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Parallel algorithms for forward and back substitution in direct solution of sparse linear systems

Anshul Gupta, Vipin Kumar
1995 Proceedings of the 1995 ACM/IEEE conference on Supercomputing (CDROM) - Supercomputing '95  
A few parallel algorithms for solving triangular systems resulting from parallel factorization of sparse linear systems have been proposed and implemented recently.  ...  Our analysis and experiments show that, although not as scalable as the best parallel sparse Cholesky factorization algorithms, parallel sparse triangular solvers can yield reasonable speedups in runtime  ...  The process of obtaining a direct solution to a sparse system of linear equations usually consists of four phases: reordering, symbolic factorization, numerical factorization, and forward elimination and  ... 
doi:10.1145/224170.224471 dblp:conf/sc/GuptaK95 fatcat:nfqpljy7wredncqddhmq44s2xm

Parallel Algorithms for Forward and Back Substitution in Linear Algebraic Equations of Finite Element Method

Sergiy Fialko
2019 Journal of Telecommunications and Information Technology  
This paper considers several algorithms for parallelizing the procedure of forward and back substitution for high-order symmetric sparse matrices on multi-core computers with shared memory.  ...  It compares the proposed approaches for various finite-element problems of structural mechanics which generate sparse matrices of different structures.  ...  Acknowledgements The author is deeply grateful to SCAD Soft for providing support and for the collection of real-life problems involving design models created by SCAD Office users.  ... 
doi:10.26636/jtit.2019.134919 fatcat:o6ctsvrbpzbcvgnyos6lhjiqia

COMPLEXITY ANALYSIS OF REORDERING ALGORITHMS FOR THE IMPLEMENTATION OF MULTIFRONTAL SOLUTION TECHNIQUE

Anchu C.
2016 International Journal of Advanced Research  
Solution of large order sparse system of equations need to be optimized on numerical computers in terms of memory usage and execution time.  ...  This thesis studies the effect of cholesky factorization of symmetric sparse matrices for reducing fill-ins and factorization time.  ...  DEEPAK P,Scientist/Engineer 'SE', SMSD/SDMG/STR, VSSC, ISRO,Thiruvananthapuram , in understanding reordering algorithms for the implementation of multifrontal solution technique.  ... 
doi:10.21474/ijar01/1325 fatcat:nmhebe2qafaxnkidvblagar45a

Page 2011 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
Summary: “In a fully parallel sparse direct solver the matrix factors are first computed and then used in forward and back substitution steps to compute the solution.  ...  For strictly el- liptic operators all steps of the algorithms employ sparse (banded) matrices; an @(N) direct method for solving systems of equations is obtained instead of the usual @(N°?) costs.  ... 

Page 165 of American Society of Civil Engineers. Collected Journals Vol. 126, Issue 4 [page]

2000 American Society of Civil Engineers. Collected Journals  
However, the forward and back substitution steps [(7)] are replaced with matrix-vector products [(8)], which are quite amenable to parallel processing.  ...  The solution to this problem is found in three steps: forward substitution, diagonal scaling and backward substitution Iz=b; Dy=z; Ux=y (7) The factorization is usually preceded by reordering rows and  ... 

Computation-free preconditioners for the parallel solution of power system problems

H. Dag, F.L. Alvarado
1997 IEEE Transactions on Power Systems  
Solution of a set of linear equations A x = b is a recurrent problem in power system analysis.  ...  Because of computational dependencies, direct methods have proven of limited value in both parallel and highly vectorized computing environments.  ...  Acknowledgements The authors acknowledge support from NSF grant ECS-9216308 and thank Dr. Harmohan Singh for his assistance with data cases for the gain matrices.  ... 
doi:10.1109/59.589609 fatcat:sauayl2uavd3tepndgnkawtim4

The parallel computation of the smallest eigenpair of an acoustic problem with damping

Martin B. Van Gijzen
1999 International Journal for Numerical Methods in Engineering  
The recently proposed Jacobi-Davidson method is well suited for parallel computing: no matrix decomposition and no back or forward substitutions are needed.  ...  The resulting quadratic eigenproblem is of very large size and for its solution with reasonable computer resources an e cient and parallelizable algorithm is required.  ...  We like to thank Marco Beltman for his extensive help with the acoustic problem. We thank Henk van der Vorst and Gerard Sleijpen for many helpful discussions on numerical details.  ... 
doi:10.1002/(sici)1097-0207(19990630)45:6<765::aid-nme607>3.0.co;2-a fatcat:pxqerf4c5baynhlrxd6345lsf4

Toward real-time diffuse optical tomography: accelerating light propagation modeling employing parallel computing on GPU and CPU

Matthaios Doulgerakis, Adam Eggebrecht, Stanislaw Wojtkiewicz, Joseph Culver, Hamid Dehghani
2017 Journal of Biomedical Optics  
The authors have no relevant financial interests in this article and no potential conflicts of interest to disclose.  ...  ACKNOWLEDGMENTS This work has been funded by the National Institutes of Health (NIH) Grant R01EB009233-2, RO1-CA132750 and Autism Speaks Meixner Translational Postdoctoral Fellowship 7962.  ...  Computational problem The solution of large sparse linear systems involved in the forward model is currently the computational bottleneck of the DOT algorithm.  ... 
doi:10.1117/1.jbo.22.12.125001 pmid:29197176 pmcid:PMC5709934 fatcat:zvbdlggtfnazhf3uyb33ev3g4e

Solving a positive definite system of linear equations via the matrix exponential

Ammar Hasan, Eric C. Kerrigan, George A. Constantinides
2011 IEEE Conference on Decision and Control and European Control Conference  
When the parallelism is fully exploited, the algorithm is shown to be more efficient in terms of computational speed in comparison to other popular methods for solving a positive definite system of linear  ...  We present a new direct algorithm for solving a system of linear equations with a positive definite matrix by discretizing a continuous-time dynamical system for a large sampling time.  ...  Since neither of these forward/back substitutions can be executed in parallel, we add the time complexities of all forward/back substitutions to get the time complexity of the Cholesky decomposition method  ... 
doi:10.1109/cdc.2011.6160593 dblp:conf/cdc/HasanKC11 fatcat:gio6iszk25aepazqx2tqwgwjqa

LU Factorization Algorithm with Minimum Degree Ordering in Power Distribution Network Problems

Alma Husagic Selman
2016 Southeast Europe Journal of Soft Computing  
When blocked, sparse matrices can be processed much more efficiently, and this made blocked sparse matrices widely used in acquiring solutions for power system problems.  ...  Power systems computations for nowadays common large distributed systems typically involve the usage of very large sparse matrices, whose analysis and verification is very time and memory consuming.  ...  METHODOLOGY: LU FACTORIZATION BASED ON MINIMUM DEGREE ORDERING The main focus of this ongoing research is to implement direct method for solution of system of linear equations in the area of power distribution  ... 
doi:10.21533/scjournal.v4i2.91 fatcat:n7f7edamqrawxima2yfzrnxupa

Kernel Solver Design of FPGA-Based Real-Time Simulator for Active Distribution Networks

Zhiying Wang, Fanpeng Zeng, Peng Li, Chengshan Wang, Xiaopeng Fu, Jianzhong Wu
2018 IEEE Access  
The framework of the solver, offline process design on PC and online process design on FPGA are proposed in detail.  ...  In this paper, a fine-grained solver of the FPGA-based real-time simulator for active distribution networks is designed to meet the computational demand.  ...  The solver is embedded in the electrical system solution module. The column operations for the forward and backward substitutions are assigned to the corresponding PEs.  ... 
doi:10.1109/access.2018.2842076 fatcat:kn6pjabdmbeihmxrmyhvb4veam

Sparse matrix inverse factors (power systems)

M.K. Enns, W.F. Tinney, F.L. Alvarado
1990 IEEE Transactions on Power Systems  
The inverses of mamx factors lend themselves to parallel operations in the direct solution phase of sparse matrix solutions.  ...  All of the mulaplications required for repeat solutions may be performed in parallel using the inverse factors, with only as many serial steps as twice the number of factors.  ...  ACKNOWLEDGEMENTS The authors acknowledge Sa0 Khai Mong for programming and producing the results given in Figures 1 through 5 and Table 1 , and Dr.  ... 
doi:10.1109/59.54554 fatcat:n3wba4scnrdkbczvtnujpiydhu

Scalable solutions to integral-equation and finite-element simulations

T. Cwik, D.S. Katz, J. Patterson
1997 IEEE Transactions on Antennas and Propagation  
Scaling of the numerical implementation depends upon many factors; for example, direct or iterative methods for solution of the linear system, as well as the computer architecture used in the simulation  ...  In this paper, scalability will be divided into two components-scalability of the numerical algorithm specifically on parallel computer systems and algorithm or sequential scalability.  ...  Zuffada and V. Jamnejad of the Jet Propulsion Laboratory, Pasadena, CA, for their contributions to the sequential version of the finite-element software discussed in Section IV-A.  ... 
doi:10.1109/8.558670 fatcat:xthd5uyasreudeofrubjlhbguy

Parallel forward and back substitution for efficient power grid simulation

Xuanxing Xiong, Jia Wang
2012 Proceedings of the International Conference on Computer-Aided Design - ICCAD '12  
In this paper, we propose two approaches to parallelize forward and back substitution.  ...  Parallelizing power grid simulation with factorization-based direct or preconditioned iterative methods is a challenging task due to the data dependency among forward and back substitution.  ...  ACKNOWLEDGMENT The authors would like to thank Xiaoming Chen for help with the NICSLU code.  ... 
doi:10.1145/2429384.2429527 dblp:conf/iccad/XiongW12 fatcat:f3kss4nyv5hrrn7et3dtsg5kfa

Parallel Direct Solver for the Finite Integration Technique in Electrokinetic Problems

Abdellatif Tinzefte, Yvonnick Le Menach, Julien Korecki, Frederic Guyomarch, Francis Piriou
2010 IEEE transactions on magnetics  
In this paper, a parallel algorithm is proposed for the direct solution of large sparse linear systems with the finite integration technique.  ...  This direct solver has the advantage of handling singularities in the matrix of linear systems.  ...  In this paper, a parallel algorithm is proposed for the direct solution of large sparse linear systems with the finite integration technique.  ... 
doi:10.1109/tmag.2010.2045886 fatcat:jpxv5cqhfjbwndm23ljcf7c6na
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