1,895 research outputs found
Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods
Large linear systems with sparse, non-symmetric matrices arise in the
modeling of Markov chains or in the discretization of convection-diffusion
problems. Due to their potential to solve sparse linear systems with an effort
that is linear in the number of unknowns, algebraic multigrid (AMG) methods are
of fundamental interest for such systems. For symmetric positive definite
matrices, fundamental theoretical convergence results are established, and
efficient AMG solvers have been developed. In contrast, for non-symmetric
matrices, theoretical convergence results have been provided only recently. A
property that is sufficient for convergence is that the matrix be an M-matrix.
In this paper, we present how the simulation of incompressible fluid flows with
particle methods leads to large linear systems with sparse, non-symmetric
matrices. In each time step, the Poisson equation is approximated by meshfree
finite differences. While traditional least squares approaches do not guarantee
an M-matrix structure, an approach based on linear optimization yields
optimally sparse M-matrices. For both types of discretization approaches, we
investigate the performance of a classical AMG method, as well as an AMLI type
method. While in the considered test problems, the M-matrix structure turns out
not to be necessary for the convergence of AMG, problems can occur when it is
violated. In addition, the matrices obtained by the linear optimization
approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure
Schnelle Löser für partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
Metropolis Methods for Quantum Monte Carlo Simulations
Since its first description fifty years ago, the Metropolis Monte Carlo
method has been used in a variety of different ways for the simulation of
continuum quantum many-body systems. This paper will consider some of the
generalizations of the Metropolis algorithm employed in quantum Monte Carlo:
Variational Monte Carlo, dynamical methods for projector monte carlo ({\it
i.e.} diffusion Monte Carlo with rejection), multilevel sampling in path
integral Monte Carlo, the sampling of permutations, cluster methods for lattice
models, the penalty method for coupled electron-ionic systems and the Bayesian
analysis of imaginary time correlation functions.Comment: Proceedings of "Monte Carlo Methods in the Physical Sciences"
Celebrating the 50th Anniversary of the Metropolis Algorith
Orbital-enriched Flat-top Partition of Unity Method for the Schr\"odinger Eigenproblem
Quantum mechanical calculations require the repeated solution of a
Schr\"odinger equation for the wavefunctions of the system. Recent work has
shown that enriched finite element methods significantly reduce the degrees of
freedom required to obtain accurate solutions. However, time to solution has
been adversely affected by the need to solve a generalized eigenvalue problem
and the ill-conditioning of associated systems matrices. In this work, we
address both issues by proposing a stable and efficient orbital-enriched
partition-of-unity method to solve the Schr\"odinger boundary-value problem in
a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In
our proposed PUM, the three-dimensional domain is covered by overlapping
patches, with a compactly-supported, non-negative weight function, that is
identically equal to unity over some finite subset of its support associated
with each patch. This so-called flat-top property provides a pathway to devise
a stable approximation over the whole domain. On each patch, we use -th
degree orthogonal polynomials that ensure -th order completeness, and in
addition include eigenfunctions of the radial solution of the Schr\"odinger
equation. Furthermore, we adopt a variational lumping approach to construct a
block-diagonal overlap matrix that yields a standard eigenvalue problem and
demonstrate accuracy, stability and efficiency of the method.Comment: 24 pages, 12 figure
Multilevel Bayesian framework for modeling the production, propagation and detection of ultra-high energy cosmic rays
Ultra-high energy cosmic rays (UHECRs) are atomic nuclei with energies over
ten million times energies accessible to human-made particle accelerators.
Evidence suggests that they originate from relatively nearby extragalactic
sources, but the nature of the sources is unknown. We develop a multilevel
Bayesian framework for assessing association of UHECRs and candidate source
populations, and Markov chain Monte Carlo algorithms for estimating model
parameters and comparing models by computing, via Chib's method, marginal
likelihoods and Bayes factors. We demonstrate the framework by analyzing
measurements of 69 UHECRs observed by the Pierre Auger Observatory (PAO) from
2004-2009, using a volume-complete catalog of 17 local active galactic nuclei
(AGN) out to 15 megaparsecs as candidate sources. An early portion of the data
("period 1," with 14 events) was used by PAO to set an energy cut maximizing
the anisotropy in period 1; the 69 measurements include this "tuned" subset,
and subsequent "untuned" events with energies above the same cutoff. Also,
measurement errors are approximately summarized. These factors are problematic
for independent analyses of PAO data. Within the context of "standard candle"
source models (i.e., with a common isotropic emission rate), and considering
only the 55 untuned events, there is no significant evidence favoring
association of UHECRs with local AGN vs. an isotropic background. The
highest-probability associations are with the two nearest, adjacent AGN,
Centaurus A and NGC 4945. If the association model is adopted, the fraction of
UHECRs that may be associated is likely nonzero but is well below 50%. Our
framework enables estimation of the angular scale for deflection of cosmic rays
by cosmic magnetic fields; relatively modest scales of to
are favored. Models that assign a large fraction of UHECRs to a
single nearby source (e.g., Centaurus A) are ruled out unless very large
deflection scales are specified a priori, and even then they are disfavored.
However, including the period 1 data alters the conclusions significantly, and
a simulation study supports the idea that the period 1 data are anomalous,
presumably due to the tuning. Accurate and optimal analysis of future data will
likely require more complete disclosure of the data.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS654 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Relieving the fermionic and the dynamical sign problem: Multilevel Blocking Monte Carlo simulations
This article gives an introduction to the multilevel blocking (MLB) approach
to both the fermion and the dynamical sign problem in path-integral Monte Carlo
simulations. MLB is able to substantially relieve the sign problem in many
situations. Besides an exposition of the method, its accuracy and several
potential pitfalls are discussed, providing guidelines for the proper choice of
certain MLB parameters. Simulation results are shown for strongly interacting
electrons in a 2D parabolic quantum dot, the real-time dynamics of several
simple model systems, and the dissipative two-state dynamics (spin-boson
problem).Comment: Review, 20 pages REVTeX, incl. 7 figure
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