46,630 research outputs found
Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption
In this paper we give new results on domain decomposition preconditioners for
GMRES when computing piecewise-linear finite-element approximations of the
Helmholtz equation , with
absorption parameter . Multigrid approximations of
this equation with are commonly used as preconditioners
for the pure Helmholtz case (). However a rigorous theory for
such (so-called "shifted Laplace") preconditioners, either for the pure
Helmholtz equation, or even the absorptive equation (), is
still missing. We present a new theory for the absorptive equation that
provides rates of convergence for (left- or right-) preconditioned GMRES, via
estimates of the norm and field of values of the preconditioned matrix. This
theory uses a - and -explicit coercivity result for the
underlying sesquilinear form and shows, for example, that if , then classical overlapping additive Schwarz will perform optimally for
the absorptive problem, provided the subdomain and coarse mesh diameters are
carefully chosen. Extensive numerical experiments are given that support the
theoretical results. The theory for the absorptive case gives insight into how
its domain decomposition approximations perform as preconditioners for the pure
Helmholtz case . At the end of the paper we propose a
(scalable) multilevel preconditioner for the pure Helmholtz problem that has an
empirical computation time complexity of about for
solving finite element systems of size , where we have
chosen the mesh diameter to avoid the pollution effect.
Experiments on problems with , i.e. a fixed number of grid points
per wavelength, are also given
Transition from Poissonian to GOE level statistics in a modified Artin's billiard
One wall of Artin's billiard on the Poincar\'e half plane is replaced by a
one-parameter () family of nongeodetic walls. A brief description of the
classical phase space of this system is given. In the quantum domain, the
continuousand gradual transition from the Poisson like to GOE level statistics
due to the small perturbations breaking the symmetry responsible for the
'arithmetic chaos' at is studied. Another GOE \rightrrow Poisson
transition due to the mixed phase space for large perturbations is also
investigated. A satisfactory description of the intermediate level statistics
by the Brody distribution was found in boh cases. The study supports the
existence of a scaling region around . A finite size scaling relation
for the Brody-parameter as a function of and the number of levels
considered can be established
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
This paper analyses the following question: let , be
the Galerkin matrices corresponding to finite-element discretisations of the
exterior Dirichlet problem for the heterogeneous Helmholtz equations
. How small must and be (in terms of -dependence) for
GMRES applied to either or
to converge in a -independent number of
iterations for arbitrarily large ? (In other words, for to be
a good left- or right-preconditioner for ?). We prove results
answering this question, give theoretical evidence for their sharpness, and
give numerical experiments supporting the estimates.
Our motivation for tackling this question comes from calculating quantities
of interest for the Helmholtz equation with random coefficients and .
Such a calculation may require the solution of many deterministic Helmholtz
problems, each with different and , and the answer to the question above
dictates to what extent a previously-calculated inverse of one of the Galerkin
matrices can be used as a preconditioner for other Galerkin matrices
Previously unpublished Odonata records from Sarawak, Borneo : part 2, Kubah National Park
Records of Odonata from Kubah National Park, near Kuching in west Sarawak, are presented. Eighty-five species are known from the national park. Notable records include Drepanosticta drusilla, Rhinocypha species cf spinifer, Bornagriolestes species, Anaciaeschna species and Macromidia genialis erratica
Indirect Signals from Dark Matter in Split Supersymmetry
We study the possibilities for the indirect detection of dark matter in Split
Supersymmetry from gamma-rays, positrons, and antiprotons. The most promising
signal is the gamma-ray line, which may be observable at the next generation of
detectors. For certain halo profiles and a high mass neutralino, the line can
even be visible in current experiments. The continuous gamma-ray signal may be
observable, if there is a central spike in the galactic halo density. The
signals are found to be similar to those in MSSM models. These indirect signals
complement other experiments, being most easily observable for regions of
parameter space, such as heavy wino and higgsino dominated neutralinos, which
are least accessible for direct detection and accelerator searches.Comment: 10 pages, 5 figures; experimental sensitivities added to figure 2,
revised version to appear in Phys. Rev.
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