142 research outputs found
Spatiospectral concentration on a sphere
We pose and solve the analogue of Slepian's time-frequency concentration
problem on the surface of the unit sphere to determine an orthogonal family of
strictly bandlimited functions that are optimally concentrated within a closed
region of the sphere, or, alternatively, of strictly spacelimited functions
that are optimally concentrated within the spherical harmonic domain. Such a
basis of simultaneously spatially and spectrally concentrated functions should
be a useful data analysis and representation tool in a variety of geophysical
and planetary applications, as well as in medical imaging, computer science,
cosmology and numerical analysis. The spherical Slepian functions can be found
either by solving an algebraic eigenvalue problem in the spectral domain or by
solving a Fredholm integral equation in the spatial domain. The associated
eigenvalues are a measure of the spatiospectral concentration. When the
concentration region is an axisymmetric polar cap the spatiospectral projection
operator commutes with a Sturm-Liouville operator; this enables the
eigenfunctions to be computed extremely accurately and efficiently, even when
their area-bandwidth product, or Shannon number, is large. In the asymptotic
limit of a small concentration region and a large spherical harmonic bandwidth
the spherical concentration problem approaches its planar equivalent, which
exhibits self-similarity when the Shannon number is kept invariant.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200
Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions
This paper presents an efficient spectral method for solving the fractional
Fredholm integro-differential equations. The non-smoothness of the solutions to
such problems leads to the performance of spectral methods based on the
classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low
order of convergence. For this reason, the development of classic numerical
methods to solve such problems becomes a challenging issue. Since the
non-smooth solutions have the same asymptotic behavior with polynomials of
fractional powers, therefore, fractional basis functions are the best candidate
to overcome the drawbacks of the accuracy of the spectral methods. On the other
hand, the fractional integration of the fractional polynomials functions is in
the class of fractional polynomials and this is one of the main advantages of
using the fractional basis functions. In this paper, an implicit spectral
collocation method based on the fractional Chelyshkov basis functions is
introduced. The framework of the method is to reduce the problem into a
nonlinear system of equations utilizing the spectral collocation method along
with the fractional operational integration matrix. The obtained algebraic
system is solved using Newton's iterative method. Convergence analysis of the
method is studied. The numerical examples show the efficiency of the method on
the problems with smooth and non-smooth solutions in comparison with other
existing methods
Criteria for Hierarchical Bases in Sobolev Spaces
AbstractSeveral approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. We are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions derived from a multiresolution analysis forms such a Riesz basis. This involves determining the smoothness of the dual system. The elements of the dual system typically consist of noncompactly supported functions, whose smoothness can be treated by extending the results of 7, 9, and 22. We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically, from spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed in the literature. For 29hierarchical basis, we find that it forms a Riesz basis in Hs(Rd) for −0.990236…<s<3/2
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
Augmented Slepians: Bandlimited Functions that Counterbalance Energy in Selected Intervals
Slepian functions provide a solution to the optimization problem of joint
time-frequency localization. Here, this concept is extended by using a
generalized optimization criterion that favors energy concentration in one
interval while penalizing energy in another interval, leading to the
"augmented" Slepian functions. Mathematical foundations together with examples
are presented in order to illustrate the most interesting properties that these
generalized Slepian functions show. Also the relevance of this novel
energy-concentration criterion is discussed along with some of its
applications
Tomographic inversion using -norm regularization of wavelet coefficients
We propose the use of regularization in a wavelet basis for the
solution of linearized seismic tomography problems , allowing for the
possibility of sharp discontinuities superimposed on a smoothly varying
background. An iterative method is used to find a sparse solution that
contains no more fine-scale structure than is necessary to fit the data to
within its assigned errors.Comment: 19 pages, 14 figures. Submitted to GJI July 2006. This preprint does
not use GJI style files (which gives wrong received/accepted dates).
Corrected typ
Multiscale wavelet analysis for integral and differential problems
2009 - 2010The object of the present research is wavelet analysis of integral and differential problems
by means of harmonic and circular wavelets. It is shown that circular wavelets
constitute a complete basis for L2[0; 1] functions, and form multiresolution analysis.
Multiresolution analysis can be briefly considered as a decomposition of L2[0; 1]
into a complete set of scale depending subspaces of wavelets. Thus, integral operators,
differential operators, and L2(R) functions were investigated as scale depending
functions through their projection onto these subspaces of wavelets. In particular:
- conditions when a certain wavelet can be applied for solution of integral or
differential problem are given;
- it is shown that the accuracy of this approach exponentially grows when increasing
the number of vanishing moments and scaling parameter;
- wavelet solutions of low-dimensional nonlinear partial differential equations are
compared with other methods;
- wavelet-based approach is applied to low-dimensional Fredholm integral equations
and the Galerkin method for two-dimensional Fredholm integral equations.[edited by author]. Oggetto della seguente ricerca `e l’analisi di problemi differenziali e integrali, utilizzando
wavelet armoniche e wavelet armoniche periodiche. Si dimostra che le wavelet
periodiche costituiscono una base completa per le funzioni L2[0; 1] e formano un’analisi
multiscala. L’analisi multirisoluzione pu`o essere brevemente considerata come la decomposizione
di L2[0; 1] in un insieme completo di sottospazi di wavelet dipendenti
da un fattore di scala. Pertanto gli operatori integrali e differenziali e le funzioni
L2(R) vengono studiati come funzioni di scala mediante le corrispondenti proiezioni
in questi sottospazi di wavelet. In particolare, vengono sviluppati quattro principali
argomenti:
- sono state individuate le condizioni per applicare una data famiglia di wavelets
alla soluzione di un data problema differenziale o integrale;
- si `e dimostrato che la precisione di questo approccio cresce esponenzialmente
quando decresce il numero dei momenti nulli e del parametro di scala;
- soluzioni wavelet di equazioni differenziali a derivate parziali nonlineari di dimensione
bassa sono state confrontate con altri metodi di soluzioni;
- l’approccio basato sull’uso delle wavelet `e stato applicato anche per ricerca
di soluzioni di alcune equazioni integrali di Fredholm e insieme al metodo di
Galerkin per risolvere equazioni integrali Fredholm di dimensioni due.[a cura dell'autore]IX n.s
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