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Adaptive precision in block-Jacobi preconditioning for iterative sparse linear system solvers

Hartwig Anzt, Jack Dongarra, Goran Flegar, Nicholas J. Higham, Enrique S. Quintana-Ortí
2018 Concurrency and Computation  
Adaptive precision in block-Jacobi preconditioning for iterative sparse linear system solvers. Concurrency and Computation Practice and Experience. 31(6):1-12.  ...  This specialized preconditioner can then be combined with any Krylov subspace method for the solution of sparse linear systems to perform all arithmetic in double precision.  ...  ACKNOWLEDGEMENT We thank Matthias Bollhöfer for fruitful discussions on flexible variants of Krylov solvers allowing for nonconstant preconditioning operators and for pointing us to the flexible version  ... 
doi:10.1002/cpe.4460 fatcat:vkh3zx2l75bbvpvnjakollnowu

HotSpot Thermal Floorplan Solver Using Conjugate Gradient to Speed Up

Zhonghua Jiang, Ning Xu
2018 Mobile Information Systems  
The iterative conjugate gradient solver is suitable for traditional sparse matrix linear systems.  ...  We also defined the relative sparse matrix in the iterative thermal floorplan of Simulated Annealing framework algorithm, and the iterative method of relative sparse matrix could be applied to other iterative  ...  Funds for the Central Universities."  ... 
doi:10.1155/2018/2921451 fatcat:7yuathojd5db3pdaxemollolbm

Variable-Size Batched LU for Small Matrices and Its Integration into Block-Jacobi Preconditioning

Hartwig Anzt, Jack Dongarra, Goran Flegar, Enrique S. Quintana-Orti
2017 2017 46th International Conference on Parallel Processing (ICPP)  
sparse linear systems.  ...  The development of these kernels is motivated by the need for tackling this embarrasingly-parallel scenario in the context of block-Jacobi preconditioning that is relevant for the iterative solution of  ...  linear solvers" (d65).  ... 
doi:10.1109/icpp.2017.18 dblp:conf/icpp/AnztDFQ17 fatcat:sryw4eagnnf3zbuzllaxhm3ebm

Enabling Next-Generation Parallel Circuit Simulation with Trilinos [chapter]

Chris Baker, Erik Boman, Mike Heroux, Eric Keiter, Siva Rajamanickam, Rich Schiek, Heidi Thornquist
2012 Lecture Notes in Computer Science  
These generally involve a higher-level partitioning of the devices [6] or lower-level partitioning of the linear system of equations [7] to facilitate the creation of a more efficient parallel matrix solver  ...  solvers [2] .  ...  and block Jacobi preconditioning (KLU used to factor the diagonal blocks).  ... 
doi:10.1007/978-3-642-29737-3_36 fatcat:hrkojy2vnrfndnywuumhh376xu

Towards Neural Sparse Linear Solvers [article]

Luca Grementieri, Paolo Galeone
2022 arXiv   pre-print
We propose neural sparse linear solvers, a deep learning framework to learn approximate solvers for sparse symmetric linear systems.  ...  Large sparse symmetric linear systems appear in several branches of science and engineering thanks to the widespread use of the finite element method (FEM).  ...  For example, we could adapt it to predict an optimal preconditioner matrix for a linear system or an improved initialization for a specific iterative method.  ... 
arXiv:2203.06944v1 fatcat:2rtfjeankbgztawvasgww5c7bu

Machine Learning-Aided Numerical Linear Algebra: Convolutional Neural Networks for the Efficient Preconditioner Generation

Markus Goetz, Hartwig Anzt
2018 2018 IEEE/ACM 9th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems (scalA)  
more than 95% prediction accuracy, and the resulting block-Jacobi preconditioner effectively accelerating an iterative GMRES solver.  ...  Generating sparsity patterns for effective block-Jacobi preconditioners is a challenging and computationally expensive task, in particular for problems with unknown origin.  ...  INTRODUCTION In scientific computing, the process of iteratively solving a linear system of equations often strongly benefits from the use of a sophisticated preconditioner that carefully adapts to the  ... 
doi:10.1109/scala.2018.00010 fatcat:erohybz6irb27bwfgasglppb2y

Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing [article]

Hartwig Anzt, Terry Cojean, Goran Flegar, Fritz Göbel, Thomas Grützmacher, Pratik Nayak, Tobias Ribizel, Yuhsiang Mike Tsai, Enrique S. Quintana-Ortí
2020 arXiv   pre-print
In this paper, we present Ginkgo, a modern C++ math library for scientific high performance computing.  ...  Ginkgo's current focus is oriented towards providing sparse linear algebra functionality for high performance GPU architectures, but given the library design, this focus can be easily extended to accommodate  ...  via a sparse matrix (incomplete sparse approximate inverse preconditioning [16] ). e block-Jacobi preconditioner available in G outperforms its competitors by automatically adapting the memory precision  ... 
arXiv:2006.16852v2 fatcat:swp3f5sglzgr3ola6ecfocp2ia

A Study of Mixed Precision Strategies for GMRES on GPUs [article]

Jennifer A. Loe, Christian A. Glusa, Ichitaro Yamazaki, Erik G. Boman, Sivasankaran Rajamanickam
2021 arXiv   pre-print
In this paper, we focus on preconditioned sparse iterative linear solvers, a key kernel in several CSE applications.  ...  We seek the best methods for incorporating multiple precisions into the GMRES linear solver; these include iterative refinement and parallelizable preconditioners.  ...  We focus on one of the expensive portions of solving PDEs, the sparse linear solve. While there are several approaches for solving sparse linear systems, we focus on sparse iterative linear solvers.  ... 
arXiv:2109.01232v1 fatcat:celwzxytdnaupad2zffcuivzw4

GPU-accelerated preconditioned iterative linear solvers

Ruipeng Li, Yousef Saad
2012 Journal of Supercomputing  
This work is an overview of our preliminary experience in developing high-performance iterative linear solver accelerated by GPU co-processors.  ...  Techniques for speeding up sparse matrix-vector product (SpMV) kernels and finding suitable preconditioning methods are discussed.  ...  The Block-ILU preconditioned GMRES method is studied for solving sparse linear systems on the NVIDIA Tesla GPUs in [27] .  ... 
doi:10.1007/s11227-012-0825-3 fatcat:cc6x2cxrbbe5tlqujevezdjm4q

Using Random Butterfly Transformations in Parallel Schur Complement-Based Preconditioning

Marc Baboulin, Aygul Jamal, Masha Sosonkina
2015 Proceedings of the 2015 Federated Conference on Computer Science and Information Systems  
of sparse linear systems.  ...  We propose to use a randomization technique based on Random Butterfly Transformations (RBT) in the Algebraic Recursive Multilevel Solver (ARMS) to improve the preconditioning phase in the iterative solution  ...  Section II presents the preconditioned Krylov subspace method (PKSM) and the parallel Algebraic Recursive Multilevel Solver (pARMS) for solving sparse linear systems.  ... 
doi:10.15439/2015f177 dblp:conf/fedcsis/BaboulinJS15 fatcat:7udjszd5yrekjotj3mm3gcju4y

Concurrent number cruncher: a GPU implementation of a general sparse linear solver

Luc Buatois, Guillaume Caumon, Bruno Lévy
2009 International Journal of Parallel, Emergent and Distributed Systems  
), to implement a sparse general-purpose linear solver.  ...  CUDA even provides a BLAS implementation, but only for dense matrices (CuBLAS). Other existing linear solvers for the GPU are also limited by their internal matrix representation.  ...  Acknowledgements The authors thank the members of the GOCAD research consortium for their support (, Xavier Cavin, Bruno Stefanizzi from AMD-ATI for providing the CTM API and the associated  ... 
doi:10.1080/17445760802337010 fatcat:6hs2z5v4svcj5k76id46s2gdr4

Concurrent Number Cruncher: An Efficient Sparse Linear Solver on the GPU [chapter]

Luc Buatois, Guillaume Caumon, Bruno Lévy
2007 Lecture Notes in Computer Science  
By combining recent GPU programming techniques with supercomputing strategies (namely block compressed row storage and register blocking), we implement a sparse generalpurpose linear solver which outperforms  ...  A wide class of geometry processing and PDE resolution methods needs to solve a linear system, where the non-zero pattern of the matrix is dictated by the connectivity matrix of the mesh.  ...  Acknowledgements The authors thank the members of the GOCAD research consortium for their support (, Xavier Cavin, and Bruno Stefanizzi from ATI for providing the CTM API and the associated  ... 
doi:10.1007/978-3-540-75444-2_37 fatcat:icrnxj5jpbh2jmizlloxn64to4

Parallel sparse matrix computations using the PINEAPL library: A performance study [chapter]

Arnold R. Krommer
1998 Lecture Notes in Computer Science  
These modules provide support for crucial computational tasks such as graph partitioning, preconditioning and iterative solution of linear systems.  ...  Additional support routines assist users in distributing and assembling the data structures used and/or generated by the sparse linear algebra modules.  ...  They provide support for crucial computational tasks such as graph partitioning, preconditioning and iterative solution of linear systems.  ... 
doi:10.1007/bfb0057934 fatcat:rsxqvtk5vvaqvdowz2rilh3uea

Towards an Exascale Enabled Sparse Solver Repository [chapter]

Jonas Thies, Martin Galgon, Faisal Shahzad, Andreas Alvermann, Moritz Kreutzer, Andreas Pieper, Melven Röhrig-Zöllner, Achim Basermann, Holger Fehske, Georg Hager, Bruno Lang, Gerhard Wellein
2016 Lecture Notes in Computational Science and Engineering  
of some prototypical iterative schemes for computing eigenvalues of sparse matrices.  ...  We discuss the development of a new 'Exascale enabled' sparse solver repository (the ESSR) that addresses these challenges-from fundamental design considerations and development processes to actual implementations  ...  We would like to thank Michael Meinel (DLR Simulation and Software Technology, software engineering group) for helpful comments on the manuscript.  ... 
doi:10.1007/978-3-319-40528-5_13 fatcat:jancdp27w5hktf5y6utn533zwi

Compressed Basis GMRES on High Performance GPUs [article]

José I. Aliaga and Hartwig Anzt and Thomas Grützmacher and Enrique S. Quintana-Ortí and Andrés E. Tomás
2020 arXiv   pre-print
Krylov methods provide a fast and highly parallel numerical tool for the iterative solution of many large-scale sparse linear systems.  ...  We develop a high performance implementation of the "compressed basis GMRES" solver in the Ginkgo sparse linear algebra library and using a large set of test problems from the SuiteSparse matrix collection  ...  Krylov solvers enhanced with some type of sophisticated preconditioning technique nowadays compound a popular approach for the iterative solution of large and sparse linear systems [24] .  ... 
arXiv:2009.12101v1 fatcat:ngvb4j3xdfbbjbdrwa5khr3u3a
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