775 research outputs found
Post-Newtonian and Numerical Calculations of the Gravitational Self-Force for Circular Orbits in the Schwarzschild Geometry
The problem of a compact binary system whose components move on circular
orbits is addressed using two different approximation techniques in general
relativity. The post-Newtonian (PN) approximation involves an expansion in
powers of v/c<<1, and is most appropriate for small orbital velocities v. The
perturbative self-force (SF) analysis requires an extreme mass ratio m1/m2<<1
for the components of the binary. A particular coordinate-invariant observable
is determined as a function of the orbital frequency of the system using these
two different approximations. The post-Newtonian calculation is pushed up to
the third post-Newtonian (3PN) order. It involves the metric generated by two
point particles and evaluated at the location of one of the particles. We
regularize the divergent self-field of the particle by means of dimensional
regularization. We show that the poles proportional to 1/(d-3) appearing in
dimensional regularization at the 3PN order cancel out from the final gauge
invariant observable. The 3PN analytical result, through first order in the
mass ratio, and the numerical SF calculation are found to agree well. The
consistency of this cross cultural comparison confirms the soundness of both
approximations in describing compact binary systems. In particular, it provides
an independent test of the very different regularization procedures invoked in
the two approximation schemes.Comment: 36 pages, 3 figures; matches the published versio
A robust implementation of the Carathéodory-Fejér method
Best rational approximations are notoriously difficult to compute. However, the difference between the best rational approximation to a function and its Carathéodory-Fejér (CF) approximation is often so small as to be negligible in practice, while CF approximations are far easier to compute. We present a robust and fast implementation of this method in the chebfun software system and illustrate its use with several examples. Our implementation handles both polynomial and rational approximation and substantially improves upon earlier published software
Post-Newtonian Theory and Dimensional Regularization
Inspiralling compact binaries are ideally suited for application of a
high-order post-Newtonian (PN) gravitational wave generation formalism. To be
observed by the LIGO and VIRGO detectors, these very relativistic systems (with
orbital velocities of the order of 0.5c in the last rotations) require
high-accuracy templates predicted by general relativity theory. Recent
calculations of the motion and gravitational radiation of compact binaries at
the 3PN approximation using the Hadamard self-field regularization have left
undetermined a few dimensionless coefficients called ambiguity parameters. In
this article we review the application of dimensional self-field
regularization, within Einstein's classical general relativity formulated in D
space-time dimensions, which finally succeeded in clearing up the problem, by
uniquely fixing the values of all the ambiguity parameters.Comment: 16 pages, to appear in the Proceedings of the "Albert Einstein
Century Conference", Paris (2005), edited by Jean-Michel Alim
Fast Computation of Fourier Integral Operators
We introduce a general purpose algorithm for rapidly computing certain types
of oscillatory integrals which frequently arise in problems connected to wave
propagation and general hyperbolic equations. The problem is to evaluate
numerically a so-called Fourier integral operator (FIO) of the form at points given on
a Cartesian grid. Here, is a frequency variable, is the
Fourier transform of the input , is an amplitude and
is a phase function, which is typically as large as ;
hence the integral is highly oscillatory at high frequencies. Because an FIO is
a dense matrix, a naive matrix vector product with an input given on a
Cartesian grid of size by would require operations.
This paper develops a new numerical algorithm which requires operations, and as low as in storage space. It operates by
localizing the integral over polar wedges with small angular aperture in the
frequency plane. On each wedge, the algorithm factorizes the kernel into two components: 1) a diffeomorphism which is
handled by means of a nonuniform FFT and 2) a residual factor which is handled
by numerical separation of the spatial and frequency variables. The key to the
complexity and accuracy estimates is that the separation rank of the residual
kernel is \emph{provably independent of the problem size}. Several numerical
examples demonstrate the efficiency and accuracy of the proposed methodology.
We also discuss the potential of our ideas for various applications such as
reflection seismology.Comment: 31 pages, 3 figure
Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries
The article reviews the current status of a theoretical approach to the
problem of the emission of gravitational waves by isolated systems in the
context of general relativity. Part A of the article deals with general
post-Newtonian sources. The exterior field of the source is investigated by
means of a combination of analytic post-Minkowskian and multipolar
approximations. The physical observables in the far-zone of the source are
described by a specific set of radiative multipole moments. By matching the
exterior solution to the metric of the post-Newtonian source in the near-zone
we obtain the explicit expressions of the source multipole moments. The
relationships between the radiative and source moments involve many non-linear
multipole interactions, among them those associated with the tails (and
tails-of-tails) of gravitational waves. Part B of the article is devoted to the
application to compact binary systems. We present the equations of binary
motion, and the associated Lagrangian and Hamiltonian, at the third
post-Newtonian (3PN) order beyond the Newtonian acceleration. The
gravitational-wave energy flux, taking consistently into account the
relativistic corrections in the binary moments as well as the various tail
effects, is derived through 3.5PN order with respect to the quadrupole
formalism. The binary's orbital phase, whose prior knowledge is crucial for
searching and analyzing the signals from inspiralling compact binaries, is
deduced from an energy balance argument.Comment: 109 pages, 1 figure; this version is an update of the Living Review
article originally published in 2002; available on-line at
http://www.livingreviews.org
A discrete approximation of Blake & Zisserman energy in image denoising and optimal choice of regularization parameters
We consider a multi-scale approach for the discrete approximation of a functional proposed by Bake and Zisserman (BZ) for solving image denoising and segmentation problems. The proposed method is based on simple and effective higher order varia-tional model. It consists of building linear discrete energies family which Γ-converges to the non-linear BZ functional. The key point of the approach is the construction of the diffusion operators in the discrete energies within a finite element adaptive procedure which approximate in the Γ-convergence sense the initial energy including the singular parts. The resulting model preserves the singularities of the image and of its gradient while keeping a simple structure of the underlying PDEs, hence efficient numerical method for solving the problem under consideration. A new point to make this approach work is to deal with constrained optimization problems that we circumvent through a Lagrangian formulation. We present some numerical experiments to show that the proposed approach allows us to detect first and second-order singularities. We also consider and implement to enhance the algorithms and convergence properties, an augmented Lagrangian method using the alternating direction method of Multipliers (ADMM)
ShearLab 3D: Faithful Digital Shearlet Transforms based on Compactly Supported Shearlets
Wavelets and their associated transforms are highly efficient when
approximating and analyzing one-dimensional signals. However, multivariate
signals such as images or videos typically exhibit curvilinear singularities,
which wavelets are provably deficient of sparsely approximating and also of
analyzing in the sense of, for instance, detecting their direction. Shearlets
are a directional representation system extending the wavelet framework, which
overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful
implementation and fast associated transforms. In this paper, we will introduce
a comprehensive carefully documented software package coined ShearLab 3D
(www.ShearLab.org) and discuss its algorithmic details. This package provides
MATLAB code for a novel faithful algorithmic realization of the 2D and 3D
shearlet transform (and their inverses) associated with compactly supported
universal shearlet systems incorporating the option of using CUDA. We will
present extensive numerical experiments in 2D and 3D concerning denoising,
inpainting, and feature extraction, comparing the performance of ShearLab 3D
with similar transform-based algorithms such as curvelets, contourlets, or
surfacelets. In the spirit of reproducible reseaerch, all scripts are
accessible on www.ShearLab.org.Comment: There is another shearlet software package
(http://www.mathematik.uni-kl.de/imagepro/members/haeuser/ffst/) by S.
H\"auser and G. Steidl. We will include this in a revisio
Decimated generalized Prony systems
We continue studying robustness of solving algebraic systems of Prony type
(also known as the exponential fitting systems), which appear prominently in
many areas of mathematics, in particular modern "sub-Nyquist" sampling
theories. We show that by considering these systems at arithmetic progressions
(or "decimating" them), one can achieve better performance in the presence of
noise. We also show that the corresponding lower bounds are closely related to
well-known estimates, obtained for similar problems but in different contexts
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