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Solving Large Sparse Lyapunov Equations on Parallel Computers [chapter]

José M. Badía, Peter Benner, Rafael Mayo, Enrique S. Quintana-Ortí
2002 Lecture Notes in Computer Science  
This paper describes the parallelization of the low-rank ADI iteration for the solution of large-scale, sparse Lyapunov equations.  ...  The only relevant operations involved in the method are matrix-vector products and the solution of linear systems.  ...  The parallelization of the solver requires parallel routines for the computation of the matrix-vector product, the solution of linear systems involving sparse (complex) matrices, and some other minor operations  ... 
doi:10.1007/3-540-45706-2_95 fatcat:vyzazexoqzcqpgjggida7a2msu

Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D

L. Giraud, A. Haidar, Y. Saad
2010 Numerical Mathematics: Theory, Methods and Applications  
In earlier works, the local Schur complements were computed exactly using a sparse direct solver.  ...  The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems; preliminary experiments on linear systems arising from structural  ...  Some preliminary experiments on linear systems arising from industrial structural mechanics computational are also reported.  ... 
doi:10.4208/nmtma.2010.33.2 fatcat:pg66gkwbfzapvchyxukhtfhefa

Using Mixed Precision for Sparse Matrix Computations to Enhance the Performance while Achieving 64-bit Accuracy

Alfredo Buttari, Jack Dongarra, Jakub Kurzak, Piotr Luszczek, Stanimir Tomov
2008 ACM Transactions on Mathematical Software  
These ideas can be applied to sparse multifrontal and supernodal direct techniques and sparse iterative techniques such as Krylov subspace methods.  ...  By using a combination of 32-bit and 64-bit floating point arithmetic, the performance of many sparse linear algebra algorithms can be significantly enhanced while maintaining the 64-bit accuracy of the  ...  VIII Time to Solution of SuperLU in Single and Double Precision Solvers for the Selected Sparse Matrices on Intel Woodcrest with Reference and Optimized BLAS Reference BLAS Goto BLAS Name t D P t SP  ... 
doi:10.1145/1377596.1377597 fatcat:5hrp5k5zy5buzlksp6jgirwrxi

On the Efficiency of Supernodal Factorization in Interior-Point Method using CPU-GPU Collaboration

Usman Ali Shah, Suhail Yousaf, Iftikhar Ahmad, Muhammad Ovais Ahmad
2020 IEEE Access  
Factorization method used in the state-of-the-art solver performs only selected operations related to large supernodes on GPU.  ...  To overcome this difficulty, the state-of-the-art hybrid (CPU-GPU) implementation of PDIPM exploits presence of supernodes in sparse matrices during factorization.  ...  We experimented with different values of and found =1×10 -14 to be a suitable value that ensured convergence to the solution in minimal number of iterations while eliminating numerical instability.  ... 
doi:10.1109/access.2020.3006353 fatcat:aj3k6kuvorhaboeni4dypfeeqq

Energy Analysis of a Solver Stack for Frequency-Domain Electromagnetics

Emmanuel Agullo, Luc Giraud, Stephane Lanteri, Gilles Marait, Anne-Cecile Orgerie, Louis Poirel
2019 2019 27th Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP)  
This solver stack combines a high order finite element discretization framework of the system of three-dimensional frequency-domain Maxwell equations with an algebraic hybrid iterative-direct sparse linear  ...  solver.  ...  In a preliminary version of this software, this system was solved using parallel sparse direct solvers such as MUMPS [7] or PaStiX [9] .  ... 
doi:10.1109/empdp.2019.8671555 dblp:conf/pdp/AgulloGLMOP19 fatcat:7lnjilmcdjaopcvtiappsxdaiq

Iterative versus direct parallel substructuring methods in semiconductor device modelling

L. Giraud, A. Marrocco, J.-C. Rioual
2005 Numerical Linear Algebra with Applications  
In this paper, we present the various ingredients of some hybrid iterative schemes that play a central role in the robustness of those solvers when they are embedded in other numerical schemes.  ...  We show that iterative solvers can be robust enough to solve the very challenging linear systems that appear in those simulations.  ...  Acknowledgments We would like to thank Parallab (Bergen, Norway) and CINES (Montpellier, France) for providing us with an access to their SGI O2000 platform.  ... 
doi:10.1002/nla.391 fatcat:ouf46jwslzgbbl7irt7wp6njae

High performance verified computing using C-XSC

Walter Krämer
2013 Computational and Applied Mathematics  
Please note that the development of sparse HPVC solvers in C-XSC is still in progress. The time measurements concerning such solvers are very promising but still preliminary.  ...  Details on the final versions of the sparse solvers will be published in Zimmer (2012) .  ...  Meanwhile, also a small set of (preliminary) C-XSC solvers for sparse linear systems is available.  ... 
doi:10.1007/s40314-013-0028-4 fatcat:3ktcbk5g5fbjjgtalz3nc3thlu

An Evaluation of Sparse Direct Symmetric Solvers: An Introduction and Preliminary Findings [chapter]

Jennifer A. Scott, Yifan Hu, Nicholas I. M. Gould
2006 Lecture Notes in Computer Science  
In recent years a number of solvers for the direct solution of large sparse, symmetric linear systems of equations have been developed.  ...  In this study, we use performance profiles as a tool for evaluating and comparing the performance of serial sparse direct solvers on an extensive set of symmetric test problems taken from a range of practical  ...  Acknowledgements We are grateful to all the authors of the solvers who supplied us with copies of their codes and documentation, helped us to use their software and answered our many queries.  ... 
doi:10.1007/11558958_98 fatcat:xllelrqtj5cnhpsbk55tkor3qm

A High Performance Dual Revised Simplex Solver [chapter]

Julian Hall, Qi Huangfu
2012 Lecture Notes in Computer Science  
NVIDIA GPU • For dense LP problems: best results Solver Type HPC Time Iterations Speed (iter/s) gurobi primal RSM serial 1357 16034 12 gurobi dual RSM serial 976 14518 15 i6 primal SSM parallel 4039 288419  ...  Operation with N k 1 k 1 k 1 0• 1 k 1 0 11111 (2010) Nvidia interested in case of sparse vector • Operations with H −Can be posed as a sparse matrix-vector product • Need to limit data transfer between  ...  alternative product form update may offer a solution • Updating representation of B −1 each iteration exploits so, using Sherman-Morrison, • Hence reversed the order of inverse and update in representation  ... 
doi:10.1007/978-3-642-31464-3_15 fatcat:372xzklr5befnf7hxy2pfi3w6i

Preliminary Implementation of PETSc Using GPUs [chapter]

Victor Minden, Barry Smith, Matthew G. Knepley
2013 Lecture Notes in Earth System Sciences  
The Krylov methods, nonlinear solvers, and integrators in PETSc run unchanged in parallel using these new subclasses.  ...  PETSc is organized as a class library with classes for vectors, sparse and dense matrices, Krylov methods, preconditioners, nonlinear solvers, and differential equation integrators.  ...  Acknowledgements We thank Nathan Bell from NVIDIA and Lisandro Dalcin for their assistance with this project.  ... 
doi:10.1007/978-3-642-16405-7_7 fatcat:yj4v7zm7yvcm7kpaoxajirjisa

Applying Parallel Direct Solver Techniques to Build Robust High Performance Preconditioners [chapter]

Pascal Hénon, François Pellegrini, Pierre Ramet, Jean Roman, Yousef Saad
2006 Lecture Notes in Computer Science  
The purpose of our work is to provide a method which exploits the parallel blockwise algorithmic approach used in the framework of high performance sparse direct solvers in order to develop robust preconditioners  ...  solvers.  ...  We provide some experiments in section 3. At last we give some conclusions in section 4.  ... 
doi:10.1007/11558958_73 fatcat:6aqzxiuldbg7hppwdqemnii25m

Performance evaluation of the Sakurai-Sugiura method with a block Krylov subspace linear solver for large dense Hermitian-definite generalized eigenvalue problems

Takahiro Yano, Yasunori Futamura, Akira Imakura, Tetsuya Sakurai
2018 JSIAM Letters  
In this paper, we consider applying a contour-integral based method to a large dense problem in conjunction with a block Krylov subspace method as an inner linear solver.  ...  Comparison of parallel performance with the contour-integral based method with a direct linear solver and a ScaLAPACK's eigensolver is shown using matrices from a practical application.  ...  The iteration counts with respect to all quadrature points are measured in a preliminary experiment.  ... 
doi:10.14495/jsiaml.10.77 fatcat:hudmtbphnbhnlcbr725rylr7xq

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.  ...  Our experiments with an NVIDIA TESLA C1060 show that for unstructured matrices SpMV kernels can be up to 10 times faster on the GPU than on the host Intel Xeon E5504 Processor.  ...  We highlighted a few alternative approaches to standard ones in the arena of iterative sparse linear system solvers.  ... 
doi:10.1007/s11227-012-0825-3 fatcat:cc6x2cxrbbe5tlqujevezdjm4q

Solving dense generalized eigenproblems on multi-threaded architectures

José I. Aliaga, Paolo Bientinesi, Davor Davidović, Edoardo Di Napoli, Francisco D. Igual, Enrique S. Quintana-Ortí
2012 Applied Mathematics and Computation  
The experimental results on a state-of-the-art 8-core platform, equipped with a graphics processing unit (GPU), reveal that in real applications, iterative Krylov-subspace methods can be a competitive  ...  of the solvers.  ...  In Experiment 1, all methods yield similar results while, in Experiment 2, the iterative solvers present slightly better accuracies.  ... 
doi:10.1016/j.amc.2012.05.020 fatcat:5rmnbtzh4nadfjcnm5ifeplq4y

The roles of sparse direct methods in large-scale simulations

X S Li, W Gao, P J R Husbands, C Yang, E G Ng
2005 Journal of Physics, Conference Series  
Most of these systems are very ill-conditioned, resulting in extremely poor convergence (sometimes no convergence) for many iterative solvers. we have successfully deployed our direct methods techniques  ...  Sparse systems of linear equations and eigen-equations arise at the heart of many large-scale, vital simulations in DOE.  ...  In particular, the large scale of the problem revealed some bottlenecks in the solver that were not detected while testing smaller problems, and our SciDAC-funded work has been mostly in parallel algorithms  ... 
doi:10.1088/1742-6596/16/1/065 fatcat:ujeo2vnbdfgbjhzzrmxhyvo3te
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