smooth boundaries are considered, but some of the techniques extend straightforwardly to the case of corners. Three are modifications of the global periodic trapezoid rule, due to Kapur-Rokhlin, to Alpert, and to Kress. The fourth is a modification to a quadrature based on Gauss-Legendre panels due to Kolm-Rokhlin; this formulation allows adaptivity. We compare in numerical experiments the convergence of the four schemes in various settings, including low- and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace problems. We also find striking differences in performance in an iterative setting. We summarize the relative advantages of the schemes.

math.NA↗

arXiv2011

Estimates on Neumann eigenfunctions at the boundary, and the "Method of Particular Solutions" for computing them

We consider the "Method of particular solutions" for numerically computing eigenvalues and eigenfunctions of the Laplacian $Δ$ on a smooth, bounded domain Omega in RR^n with either Dirichlet or Neumann boundary conditions. This method constructs approximate eigenvalues E, and approximate eigenfunctions u that satisfy $Δu=Eu$ in Omega, but not the exact boundary condition. An inclusion bound is then an estimate on the distance of E from the actual spectrum of the Laplacian, in terms of (boundary data of) u. We prove operator norm estimates on certain operators on $L^2(\partial Ω)$ constructed from the boundary values of the true eigenfunctions, and show that these estimates lead to sharp inclusion bounds in the sense that their scaling with $E$ is optimal. This is advantageous for the accurate computation of large eigenvalues. The Dirichlet case can be treated using elementary arguments and has appeared in SIAM J. Num. Anal. 49 (2011), 1046-1063, while the Neumann case seems to require much more sophisticated technology. We include preliminary numerical examples for the Neumann case.

math.SP↗

arXiv2010

Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $Ω\subset\RR^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(Δ+ E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.

math.AP↗

arXiv2007

Quantum mushroom billiards

We report the first large-scale statistical study of very high-lying eigenmodes (quantum states) of the mushroom billiard proposed by L. Bunimovich in this journal, vol. 11, 802 (2001). The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly-regular modes. Our model explains our observed density of such superpositions dying as E^{-1/3} (E is the eigenvalue). We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16000 odd modes (an order of magnitude more than any existing study). We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and such high mode numbers orders of magnitude faster than with competing methods.

nlin.CD↗

arXiv2007

Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains

The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can be represented by the MFS basis in a numerically stable way. For Helmholtz problems on the unit disc we give a full analysis which includes the high frequency (short wavelength) limit. For more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued, which in turn depends on both domain shape and boundary data. Using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10^{-11} on a wide variety of analytic domains. At high frequencies of order only 3 points per wavelength are needed, which compares very favorably to boundary integral methods.

math.NA↗

arXiv2007

Analytic steady-state space use patterns and rapid computations in mechanistic home range analysis

Mechanistic home range models are important tools in modeling animal dynamics in spatially-complex environments. We introduce a class of stochastic models for animal movement in a habitat of varying preference. Such models interpolate between spatially-implicit resource selection analysis (RSA) and advection-diffusion models, possessing these two models as limiting cases. We find a closed-form solution for the steady-state (equilibrium) probability distribution u* using a factorization of the redistribution operator into symmetric and diagonal parts. How space use is controlled by the preference function w then depends on the characteristic width of the redistribution kernel: when w changes rapidly compared to this width, u* ~ w, whereas on global scales large compared to this width, u* ~ w^2. We analyse the behavior at discontinuities in w which occur at habitat type boundaries. We simulate the dynamics of space use given two-dimensional prey-availability data and explore the effect of the redistribution kernel width. Our factorization allows such numerical simulations to be done extremely fast; we expect this to aid the computationally-intensive task of model parameter fitting and inverse modeling.

q-bio.QM↗

arXiv2006

Mechanistic home range models and resource selection analysis: a reconciliation and unification

In the three decades since its introduction, resource selection analysis (RSA) has become a widespread method for analyzing spatial patterns of animal relocations obtained from telemetry studies. Recently, mechanistic home range models have been proposed as an alternative framework for studying patterns of animal space use. In contrast to RSA models, mechanistic home range models are derived from underlying mechanistic descriptions of individual movement behavior and yield spatially-explicit predictions for patterns of animal space-use. In addition, their mechanistic underpinning means that, unlike RSA, mechanistic home range models can also be used to predict changes in space-use following perturbation. In this paper, we develop a formal reconciliation between these two methods of home range analysis, showing how differences in the habitat preferences of individuals give rise to spatially-explicit patterns of space-use. The resulting unified framework combines the simplicity of resource selection analysis with the spatially-explicit and predictive capabilities of mechanistic home range models.

q-bio.PE↗

arXiv2001

Mesoscopic scattering in the half-plane: squeezing conductance through a small hole

We model the 2-probe conductance of a quantum point contact (QPC), in linear response. If the QPC is highly non-adiabatic or near to scatterers in the open reservoir regions, then the usual distinction between leads and reservoirs breaks down and a technique based on scattering theory in the full two-dimensional half-plane is more appropriate. Therefore we relate conductance to the transmission cross section for incident plane waves. This is equivalent to the usual Landauer formula using a radial partial-wave basis. We derive the result that an arbitrarily small (tunneling) QPC can reach a p-wave channel conductance of 2e^2/h when coupled to a suitable reflector. If two or more resonances coincide the total conductance can even exceed this. This relates to recent mesoscopic experiments in open geometries. We also discuss reciprocity of conductance, and the possibility of its breakdown in a proposed QPC for atom waves.

cond-mat.mes-hall↗

arXiv2000

Substrate-based atom waveguide using guided two-color evanescent light fields

We propose a dipole-force linear waveguide which confines neutral atoms up to lambda/2 above a microfabricated single-mode dielectric optical guide. The optical guide carries far blue-detuned light in the horizontally-polarized TE mode and far red-detuned light in the vertically-polarized TM mode, with both modes close to optical cut-off. A trapping minimum in the transverse plane is formed above the optical guide due to the differing evanescent decay lengths of the two modes. This design allows manufacture of mechanically stable atom-optical elements on a substrate. We calculate the full vector bound modes for an arbitrary guide shape using two-dimensional non-uniform finite elements in the frequency-domain, allowing us to optimize atom waveguide properties. We find that a rectangular optical guide of 0.8um by 0.2um carrying 6mW of total laser power (detuning +-15nm about the D2 line) gives a trap depth of 200uK for cesium atoms (m_F = 0), transverse oscillation frequencies of f_x = 40kHz and f_y = 160kHz, collection area ~ 1um^2 and coherence time of 9ms. We discuss the effects of non-zero m_F, surface interactions, heating rate, the substrate refractive index, and the limits on waveguide bending radius.

physics.atom-ph↗