5,356 research outputs found
Spectral filtering for the reduction of the Gibbs phenomenon of polynomial approximation methods on Lissajous curves with applications in MPI
Polynomial interpolation and approximation methods on sampling points along Lissajous curves using Chebyshev series is an effective way for a fast image reconstruction in Magnetic Particle Imaging. Due to the nature of spectral methods, a Gibbs phenomenon occurs in the reconstructed image if the underlying function has discontinuities. A possible solution for this problem are spectral filtering methods acting on the coefficients of the approximating polynomial. In this work, after a description of the Gibbs phenomenon and classical filtering techniques in one and several dimensions, we present an adaptive spectral filtering process for the resolution of this phenomenon and for an improved approximation of the underlying function or image. In this adaptive filtering technique, the spectral filter depends on the distance of a spatial point to the nearest discontinuity. We show the effectiveness of this filtering approach in theory, in numerical simulations as well as in the application in Magnetic Particle Imaging
Algebraic Fourier reconstruction of piecewise smooth functions
Accurate reconstruction of piecewise-smooth functions from a finite number of
Fourier coefficients is an important problem in various applications. The
inherent inaccuracy, in particular the Gibbs phenomenon, is being intensively
investigated during the last decades. Several nonlinear reconstruction methods
have been proposed, and it is by now well-established that the "classical"
convergence order can be completely restored up to the discontinuities. Still,
the maximal accuracy of determining the positions of these discontinuities
remains an open question. In this paper we prove that the locations of the
jumps (and subsequently the pointwise values of the function) can be
reconstructed with at least "half the classical accuracy". In particular, we
develop a constructive approximation procedure which, given the first
Fourier coefficients of a piecewise- function, recovers the locations
of the jumps with accuracy , and the values of the function
between the jumps with accuracy (similar estimates are
obtained for the associated jump magnitudes). A key ingredient of the algorithm
is to start with the case of a single discontinuity, where a modified version
of one of the existing algebraic methods (due to K.Eckhoff) may be applied. It
turns out that the additional orders of smoothness produce a highly correlated
error terms in the Fourier coefficients, which eventually cancel out in the
corresponding algebraic equations. To handle more than one jump, we propose to
apply a localization procedure via a convolution in the Fourier domain
Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data
In this paper we provide a reconstruction algorithm for piecewise-smooth
functions with a-priori known smoothness and number of discontinuities, from
their Fourier coefficients, posessing the maximal possible asymptotic rate of
convergence -- including the positions of the discontinuities and the pointwise
values of the function. This algorithm is a modification of our earlier method,
which is in turn based on the algebraic method of K.Eckhoff proposed in the
1990s. The key ingredient of the new algorithm is to use a different set of
Eckhoff's equations for reconstructing the location of each discontinuity.
Instead of consecutive Fourier samples, we propose to use a "decimated" set
which is evenly spread throughout the spectrum
Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: insights into spectral vanishing viscosity
AbstractThis study addresses linear dispersion–diffusion analysis for the spectral/hp continuous Galerkin (CG) formulation in one dimension. First, numerical dispersion and diffusion curves are obtained for the advection–diffusion problem and the role of multiple eigencurves peculiar to spectral/hp methods is discussed. From the eigencurves' behaviour, we observe that CG might feature potentially undesirable non-smooth dispersion/diffusion characteristics for under-resolved simulations of problems strongly dominated by either convection or diffusion. Subsequently, the linear advection equation augmented with spectral vanishing viscosity (SVV) is analysed. Dispersion and diffusion characteristics of CG with SVV-based stabilization are verified to display similar non-smooth features in flow regions where convection is much stronger than dissipation or vice-versa, owing to a dependency of the standard SVV operator on a local Péclet number. First a modification is proposed to the traditional SVV scaling that enforces a globally constant Péclet number so as to avoid the previous issues. In addition, a new SVV kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher-order discretizations, as measured by the extent of the wavenumber range where numerical errors are negligible. The dissipation characteristics of CG with the SVV modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous Galerkin (DG) scheme. Nevertheless, for the kernel function proposed, the full upwind DG scheme is found to have a slightly higher resolution power for the same dissipation levels. These results show that improved CG-SVV characteristics can be pursued via different kernel functions with the aid of optimization algorithms
Moment inversion problem for piecewise D-finite functions
We consider the problem of exact reconstruction of univariate functions with
jump discontinuities at unknown positions from their moments. These functions
are assumed to satisfy an a priori unknown linear homogeneous differential
equation with polynomial coefficients on each continuity interval. Therefore,
they may be specified by a finite amount of information. This reconstruction
problem has practical importance in Signal Processing and other applications.
It is somewhat of a ``folklore'' that the sequence of the moments of such
``piecewise D-finite''functions satisfies a linear recurrence relation of
bounded order and degree. We derive this recurrence relation explicitly. It
turns out that the coefficients of the differential operator which annihilates
every piece of the function, as well as the locations of the discontinuities,
appear in this recurrence in a precisely controlled manner. This leads to the
formulation of a generic algorithm for reconstructing a piecewise D-finite
function from its moments. We investigate the conditions for solvability of the
resulting linear systems in the general case, as well as analyze a few
particular examples. We provide results of numerical simulations for several
types of signals, which test the sensitivity of the proposed algorithm to
noise
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