3,853 research outputs found
Truncation error analysis of multipole expansion
The multilevel fast multipole algorithm is based on the multipole expansion, which has numerical error sources such as truncation of the addition theorem, numerical integration, and interpolation/anterpolation. Of these, we focus on the truncation error and discuss its control precisely. The conventional selection rule fails when the buffer size is small compared to the desired numerical accuracy. We propose a new approach and show that the truncation error can be controlled and predicted regardless of the number of buffer sizes.published_or_final_versio
Fourier Based Fast Multipole Method for the Helmholtz Equation
The fast multipole method (FMM) has had great success in reducing the
computational complexity of solving the boundary integral form of the Helmholtz
equation. We present a formulation of the Helmholtz FMM that uses Fourier basis
functions rather than spherical harmonics. By modifying the transfer function
in the precomputation stage of the FMM, time-critical stages of the algorithm
are accelerated by causing the interpolation operators to become
straightforward applications of fast Fourier transforms, retaining the
diagonality of the transfer function, and providing a simplified error
analysis. Using Fourier analysis, constructive algorithms are derived to a
priori determine an integration quadrature for a given error tolerance. Sharp
error bounds are derived and verified numerically. Various optimizations are
considered to reduce the number of quadrature points and reduce the cost of
computing the transfer function.Comment: 24 pages, 13 figure
Recoil Polarization Measurements for Neutral Pion Electroproduction at Q^2=1 (GeV/c)^2 Near the Delta Resonance
We measured angular distributions of differential cross section, beam
analyzing power, and recoil polarization for neutral pion electroproduction at
Q^2 = 1.0 (GeV/c)^2 in 10 bins of W across the Delta resonance. A total of 16
independent response functions were extracted, of which 12 were observed for
the first time. Comparisons with recent model calculations show that response
functions governed by real parts of interference products are determined
relatively well near 1.232 GeV, but variations among models is large for
response functions governed by imaginary parts and for both increases rapidly
with W. We performed a nearly model-independent multipole analysis that adjusts
complex multipoles with high partial waves constrained by baseline models.
Parabolic fits to the W dependence of the multipole analysis around the Delta
mass gives values for SMR = (-6.61 +/- 0.18)% and EMR = (-2.87 +/- 0.19)% that
are distinctly larger than those from Legendre analysis of the same data.
Similarly, the multipole analysis gives Re(S0+/M1+) = (+7.1 +/- 0.8)% at
W=1.232 GeV, consistent with recent models, while the traditional Legendre
analysis gives the opposite sign because its truncation errors are quite
severe. Finally, using a unitary isobar model (UIM), we find that excitation of
the Roper resonance is dominantly longitudinal with S1/2 = (0.05 +/- 0.01)
GeV^(-1/2) at Q^2=1. The ReS0+ and ReE0+ multipoles favor pseudovector coupling
over pseudoscalar coupling or a recently proposed mixed-coupling scheme, but
the UIM does not reproduce the imaginary parts of 0+ multipoles well.Comment: 60 pages, 54 figure
Bound on global error of the fast multipole method for Helmholtz equation in 2-D
This paper analyze the global error of the fast multipole method(FMM) for
two-dimensional Helmholtz equation. We first propose the global error of the
FMM for the discretized boundary integral operator. The error is caused by
truncating Graf's addition theorem, according to the limiting forms of Bessel
and Neumann functions, we provide sharper and more precise estimates for the
truncations of Graf's addition theorem. Finally, using the estimates we derive
the explicit bound and convergence rate for the global error of the FMM for
Helmholtz equation, numerical experiments show that the results are valid. The
method in this paper can also be applied to the FMM for other problems such as
potential problems, elastostatic problems, Stokes flow problems and so on
Signatures of Chiral Dynamics in Low Energy Compton Scattering off the Nucleon
We present a projector formalism which allows to define dynamical
polarizabilities of the nucleon from a multipole expansion of the nucleon
Compton amplitudes. We give predictions for the energy dependence of these
dynamical polarizabilities both from dispersion theory and from
leading-one-loop chiral effective field theory. Based on the good agreement
between the two theoretical frameworks, we conclude that the energy dependence
of the dynamical polarizabilities is dominated by chiral dynamics, except in
those multipole channels where the first nucleon resonance Delta(1232) can be
excited. Both the dispersion theory framework and a chiral effective field
theory with explicit Delta(1232) degrees of freedom lead to a very good
description of the available low energy proton Compton data. We discuss the
sensitivity of the proton Compton cross section to dynamical polarizabilities
of different multipole content and present a fit of the static electric and
magnetic dipole polarizabilities from low-energy Compton data up to omega=170
MeV, finding alpha_E=(11.04+-1.36)*10^(-4) fm^3, beta_M =(2.76-+1.36)*10^(-4)
fm^3.Comment: 43 pages, 13 figure
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