6,658 research outputs found
Scalar and vector Slepian functions, spherical signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and, particularly for applications in the
geosciences, for scalar and vectorial signals defined on the surface of a unit
sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics,
edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be
published by Springer Verlag. This is a slightly modified but expanded
version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the
Handbook, when it was called: Slepian functions and their use in signal
estimation and spectral analysi
Slepian functions and their use in signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden,
Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla
The Beylkin-Cramer Summation Rule and A New Fast Algorithm of Cosmic Statistics for Large Data Sets
Based on the Beylkin-Cramer summation rule, we introduce a new fast algorithm
that enable us to explore the high order statistics efficiently in large data
sets. Central to this technique is to make decomposition both of fields and
operators within the framework of multi-resolution analysis (MRA), and realize
theirs discrete representations. Accordingly, a homogenous point process could
be equivalently described by a operation of a Toeplitz matrix on a vector,
which is accomplished by making use of fast Fourier transformation. The
algorithm could be applied widely in the cosmic statistics to tackle large data
sets. Especially, we demonstrate this novel technique using the spherical,
cubic and cylinder counts in cells respectively. The numerical test shows that
the algorithm produces an excellent agreement with the expected results.
Moreover, the algorithm introduces naturally a sharp-filter, which is capable
of suppressing shot noise in weak signals. In the numerical procedures, the
algorithm is somewhat similar to particle-mesh (PM) methods in N-body
simulations. As scaled with , it is significantly faster than the
current particle-based methods, and its computational cost does not relies on
shape or size of sampling cells. In addition, based on this technique, we
propose further a simple fast scheme to compute the second statistics for
cosmic density fields and justify it using simulation samples. Hopefully, the
technique developed here allows us to make a comprehensive study of
non-Guassianity of the cosmic fields in high precision cosmology. A specific
implementation of the algorithm is publicly available upon request to the
author.Comment: 27 pages, 9 figures included. revised version, changes include (a)
adding a new fast algorithm for 2nd statistics (b) more numerical tests
including counts in asymmetric cells, the two-point correlation functions and
2nd variances (c) more discussions on technic
General relativistic neutrino transport using spectral methods
We present a new code, Lorene's Ghost (for Lorene's gravitational handling of
spectral transport) developed to treat the problem of neutrino transport in
supernovae with the use of spectral methods. First, we derive the expression
for the nonrelativistic Liouville operator in doubly spherical coordinates (r,
theta, phi, epsilon, Theta, Phi)$, and further its general relativistic
counterpart. We use the 3 + 1 formalism with the conformally flat approximation
for the spatial metric, to express the Liouville operator in the Eulerian
frame. Our formulation does not use any approximations when dealing with the
angular arguments (theta, phi, Theta, Phi), and is fully energy-dependent. This
approach is implemented in a spherical shell, using either Chebyshev
polynomials or Fourier series as decomposition bases. It is here restricted to
simplified collision terms (isoenergetic scattering) and to the case of a
static fluid. We finish this paper by presenting test results using basic
configurations, including general relativistic ones in the Schwarzschild
metric, in order to demonstrate the convergence properties, the conservation of
particle number and correct treatment of some general-relativistic effects of
our code. The use of spectral methods enables to run our test cases in a
six-dimensional setting on a single processor.Comment: match published versio
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