Parallel Algebraic Multigrid Methods on Distributed Memory Computers

Haase, Gundolf; Kuhn, Michael; Reitzinger, Stefan

doi:10.1137/s1064827501386237

Cited by 39 publications

(31 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The iteration method resulting from (5.9) is called consistent, if the exact solution x (∞) is a fixed point [153]. Introducing a damping parameter ω bŷ 17) it is possible to damp the high-frequency error components to obtain a better approximative solution. Starting from (5.12), the error of the iterative algorithm in step k can be computed by…”

Section: Iterative Methods In Circuit-level Analysismentioning

confidence: 99%

“…In numerical circuit-based EMC simulation an efficient successive solution of a series of large linear systems of equations has to be realized [17]. In some cases, factorizations of the system matrix can be reused for following simulation steps (e.g.…”

Section: Modern Solution Approachesmentioning

confidence: 99%

“…Applying the ACA-algorithm from Section 4.3.2 to compute a data-sparse representation of the BEM matrix, the prolongation process can be realized very simple and efficient. In principle, an ACA-compressed BEM matrix A h can be separated into nearfield and farfield part 17) as described in Section 4.3. 1.6, with the sparse matrix A near h and the approximate part A far h .…”

Section: Amg Preconditioners For Peec Matrices 115mentioning

confidence: 99%

“…For a parallelization of algebraic multigrid algorithms the application of a domain decomposition approach proved to be useful [17]. In contrast to geometric multgrid methods the coarsening hierarchy and the corresponding coarse-grid operators in AMG develop dynamically from one level to another why a parallelization is not trivial, in general [166].…”

Section: Parallelization Of Amgmentioning

confidence: 99%

“…In der numerischen schaltkreisbasierten EMV-Simulation ist es notwendig, eine effiziente aufeinander folgende Lösung großer linearer Gleichungssysteme zu realisieren [17]. Eine Effizienzsteigerung kann unter Umständen bereits dadurch erzielt werden, dass Faktorisierungen der Systemmatrix für nachfolgende Simulationsschritte wieder verwendet werden können (z.B.…”

Section: Appendix a ----------------------------------------------unclassified

See 4 more Smart Citations

Fast and Efficient Methods for Circuit-based Automotive EMC Simulation

Zitzmann

Weigel

2008

Springer Proceedings in Physics

View full text Add to dashboard Cite

Section: Iterative Methods In Circuit-level Analysismentioning

confidence: 99%

Section: Modern Solution Approachesmentioning

confidence: 99%

Section: Amg Preconditioners For Peec Matrices 115mentioning

confidence: 99%

Section: Parallelization Of Amgmentioning

confidence: 99%

Section: Appendix a ----------------------------------------------unclassified

See 3 more Smart Citations

Fast and Efficient Methods for Circuit-based Automotive EMC Simulation

Zitzmann

Weigel

2008

Springer Proceedings in Physics

View full text Add to dashboard Cite

Algebraic interface‐based coarsening AMG preconditioner for multi‐scale sparse matrices with applications to radiation hydrodynamics computation

2016

Numerical Linear Algebra App

View full text Add to dashboard Cite

Coarsening is a crucial component of algebraic multigrid (AMG) methods for iteratively solving sparse linear systems arising from scientific and engineering applications. Its application largely determines the complexity of the AMG iteration operator. Usually, high operator complexities lead to fast convergence of the AMG method; however, they require additional memory and as such do not scale as well in parallel computation. In contrast, although low operator complexities improve parallel scalability, they often lead to deterioration in convergence. This study introduces a new type of coarsening strategy called algebraic interface-based coarsening that yields a better balance between convergence and complexity for a class of multi-scale sparse matrices. Numerical results for various model-type problems and a radiation hydrodynamics practical application are provided to show the effectiveness of the proposed AMG solver. KEYWORDS algebraic multigrid (AMG), coarsening, preconditioning, parallel computation, multiscale sparse matrices, radiation hydrodynamics | INTRODUCTIONMany challenging computational problems in science and engineering need the solutions of large-scale sparse linear systems, that is,where A = (a ij ) n × n ∈ R n × n is a sparse matrix with entries a ij and u , f ∈ R n are the unknown and right-side vectors, respectively. The solver for these linear systems is a crucial component of simulation codes in diverse realistic applications and usually dominates the time consumption of simulations. Algebraic multigrid (AMG) 1-3 is one of the most efficient methods for iteratively solving linear systems arising from discretization of partial differential equations. In the past decades, challenges in real applications have sparked great interest in and further development of AMG methods. Many algorithm and theory variants have been developed to handle various problems (e.g., ). In particular, parallel algorithm designs and implementations of AMG have received considerable attention in the past 15 years as a means of scaling up to massively parallel machines, see, for example. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] Overall, AMG plays an increasingly important role in large-scale simulation for practical applications. 23,[48][49][50][51][52][53][54] The operator complexity (C op ) is a key performance measure for AMG methods. It is used as an important indicator of the computational cost of each iteration and storage requirements and is defined as follows:( 1:2) where A l denotes a matrix of level l and |A l | denotes the number of nonzero entries in A l . A 0 = A is the matrix of the finest * These authors contributed equally to this work.level (l = 0). The coarse-level matrix A l (l >0) usually comes from an application of the Galerkin approach (see Section 2). For large-scale computations on modern supercomputers, both low operator complexity and fast convergence are required. The classical AMG methods 3,32,39 are designed for fast convergence; however, they often...

show abstract

A distributed memory parallel implementation of the multigrid method for solving three‐dimensional implicit solid mechanics problems

Namazifard

Parsons

2004

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYWe describe the parallel implementation of a multigrid method for unstructured finite element discretizations of solid mechanics problems. We focus on a distributed memory programming model and use the MPI library to perform the required interprocessor communications. We present an algebraic framework for our parallel computations, and describe an object-based programming methodology using Fortran90. The performance of the implementation is measured by solving both fixed-and scaled-size problems on three different parallel computers (an SGI Origin2000, an IBM SP2 and a Cray T3E). The code performs well in terms of speedup, parallel efficiency and scalability. However, the floating point performance is considerably below the peak values attributed to these machines. Lazy processors are documented on the Origin that produce reduced performance statistics. The solution of two problems on an SGI Origin2000, an IBM PowerPC SMP and a Linux cluster demonstrate that the algorithm performs well when applied to the unstructured meshes required for practical engineering analysis.

show abstract

Parallel Algebraic Multigrid Methods on Distributed Memory Computers

Cited by 39 publications

References 15 publications

Fast and Efficient Methods for Circuit-based Automotive EMC Simulation

Fast and Efficient Methods for Circuit-based Automotive EMC Simulation

Algebraic interface‐based coarsening AMG preconditioner for multi‐scale sparse matrices with applications to radiation hydrodynamics computation

A distributed memory parallel implementation of the multigrid method for solving three‐dimensional implicit solid mechanics problems

Contact Info

Product

Resources

About