843 research outputs found
Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards
The Quantum Unique Ergodicity (QUE) conjecture of Rudnick-Sarnak is that
every eigenfunction phi_n of the Laplacian on a manifold with
uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical
limit (eigenvalue E_n -> infinity), that is, `strong scars' are absent. We
study numerically the rate of equidistribution for a uniformly-hyperbolic
Sinai-type planar Euclidean billiard with Dirichlet boundary condition (the
`drum problem') at unprecedented high E and statistical accuracy, via the
matrix elements of a piecewise-constant test function A. By
collecting 30000 diagonal elements (up to level n ~ 7*10^5) we find that their
variance decays with eigenvalue as a power 0.48 +- 0.01, close to the estimate
1/2 of Feingold-Peres (FP). This contrasts the results of existing studies,
which have been limited to E_n a factor 10^2 smaller. We find strong evidence
for QUE in this system. We also compare off-diagonal variance, as a function of
distance from the diagonal, against FP at the highest accuracy (0.7%) thus far
in any chaotic system. We outline the efficient scaling method used to
calculate eigenfunctions.Comment: 38 pages, 11 figures, version of Jan '06, in review, Comm. Pure Appl.
Mat
Quasi-orthogonality on the boundary for Euclidean Laplace eigenfunctions
Consider the Laplacian in a bounded domain in R^d with general (mixed)
homogeneous boundary conditions. We prove that its eigenfunctions are
`quasi-orthogonal' on the boundary with respect to a certain norm. Boundary
orthogonality is proved asymptotically within a narrow eigenvalue window of
width o(E^{1/2}) centered about E, as E->infinity. For the special case of
Dirichlet boundary conditions, the normal-derivative functions are
quasi-orthogonal on the boundary with respect to the geometric weight function
r.n. The result is independent of any quantum ergodicity assumptions and hence
of the nature of the domain's geodesic flow; however if this is ergodic then
heuristic semiclassical results suggest an improved asymptotic estimate.
Boundary quasi-orthogonality is the key to a highly efficient `scaling method'
for numerical solution of the Laplace eigenproblem at large eigenvalue. One of
the main results of this paper is then to place this method on a more rigorous
footing.Comment: 21 pages, 2 figures; preprint version of 7/4/0
Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects
We present a high-order accurate boundary-based solver for three-dimensional
(3D) frequency-domain scattering from a doubly-periodic grating of smooth
axisymmetric sound-hard or transmission obstacles. We build the one-obstacle
solution operator using separation into P azimuthal modes via the FFT, the
method of fundamental solutions (with N proxy points lying on a curve), and
dense direct least-squares solves; the effort is O(N^3P) with a small constant.
Periodizing then combines fast multipole summation of nearest neighbors with an
auxiliary global Helmholtz basis expansion to represent the distant
contributions, and enforcing quasi-periodicity and radiation conditions on the
unit cell walls. Eliminating the auxiliary coefficients, and preconditioning
with the one-obstacle solution operator, leaves a well-conditioned square
linear system that is solved iteratively. The solution time per incident wave
is then O(NP) at fixed frequency. Our scheme avoids singular quadratures,
periodic Green's functions, and lattice sums, and its convergence rate is
unaffected by resonances within obstacles. We include numerical examples such
as scattering from a grating of period 13 {\lambda} x 13{\lambda} of
highly-resonant sound-hard "cups" each needing NP = 64800 surface unknowns, to
10-digit accuracy, in half an hour on a desktop.Comment: 22 pages, 9 figures, submitted to Journal of Computational Physic
Unimodal clustering using isotonic regression: ISO-SPLIT
A limitation of many clustering algorithms is the requirement to tune
adjustable parameters for each application or even for each dataset. Some
techniques require an \emph{a priori} estimate of the number of clusters while
density-based techniques usually require a scale parameter. Other parametric
methods, such as mixture modeling, make assumptions about the underlying
cluster distributions. Here we introduce a non-parametric clustering method
that does not involve tunable parameters and only assumes that clusters are
unimodal, in the sense that they have a single point of maximal density when
projected onto any line, and that clusters are separated from one another by a
separating hyperplane of relatively lower density. The technique uses a
non-parametric variant of Hartigan's dip statistic using isotonic regression as
the kernel operation repeated at every iteration. We compare the method against
k-means++, DBSCAN, and Gaussian mixture methods and show in simulations that it
performs better than these standard methods in many situations. The algorithm
is suited for low-dimensional datasets with a large number of observations, and
was motivated by the problem of "spike sorting" in neural electrical
recordings. Source code is freely available
High-order discretization of a stable time-domain integral equation for 3D acoustic scattering
We develop a high-order, explicit method for acoustic scattering in three
space dimensions based on a combined-field time-domain integral equation. The
spatial discretization, of Nystr\"om type, uses Gaussian quadrature on panels
combined with a special treatment of the weakly singular kernels arising in
near-neighbor interactions. In time, a new class of convolution splines is used
in a predictor-corrector algorithm. Experiments on a torus and a perturbed
torus are used to explore the stability and accuracy of the proposed scheme.
This involved around one thousand solver runs, at up to 8th order and up to
around 20,000 spatial unknowns, demonstrating 5-9 digits of accuracy. In
addition we show that parameters in the combined field formulation, chosen on
the basis of analysis for the sphere and other convex scatterers, work well in
these cases.Comment: 24 pages, 10 figure
How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?
We show that the condition number of any cyclically contiguous
submatrix of the discrete Fourier transform (DFT) matrix is at
least up to algebraic prefactors. That is, fixing any shape parameters
, the growth is as
with rate . Such Vandermonde system matrices arise in many applications,
such as Fourier continuation, super-resolution, and diffraction imaging. Our
proof uses the Kaiser-Bessel transform pair (of which we give a self-contained
proof), and estimates on sums over distorted sinc functions, to construct a
localized trial vector whose DFT is also localized. We warm up with an
elementary proof of the above but with half the rate, via a periodized Gaussian
trial vector. Using low-rank approximation of the kernel , we also
prove another lower bound , up to algebraic prefactors,
which is stronger than the above for small . When combined, the
bounds are within a factor of two of the numerically-measured empirical
asymptotic rate, uniformly over , and they become sharp in certain
regions. However, the results are not asymptotic: they apply to essentially all
, , and , and with all constants explicit.Comment: 24 pages, 4 figures. v3 slightly strengthens results via p not q in
\sigma_1 bound, corrects minor typos (eg e\pi/4 inversions), updates abstract
re K-B proo
Efficient high-order accurate Fresnel diffraction via areal quadrature and the nonuniform FFT
We present a fast algorithm for computing the diffracted field from arbitrary
binary (sharp-edged) planar apertures and occulters in the scalar Fresnel
approximation, for up to moderately high Fresnel numbers (). It
uses a high-order areal quadrature over the aperture, then exploits a single 2D
nonuniform fast Fourier transform (NUFFT) to evaluate rapidly at target points
(of order such points per second, independent of aperture complexity).
It thus combines the high accuracy of edge integral methods with the high speed
of Fourier methods. Its cost is , where is the
linear resolution required in source and target planes, to be compared with
for edge integral methods. In tests with several aperture
shapes, this translates to between 2 and 5 orders of magnitude acceleration. In
starshade modeling for exoplanet astronomy, we find that it is roughly faster than the state of the art in accurately computing the set of
telescope pupil wavefronts. We provide a documented, tested MATLAB/Octave
implementation.
An appendix shows the mathematical equivalence of the boundary diffraction
wave, angular integration, and line integral formulae, then analyzes a new
non-singular reformulation that eliminates their common difficulties near the
geometric shadow edge. This supplies a robust edge integral reference against
which to validate the main proposal.Comment: 21 pages, 7 figures, revised version, to appear, J. Astron. Telesc.
Instrum. Syst. (JATIS
High-order boundary integral equation solution of high frequency wave scattering from obstacles in an unbounded linearly stratified medium
We apply boundary integral equations for the first time to the
two-dimensional scattering of time-harmonic waves from a smooth obstacle
embedded in a continuously-graded unbounded medium. In the case we solve the
square of the wavenumber (refractive index) varies linearly in one coordinate,
i.e. where is a constant; this models
quantum particles of fixed energy in a uniform gravitational field, and has
broader applications to stratified media in acoustics, optics and seismology.
We evaluate the fundamental solution efficiently with exponential accuracy via
numerical saddle-point integration, using the truncated trapezoid rule with
typically 100 nodes, with an effort that is independent of the frequency
parameter . By combining with high-order Nystrom quadrature, we are able to
solve the scattering from obstacles 50 wavelengths across to 11 digits of
accuracy in under a minute on a desktop or laptop.Comment: 22 pages, 9 figures, submitted to J. Comput. Phy
Validation of neural spike sorting algorithms without ground-truth information
We describe a suite of validation metrics that assess the credibility of a
given automatic spike sorting algorithm applied to a given electrophysiological
recording, when ground-truth is unavailable. By rerunning the spike sorter two
or more times, the metrics measure stability under various perturbations
consistent with variations in the data itself, making no assumptions about the
noise model, nor about the internal workings of the sorting algorithm. Such
stability is a prerequisite for reproducibility of results. We illustrate the
metrics on standard sorting algorithms for both in vivo and ex vivo recordings.
We believe that such metrics could reduce the significant human labor currently
spent on validation, and should form an essential part of large-scale automated
spike sorting and systematic benchmarking of algorithms.Comment: 22 pages, 7 figures; submitted to J. Neurosci. Met
Explicit unconditionally stable methods for the heat equation via potential theory
We study the stability properties of explicit marching schemes for
second-kind Volterra integral equations that arise when solving boundary value
problems for the heat equation by means of potential theory. It is well known
that explicit finite difference or finite element schemes for the heat equation
are stable only if the time step is of the order ,
where is the finest spatial grid spacing. In contrast, for the
Dirichlet and Neumann problems on the unit ball in all dimensions , we
show that the simplest Volterra marching scheme, i.e., the forward Euler
scheme, is unconditionally stable. Our proof is based on an explicit spectral
radius bound of the marching matrix, leading to an estimate that an -norm
of the solution to the integral equation is bounded by times the
norm of the right hand side. For the Robin problem on the half space in any
dimension, with constant Robin (heat transfer) coefficient , we exhibit
a constant such that the forward Euler scheme is stable if , independent of any spatial discretization. This relies on new
lower bounds on the spectrum of real symmetric Toeplitz matrices defined by
convex sequences. Finally, we show that the forward Euler scheme is
unconditionally stable for the Dirichlet problem on any smooth convex domain in
any dimension, in -norm.Comment: 37 page
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