4,646 research outputs found
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc
We describe our software package Block Locally Optimal Preconditioned
Eigenvalue Xolvers (BLOPEX) publicly released recently. BLOPEX is available as
a stand-alone serial library, as an external package to PETSc (``Portable,
Extensible Toolkit for Scientific Computation'', a general purpose suite of
tools for the scalable solution of partial differential equations and related
problems developed by Argonne National Laboratory), and is also built into {\it
hypre} (``High Performance Preconditioners'', scalable linear solvers package
developed by Lawrence Livermore National Laboratory). The present BLOPEX
release includes only one solver--the Locally Optimal Block Preconditioned
Conjugate Gradient (LOBPCG) method for symmetric eigenvalue problems. {\it
hypre} provides users with advanced high-quality parallel preconditioners for
linear systems, in particular, with domain decomposition and multigrid
preconditioners. With BLOPEX, the same preconditioners can now be efficiently
used for symmetric eigenvalue problems. PETSc facilitates the integration of
independently developed application modules with strict attention to component
interoperability, and makes BLOPEX extremely easy to compile and use with
preconditioners that are available via PETSc. We present the LOBPCG algorithm
in BLOPEX for {\it hypre} and PETSc. We demonstrate numerically the scalability
of BLOPEX by testing it on a number of distributed and shared memory parallel
systems, including a Beowulf system, SUN Fire 880, an AMD dual-core Opteron
workstation, and IBM BlueGene/L supercomputer, using PETSc domain decomposition
and {\it hypre} multigrid preconditioning. We test BLOPEX on a model problem,
the standard 7-point finite-difference approximation of the 3-D Laplacian, with
the problem size in the range .Comment: Submitted to SIAM Journal on Scientific Computin
Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation
Spatially distributed problems are often approximately modelled in terms of
partial differential equations (PDEs) for appropriate coarse-grained quantities
(e.g. concentrations). The derivation of accurate such PDEs starting from finer
scale, atomistic models, and using suitable averaging, is often a challenging
task; approximate PDEs are typically obtained through mathematical closure
procedures (e.g. mean-field approximations). In this paper, we show how such
approximate macroscopic PDEs can be exploited in constructing preconditioners
to accelerate stochastic simulations for spatially distributed particle-based
process models. We illustrate how such preconditioning can improve the
convergence of equation-free coarse-grained methods based on coarse
timesteppers. Our model problem is a stochastic reaction-diffusion model
capable of exhibiting Turing instabilities.Comment: 8 pages, 6 figures, submitted to Journal of Chemical Physic
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