Abstract
The concept of supernodes, originally developed to accelerate direct solution methods for linear systems, is generalized to block factorized sparse approximate inverse (Block FSAI) preconditioning of non-symmetric linear systems. It is shown that aggregating the unknowns in clusters that are processed together is particularly useful both to reduce the cost for the preconditioner setup and accelerate the convergence of the iterative solver. A set of numerical experiments performed on matrices arising from the meshfree discretization of 2D and 3D potential problems, where a very large number of nodal contacts is usually found, shows that the supernodal Block FSAI preconditioner outperforms the native algorithm and exhibits a much more stable behavior with respect to the variation of the user-specified parameters.
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References
Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182, 418–477 (2002)
Ferronato, M.: Preconditioning for sparse linear systems at the dawn of the 21st century: history, current developments, and future perspectives. ISRN Appl. Math. doi:10.5402/2012/127647 (2012)
Raghavan, P., Teranishi, K.: Parallel hybrid preconditioning: incomplete factorization with selective sparse approximate inversion. SIAM J. Sci. Comput. 32, 1323–1345 (2010)
Helfenstein, R., Koko, J.: Parallel preconditioned conjugate gradient algorithm on GPU. J. Comput. Appl. Math. 236, 3584–3590 (2012)
Janna, C., Ferronato, M., Gambolati, G.: Enhanced block FSAI preconditioning using domain decomposition techniques. SIAM J. Sci. Comput. 35, S229–S249 (2013)
Chow, E., Patel, A.: Fine-grained parallel incomplete LU factorization. SIAM J. Sci. Comput. 37, C169–C193 (2015)
Janna, C., Ferronato, M., Sartoretto, F., Gambolati, G.: FSAIPACK: A software package for high performance FSAI preconditioning. ACM Transactions on Mathematical Software 41, paper no. 10 (2015)
Grigori, L., Moufawad, S.: Communication avoiding ILU0 preconditioner. SIAM J. Sci. Comput. 37, C217–C246 (2015)
Janna, C., Ferronato, M., Gambolati, G.: A Block FSAI-ILU parallel preconditioner for symmetric positive definite linear systems. SIAM J. Sci. Comput. 32, 2468–2484 (2010)
Ferronato, M., Janna, C., Pini, G.: A generalized Block FSAI preconditioner for nonsymmetric linear systems. J. Comput. Appl. Math. 256, 230–241 (2014)
Ferronato, M., Janna, C., Pini, G.: Efficient parallel solution to large-size sparse eigenproblems with block FSAI preconditioning. Numer. Linear Algebra Appl. 19, 797–815 (2012)
Ferronato, M., Janna, C., Pini, G.: Parallel Jacobi-Davidson with block FSAI preconditioning and controlled inner iterations. Numer. Linear Algebra Appl. 23, 394–409 (2016)
Janna, C., Ferronato, M.: Adaptive pattern research for Block FSAI preconditioning. SIAM J. Sci. Comput. 33, 3357–3380 (2011)
Chow, E.: A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804–1822 (2000)
Duff, I.S., Reid, J.K.: The multifrontal solution of indefinite sparse symmetric linear equations. ACM Trans. Math. Softw. 9, 302–325 (1983)
Ashcraft, C., Grimes, R.G.: The influence of relaxed supernode partitions on the multifrontal method. ACM Trans. Math. Softw. 15, 291–309 (1989)
Ashcraft, C.: Compressed graphs and the minimum degree algorithm. SIAM J. Sci. Comput. 16, 1404–1411 (1995)
Gupta, A., George, T.: Adaptive techniques for improving the performance of incomplete factorization preconditioning. SIAM J. Sci. Comput. 32, 84–110 (2010)
Vannieuwenhoven, N., Meerbergen, K.: IMF: An Incomplete multifrontal LU-factorization for element-structured sparse linear systems. SIAM J. Sci. Comput. 35, A270–A293 (2013)
Janna, C., Ferronato, M., Gambolati, G.: The use of supernodes in factored sparse approximate inverse preconditioning. SIAM J. Sci. Comput. 37, C72–C94 (2015)
Huckle, T.: Approximate sparsity patterns for the inverse of a matrix and preconditioning. Appl. Numer. Math. 30, 291–303 (1999)
Lin, C., Moré, J.J.: Incomplete Cholesky factorizations with limited memory. SIAM J. Sci. Comput. 21, 24–45 (1999)
Janna, C., Castelletto, N., Ferronato, M.: The effect of graph partitioning techniques on parallel Block FSAI preconditioning: a computational study. Numer. Algorithms 68, 813–836 (2015)
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Mirzaei, D., Schaback, R.: Direct meshless local Petrov-Galerkin (DMLPG) method: A generalized MLS approximation. Appl. Numer. Math. 68, 73–82 (2013)
Atluri, S.N., Zhu, T.: The meshless local Petrov-Galerkin (MLPG) method: A simple and less costly alternative to the finite element methods. Comput. Model. Eng. Sci. 3, 11–51 (2002)
Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32, 983–1000 (2012)
Mazzia, A., Pini, G., Sartoretto, F.: A DMPLG refinement technique for 2D and 3D potential problems. Comput. Model. Eng. Sci. 108, 239–262 (2015)
Davis, T.A., Hu, Y.: The university of florida sparse matrix collection. ACM Trans. Math. Softw. 38, 1–25 (2011)
van der Vorst, H.A.: Bi-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)
Acknowledgements
This work has been supported by the University of Padova project “Stable and efficient discretization of the mechanics of faults” and by the ISCRA project IsC36_PRECISO. The authors are indebted to Carlo Janna for his contribution in the code implementation.
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Ferronato, M., Pini, G. A supernodal block factorized sparse approximate inverse for non-symmetric linear systems. Numer Algor 78, 333–354 (2018). https://doi.org/10.1007/s11075-017-0378-x
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DOI: https://doi.org/10.1007/s11075-017-0378-x