6,872 research outputs found
Hybrid spherical approximation
In this paper a local approximation method on the sphere is presented. As
interpolation scheme we consider a partition of unity method, such as the
modified spherical Shepard's method, which uses zonal basis functions (ZBFs)
plus spherical harmonics as local approximants. Moreover, a spherical zone
algorithm is efficiently implemented, which works well also when the amount of
data is very large, since it is based on an optimized searching procedure.
Numerical results show good accuracy of the method, also on real geomagnetic
data
Spectral Methods for Numerical Relativity. The Initial Data Problem
Numerical relativity has traditionally been pursued via finite differencing.
Here we explore pseudospectral collocation (PSC) as an alternative to finite
differencing, focusing particularly on the solution of the Hamiltonian
constraint (an elliptic partial differential equation) for a black hole
spacetime with angular momentum and for a black hole spacetime superposed with
gravitational radiation. In PSC, an approximate solution, generally expressed
as a sum over a set of orthogonal basis functions (e.g., Chebyshev
polynomials), is substituted into the exact system of equations and the
residual minimized. For systems with analytic solutions the approximate
solutions converge upon the exact solution exponentially as the number of basis
functions is increased. Consequently, PSC has a high computational efficiency:
for solutions of even modest accuracy we find that PSC is substantially more
efficient, as measured by either execution time or memory required, than finite
differencing; furthermore, these savings increase rapidly with increasing
accuracy. The solution provided by PSC is an analytic function given
everywhere; consequently, no interpolation operators need to be defined to
determine the function values at intermediate points and no special
arrangements need to be made to evaluate the solution or its derivatives on the
boundaries. Since the practice of numerical relativity by finite differencing
has been, and continues to be, hampered by both high computational resource
demands and the difficulty of formulating acceptable finite difference
alternatives to the analytic boundary conditions, PSC should be further pursued
as an alternative way of formulating the computational problem of finding
numerical solutions to the field equations of general relativity.Comment: 15 pages, 5 figures, revtex, submitted to PR
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
A pseudospectral matrix method for time-dependent tensor fields on a spherical shell
We construct a pseudospectral method for the solution of time-dependent,
non-linear partial differential equations on a three-dimensional spherical
shell. The problem we address is the treatment of tensor fields on the sphere.
As a test case we consider the evolution of a single black hole in numerical
general relativity. A natural strategy would be the expansion in tensor
spherical harmonics in spherical coordinates. Instead, we consider the simpler
and potentially more efficient possibility of a double Fourier expansion on the
sphere for tensors in Cartesian coordinates. As usual for the double Fourier
method, we employ a filter to address time-step limitations and certain
stability issues. We find that a tensor filter based on spin-weighted spherical
harmonics is successful, while two simplified, non-spin-weighted filters do not
lead to stable evolutions. The derivatives and the filter are implemented by
matrix multiplication for efficiency. A key technical point is the construction
of a matrix multiplication method for the spin-weighted spherical harmonic
filter. As example for the efficient parallelization of the double Fourier,
spin-weighted filter method we discuss an implementation on a GPU, which
achieves a speed-up of up to a factor of 20 compared to a single core CPU
implementation.Comment: 33 pages, 9 figure
Natural preconditioners for saddle point systems
The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or discrete setting, so saddle point systems arising from discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as does, for example, the widely used sequential quadratic programming approach to nonlinear optimization.\ud
This article concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee rapid convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness -- in terms of rapidity of convergence -- is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends
nu=1/2 quantum Hall effect in the Aharonov-Casher geometry in a mesoscopic ring
We study the effect of an electric charge in the middle of a ring of
electrons in a magnetic field such as . In the absence of the
central charge, a residual current should appear due to an Aharanov-Bohm
effect. As the charge varies, periodic currents should appear in the ring. We
evaluate the amplitude of these currents, as well as their period as the
central charge varies. The presence of these currents should be a direct
signature of the existence of a statistical gauge field in the
quantum Hall effect. Numerical diagonalizations for a small number of electrons
on the sphere are also carried out. The numerical results up to 9 electrons are
qualitatively consistent with the mean field picture.Comment: 23 pages,14 included postscript figures, submitted to Phys. Rev.
Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations
The purpose of this work is to analyse a family of mutually orthogonal
polynomials on the unit ball with respect to an inner product which includes an
additional term on the sphere. First, we will get connection formulas relating
classical multivariate orthogonal polynomials on the ball with our family of
orthogonal polynomials. Then, using the representation of these polynomials in
terms of spherical harmonics, algebraic and differential properties will be
deduced
Complex (super)-matrix models with external sources and -ensembles of Chern-Simons and ABJ(M) type
The Langmann-Szabo-Zarembo (LSZ) matrix model is a complex matrix model with
a quartic interaction and two external matrices. The model appears in the study
of a scalar field theory on the non-commutative plane. We prove that the LSZ
matrix model computes the probability of atypically large fluctuations in the
Stieltjes-Wigert matrix model, which is a -ensemble describing
Chern-Simons theory on the three-sphere. The correspondence holds in a
generalized sense: depending on the spectra of the two external matrices, the
LSZ matrix model either describes probabilities of large fluctuations in the
Chern-Simons partition function, in the unknot invariant or in the two-unknot
invariant. We extend the result to supermatrix models, and show that a
generalized LSZ supermatrix model describes the probability of atypically large
fluctuations in the ABJ(M) matrix model.Comment: 30 pages, 2 figures. v2: A correction made and several new results
added; title changed. v3: Presentation reorganized, new results and
references added, final versio
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