662 research outputs found
Discontinuous collocation methods and gravitational self-force applications
Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure
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
Pointwise Convergence of Jacobi Polynomials
The goal of this thesis is to numerically study a pointwise Jacobi convergence theorem for piecewise analytic functions based on the theorem on Legendre error. The convergence rates were examined on the boundary, singularity, and interior points. Results revealed that pointwise error convergence rates depend on the point of singularity, Jacobi polynomial coefficients α and β, and, lastly, on type of the piecewise analytic function
Gibbs Phenomenon for Jacobi Approximations
The classical Gibbs phenomenon is a peculiarity that arises when approximating functions near a jump discontinuity with the Fourier series. Namely, the Fourier series overshoots (and undershoots ) the discontinuity by approximately 9% of the total jump. This same phenomenon, with the same value of the overshoot, has been shown to occur when approximating jump-discontinuous functions using specific families of orthogonal polynomials. In this paper, we extend these results and prove that the Gibbs phenomenon exists for approximations of functions with interior jump discontinuities with the two-parameter family of Jacobi polynomials Pn(a,b)(x). In particular, we show that for all a, b the approximation overshoots and undershoots the function by the same value as in the classical case – approximately 9% of the jump
Reduction of the Gibbs Phenomenon via Interpolation Using Chebyshev Polynomials, Filtering and Chebyshev-Pade\u27 Approximations
In this manuscript, we will examine several methods of interpolation, with an emphasis on Chebyshev polynomials and the removal of the Gibbs Phenomenon. Included as an appendix are the author’s Mat- Lab implementations of Lagrange, Chebyshev, and rational interpolation methods
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
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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