8,886 research outputs found
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
This work develops a nonlinear multigrid method for diffusion problems
discretized by cell-centered finite volume methods on general unstructured
grids. The multigrid hierarchy is constructed algebraically using aggregation
of degrees of freedom and spectral decomposition of reference linear operators
associated with the aggregates. For rapid convergence, it is important that the
resulting coarse spaces have good approximation properties. In our approach,
the approximation quality can be directly improved by including more spectral
degrees of freedom in the coarsening process. Further, by exploiting local
coarsening and a piecewise-constant approximation when evaluating the nonlinear
component, the coarse level problems are assembled and solved without ever
re-visiting the fine level, an essential element for multigrid algorithms to
achieve optimal scalability. Numerical examples comparing relative performance
of the proposed nonlinear multigrid solvers with standard single-level
approaches -- Picard's and Newton's methods -- are presented. Results show that
the proposed solver consistently outperforms the single-level methods, both in
efficiency and robustness
PORTA: A three-dimensional multilevel radiative transfer code for modeling the intensity and polarization of spectral lines with massively parallel computers
The interpretation of the intensity and polarization of the spectral line
radiation produced in the atmosphere of the Sun and of other stars requires
solving a radiative transfer problem that can be very complex, especially when
the main interest lies in modeling the spectral line polarization produced by
scattering processes and the Hanle and Zeeman effects. One of the difficulties
is that the plasma of a stellar atmosphere can be highly inhomogeneous and
dynamic, which implies the need to solve the non-equilibrium problem of the
generation and transfer of polarized radiation in realistic three-dimensional
(3D) stellar atmospheric models. Here we present PORTA, an efficient multilevel
radiative transfer code we have developed for the simulation of the spectral
line polarization caused by scattering processes and the Hanle and Zeeman
effects in 3D models of stellar atmospheres. The numerical method of solution
is based on the non-linear multigrid iterative method and on a novel
short-characteristics formal solver of the Stokes-vector transfer equation
which uses monotonic B\'ezier interpolation. Therefore, with PORTA the
computing time needed to obtain at each spatial grid point the self-consistent
values of the atomic density matrix (which quantifies the excitation state of
the atomic system) scales linearly with the total number of grid points.
Another crucial feature of PORTA is its parallelization strategy, which allows
us to speed up the numerical solution of complicated 3D problems by several
orders of magnitude with respect to sequential radiative transfer approaches,
given its excellent linear scaling with the number of available processors. The
PORTA code can also be conveniently applied to solve the simpler 3D radiative
transfer problem of unpolarized radiation in multilevel systems.Comment: 15 pages, 15 figures, to appear in Astronomy and Astrophysic
A Fast Parallel Poisson Solver on Irregular Domains Applied to Beam Dynamic Simulations
We discuss the scalable parallel solution of the Poisson equation within a
Particle-In-Cell (PIC) code for the simulation of electron beams in particle
accelerators of irregular shape. The problem is discretized by Finite
Differences. Depending on the treatment of the Dirichlet boundary the resulting
system of equations is symmetric or `mildly' nonsymmetric positive definite. In
all cases, the system is solved by the preconditioned conjugate gradient
algorithm with smoothed aggregation (SA) based algebraic multigrid (AMG)
preconditioning. We investigate variants of the implementation of SA-AMG that
lead to considerable improvements in the execution times. We demonstrate good
scalability of the solver on distributed memory parallel processor with up to
2048 processors. We also compare our SAAMG-PCG solver with an FFT-based solver
that is more commonly used for applications in beam dynamics
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
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