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On long-range interpolation operators for aggressive coarsening

Ulrike Meier Yang
2009 Numerical Linear Algebra with Applications  
Two new long range interpolation operators to be used in combination with aggressive coarsening are presented.  ...  Algebraic multigrid (AMG) is a very efficient scalable preconditioner for solving sparse linear systems on unstructured grids.  ...  Acknowledgments The author thanks Joshua Nolting for implementing the version of the distance-two multipass interpolation routine that uses option 2 to choose the points for the first pass and Tzanio Kolev  ... 
doi:10.1002/nla.689 fatcat:37itoaky6je57jreu373komsrm

An algebraic distances measure of AMG strength of connection [article]

Achi Brandt, James Brannick, Karsten Kahl, Ira Livshits
2011 arXiv   pre-print
The current coarsening methodology is based on measuring how a so-called algebraically smooth error value at one point depends on the error values at its neighbors.  ...  Such a concept of strength of connection is well understood for operators whose principal part is an M-matrix; however, the strength concept for more general matrices is not yet clearly understood, and  ...  A main new feature of our coarsening algorithm is that it allows for aggressive and problem-oriented coarsening as well as long-range interpolation in a straightforward way.  ... 
arXiv:1106.5990v1 fatcat:emffnegc75fa3bljkrlc3kwxd4

High-performance algebraic multigrid solver optimized for multi-core based distributed parallel systems

Jongsoo Park, Mikhail Smelyanskiy, Ulrike Meier Yang, Dheevatsa Mudigere, Pradeep Dubey
2015 Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC '15  
Algebraic Multigrid (amg) is a linear solver, well known for its linear computational complexity and excellent parallelization scalability.  ...  As a result, amg is expected to be a solver of choice for emerging extreme scale systems capable of delivering hundred Pflops and beyond.  ...  We also thank Intel Endeavor team for their competent cluster support.  ... 
doi:10.1145/2807591.2807603 dblp:conf/sc/ParkSYMD15 fatcat:oss46ojlhrdsno2hmbibeyfk3q

Multi-Level Ewald: A Hybrid Multigrid/Fast Fourier Transform Approach to the Electrostatic Particle-Mesh Problem

David S. Cerutti, David A. Case
2010 Journal of Chemical Theory and Computation  
We present a new method for decomposing the one convolution required by standard Particle-Particle Particle-Mesh (P 3 M) electrostatic methods into a series of convolutions over slab-shaped subregions  ...  We anticipate that the MLE method's ability to break a single convolution into many independent subproblems will be useful for extending the parallel scaling of molecular simulations.  ...  David Hardy for helpful conversations. This work was funded by NIH grant RR12255.  ... 
doi:10.1021/ct900522g pmid:22039358 pmcid:PMC3202610 fatcat:7o2fzarenbgtvbbwt7mys5afa4

Reducing Complexity in Parallel Algebraic Multigrid Preconditioners

Hans De Sterck, Ulrike Meier Yang, Jeffrey J. Heys
2006 SIAM Journal on Matrix Analysis and Applications  
Two new parallel AMG coarsening schemes are proposed, that are based on solely enforcing a maximum independent set property, resulting in sparser coarse grids.  ...  The new coarsening techniques remedy memory and execution time complexity growth for various large three-dimensional (3D) problems.  ...  Additional C-points or long-range interpolation may be required here.  ... 
doi:10.1137/040615729 fatcat:jfkdhihsyvdzbmiwrxzstbwdje

A highly parallel multilevel Newton-Krylov-Schwarz method with subspace-based coarsening and partition-based balancing for the multigroup neutron transport equations on 3D unstructured meshes [article]

Fande Kong, Yaqi Wang, Derek R. Gaston, Cody J. Permann, Andrew E. Slaughter, Alexander D. Lindsay, Richard C. Martineau
2019 arXiv   pre-print
We numerically show that the proposed algorithm is scalable with more than 10,000 processor cores for a realistic application with a few billions unknowns on 3D unstructured meshes.  ...  The multilevel domain decomposition methods is one of the most popular algorithms for solving the multigroup neutron transport equations, but the construction of coarse spaces is expensive and often not  ...  If an aggressive coarsening scheme such as Alg. 4.4 is adopted, it is necessary to use a long range interpolation, such as a "multipass interpolation" as described in [28] , in order to achieve a reasonable  ... 
arXiv:1903.03659v1 fatcat:j667z67y4ndh3onhsy5uvfa4ha

A comparison of element agglomeration algorithms for unstructured geometric multigrid [article]

S. Dargaville, A.G. Buchan, R.P. Smedley-Stevenson, P.N. Smith, C.C. Pain
2020 arXiv   pre-print
In two dimensions all coarsening algorithms result in multigrid methods which perform similarly, but in three dimensions aggressive element agglomeration performed by METIS produces the shortest runtimes  ...  The resulting multigrid schemes are tested matrix-free on two problems in 2D and 3D taken from radiation transport applications; one of which is in the diffusion limit.  ...  force the coarsening on the top grid to be aggressive by agglomerating many fine elements, while decreasing the aggressiveness on lower grids.  ... 
arXiv:2005.09104v1 fatcat:iv4fzei2dnh7dgaf73umjg2obi

A review of algebraic multigrid

K. Stüben
2001 Journal of Computational and Applied Mathematics  
AMG can directly be applied, for instance, to e ciently solve various types of elliptic partial di erential equations discretized on unstructured meshes, both in 2D and 3D.  ...  For practically relevant problem sizes, classical one-level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an e cient solution of even larger  ...  Such "long-range" interpolation [48] generally allows a much faster coarsening and drastically increases the sparsity on coarser levels.  ... 
doi:10.1016/s0377-0427(00)00516-1 fatcat:q2kwtxvbmbakpgnbxihbplrece

Bootstrap Algebraic Multigrid: status report, open problems, and outlook [article]

Achi Brandt, James Brannick, Karsten Kahl, Ira Livshits
2014 arXiv   pre-print
, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process.  ...  This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies  ...  We refer to the paper [8] in which algebraic distance and long-range LS interpolation were studied for non-grid-aligned anisotropic problems for additional details and results on this topic.  ... 
arXiv:1406.1819v1 fatcat:6t4yjt46djepfbluil4vdc4oam

Improving algebraic multigrid interpolation operators for linear elasticity problems

A. H. Baker, Tz. V. Kolev, U. M. Yang
2009 Numerical Linear Algebra with Applications  
In this paper we investigate several approaches for improving AMG convergence on linear elasticity problems by explicitly incorporating the near-nullspace modes in the range of the interpolation.  ...  We demonstrate the effectiveness of the new interpolation operators on several 2D and 3D elasticity problems.  ...  ACKNOWLEDGEMENT We thank John Ruge for many insightful discussions and for inspiring variant 2 of the GM approach, and we thank Panayot Vassilevski for his helpful input and advice.  ... 
doi:10.1002/nla.688 fatcat:cnqkuufj3nhaxl5mahin3emppq

An introduction to algebraic multigrid

R.D. Falgout
2006 Computing in science & engineering (Print)  
Algebraic multigrid (AMG) solves linear systems based on multigrid principles, but in a way that only depends on the coefficients in the underlying matrix.  ...  The algorithm indicated by circles achieves decent scaling results by controlling complexity through more aggressive coarsening and the use of long-range interpolation with no second pass.  ...  In the theorems, E is the multigrid operator, M is an operator derived from the smoother, P is interpolation, and R is a restriction-like operator such that RP = I (so that P R is a projection onto range  ... 
doi:10.1109/mcse.2006.105 fatcat:soltxkqhtrde5kqsuujlp6ddxy

Algebraic distance for anisotropic diffusion problems: multilevel results [article]

A. Brandt, J. Brannick, K. Kahl, I. Livshits
2014 arXiv   pre-print
This algebraic distance measure, combined with compatible relaxation, is used to choose suitable coarse grids and accurate interpolation operators for algebraic multigrid algorithms.  ...  In this paper we motivate, discuss the implementation and present the resulting numerics for a new definition of strength of connection which is based on the notion of algebraic distance.  ...  long-range interpolation with caliber c > 2, as needed to accurately capture general anisotropies, that at the same time maintains low grid and operator complexities.  ... 
arXiv:1409.4702v1 fatcat:i6v7yfqms5ethnxvsmjtlfm7mm

Algebraic multigrid for the finite pointset method

Bram Metsch, Fabian Nick, Jörg Kuhnert
2020 Computing and Visualization in Science  
The discretization of the differential operators used in FPM leads to non-symmetric matrices that do not have the M-matrix property.  ...  In the segregated approach, three pressure systems and one velocity system need to be solved.  ...  Furthermore, we want to thank John Ruge from FRSC for many helpful hints and advice.  ... 
doi:10.1007/s00791-020-00324-3 fatcat:ihozorekyzg2nkknvnqc6wmdd4

A Matrix Dependent/Algebraic Multigrid Approach for Extruded Meshes with Applications to Ice Sheet Modeling

R. Tuminaro, M. Perego, I. Tezaur, A. Salinger, S. Price
2016 SIAM Journal on Scientific Computing  
Additionally, an aggressive coarsening version of the semicoarsening algorithm is developed so that the ratio of the number of unknowns between consecutive levels is much greater than two.  ...  When the resulting coarse mesh contains only one vertical layer, further coarsening is applied in the horizontal direction using standard AMG techniques.  ...  We thank Ali Pinar for suggesting the idea of combining coloring and connected component algorithms to detect hinged peninsulas within graphs.  ... 
doi:10.1137/15m1040839 fatcat:pvpdnaoh2rcoldrob5g4pbdr4m

Scaling Hypre's Multigrid Solvers to 100,000 Cores [chapter]

Allison H. Baker, Robert D. Falgout, Tzanio V. Kolev, Ulrike Meier Yang
2012 High-Performance Scientific Computing  
The hypre software library [15] is a collection of high performance preconditioners and solvers for large sparse linear systems of equations on massively parallel machines.  ...  We present scaling results on over 100,000 cores and even solve a problem with over a trillion unknowns.  ...  [22] , or even more aggressive coarsening schemes, which need interpolation with an even longer range [24, 26] .  ... 
doi:10.1007/978-1-4471-2437-5_13 fatcat:j4ge3obbjrenzex37j6hifd5gm
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