264 research outputs found
Mixed-precision AMG as linear equation solver for definite systems
AbstractThe performance of algebraic multigrid (AMG) algorithms, implemented in 4-byte floating point arithmetic, is investigated on modern cluster architecture with multi-core CPUs. The algorithmic considerations comprise the effect of preconditioning in 4-byte floating point arithmetic on Krylov solvers using standard 8-byte floating point arithmetic. The data of basic linear algebra benchmarks are used to interpret the performance of AMG algorithms employed as linear solvers in computational fluid dynamics simulation tools
Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of
large-scale linear systems that arise from the discretization of elliptic PDEs
amenable to rank compression. The preconditioner is based on hierarchical
low-rank approximations and the cyclic reduction method. The setup and
application phases of the preconditioner achieve log-linear complexity in
memory footprint and number of operations, and numerical experiments exhibit
good weak and strong scalability at large processor counts in a distributed
memory environment. Numerical experiments with linear systems that feature
symmetry and nonsymmetry, definiteness and indefiniteness, constant and
variable coefficients demonstrate the preconditioner applicability and
robustness. Furthermore, it is possible to control the number of iterations via
the accuracy threshold of the hierarchical matrix approximations and their
arithmetic operations, and the tuning of the admissibility condition parameter.
Together, these parameters allow for optimization of the memory requirements
and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics,
Dec 201
Exploiting spatial symmetries for solving Poisson's equation
This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetriesâ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystemsâ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid pointsâ ordering allows reducing the operatorsâ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solversâ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.This work has been financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Ădel Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively.Peer ReviewedPostprint (published version
Schnelle Löser fĂŒr partielle Differentialgleichungen
The workshop Schnelle LoÌser fuÌr 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
HAZniCS -- Software Components for Multiphysics Problems
We introduce the software toolbox HAZniCS for solving interface-coupled
multiphysics problems. HAZniCS is a suite of modules that combines the
well-known FEniCS framework for finite element discretization with solver and
graph library HAZmath. The focus of the paper is on the design and
implementation of a pool of robust and efficient solver algorithms which tackle
issues related to the complex interfacial coupling of the physical problems
often encountered in applications in brain biomechanics. The robustness and
efficiency of the numerical algorithms and methods is shown in several
numerical examples, namely the Darcy-Stokes equations that model flow of
cerebrospinal fluid in the human brain and the mixed-dimensional model of
electrodiffusion in the brain tissue
- âŠ