Abstract
Solving large sparse linear systems, efficiently, on supercomputing infrastructures is a time-consuming component for a wide variety of simulation processes. An effective parallel solver should meet the required specifications, concerning both convergence behavior and scalability. Herewith, a class of two-stage algebraic domain decomposition preconditioning schemes based on the upper Schur complement method is proposed, in order to exploit appropriately distributed memory systems with multicore processors. The design of the method has been focused on homogeneous hybrid parallel systems, i.e., distributed and shared memory systems. However, the proposed method can also be applied to heterogeneous systems, such as cloud infrastructures, or hybrid parallel systems with accelerators, by modifying the workload distribution algorithm and taking into account the different network latencies and bandwidths. The first stage of the proposed schemes is related to the assignment of the subdomains among the workstations of the distributed system, whereas the second stage concerns the further redistribution of the subdomains to each core of a processor. The proposed method utilizes multiprojection techniques, based on semi-aggregated subdomains, leading to improved convergence behavior as the number of subdomains increases. Moreover, a subspace compression technique is used, in order to improve the performance of the preprocessing phase and reduce the memory requirements of the proposed scheme. The preconditioning schemes were combined with a parallel Krylov subspace method, i.e., the parallel preconditioned GMRES(m) method. The convergence behavior, the performance and the scalability of the proposed preconditioning schemes are examined and compared to existing state-of-the-art methods, by conducting several numerical experiments on supercomputing infrastructures.
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References
Agullo E, Giraud L, Guermouche A, Haidar A, Roman J (2013) Parallel algebraic domain decomposition solver for the solution of augmented systems. Adv Eng Softw 60–61:23–30. https://doi.org/10.1016/j.advengsoft.2012.07.004
Axelsson O (1996) Iterative solution methods. Cambridge University Press, London
Bank R, Falgout R, Jones T, Manteuffel TA, McCormick SF, Ruge JW (2015) Algebraic multigrid domain and range decomposition (amg-dd/amg-rd). SIAM J Sci Comput 37(5):S113–S136
Bank RE, Smith RK (2002) An algebraic multilevel multigraph algorithm. SIAM J Sci Comput 23(5):1572–1592
Benzi M (2002) Preconditioning techniques for large linear systems: a survey. J Comput Phys 182(2):418–477
Bramble JH, Pasciak JE, Schatz AH (1986) The construction of preconditioners for elliptic problems by substructuring. i. Math Comput 47(175):103–134
Brezina M, Vaněk P (1999) A black-box iterative solver based on a two-level schwarz method. Computing 63(3):233–263
Cai XC, Sarkis M (1999) A restricted additive schwarz preconditioner for general sparse linear systems. SIAM J Sci Comput 21(2):792–797
Chan TF, Mathew TP (1994) Domain decomposition algorithms. Acta Numer 3:61–143
Chevalier C, Pellegrini F (2008) Pt-scotch: a tool for efficient parallel graph ordering. Parallel Comput 34(6–8):318–331
Davis TA (2004) Algorithm 832: umfpack, anunsymmetricpattern multifrontal method with a column preordering strategy. ACM Trans Math Softw 30(2):196–199
Davis TA (2006) Direct methods for sparse linear systems, vol 2. SIAM, New Delhi
Erlangga YA, Nabben R (2009) Algebraic multilevel krylov methods. SIAM J Sci Comput 31(5):3417–3437
Ferronato M, Janna C, Pini G (2014) A generalized Block FSAI preconditioner for nonsymmetric linear systems. J Comput Appl Math 256:230–241
Filelis-Papadopoulos CK, Gravvanis GA (2016) A class of generic factored and multi-level recursive approximate inverse techniques for solving general sparse systems. Eng Comput 33(1):74–99
Giraud L, Haidar A, Saad Y (2010) Sparse approximations of the schur complement for parallel algebraic hybrid linear solvers in 3D. Numer Math Theory Methods Appl 3(3):276–294
Grote MJ, Huckle T (1997) Parallel preconditioning with sparse approximate inverses. SIAM J Sci Comput 18(3):838–853
Have P, Masson R, Nataf F, Szydlarski M, Xiang H, Zhao T (2013) Algebraic domain decomposition methods for highly heterogeneous problems. SIAM J Sci Comput 35(3):C284–C302
Janna C, Ferronato M, Gambolati G (2013) Enhanced block fsai preconditioning using domain decomposition techniques. SIAM J Sci Comput 35(5):S229–S249
Karypis G, Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392
Karypis G, Kumar V (1998) Multilevel k-way partitioning scheme for irregular graphs. J Parallel Distrib Comput 48(1):96–129
Li R, Saad Y (2017) Low-rank correction methods for algebraic domain decomposition preconditioners. SIAM J Matrix Anal Appl 38(3):807–828
Li Z, Saad Y (2006) Schurras: a restricted version of the overlapping schur complement preconditioner. SIAM J Sci Comput 27(5):1787–1801
Li Z, Saad Y, Sosonkina M (2003) pARMS: a parallel version of the algebraic recursive multilevel solver. Numer Linear Algebra Appl 10(5–6):485–509
Mandel J (1990) Two-level domain decomposition preconditioning for the p-version finite element method in three dimensions. Int J Numer Methods Eng 29(5):1095–1108
Manguoglu M (2012) Parallel solution of sparse linear systems. In: Berry M et al (eds) High-performance scientific computing. Springer, Berlin, pp 171–184
Mitchell WF (1997) A parallel multigrid method using the full domain partition. Electron Trans Numer Anal 6:224–233
Moutafis BE, Filelis-Papadopoulos CK, Gravvanis GA (2017) Parallel multi-projection preconditioned methods based on semi-aggregation techniques. J Comput Sci 22:45–53
Moutafis BE, Filelis-Papadopoulos CK, Gravvanis GA (2017) Parallel multi-projection preconditioned methods based on subspace compression. Math Problems Eng 2017, 2580820. https://doi.org/10.1155/2017/2580820
Moutafis BE, Filelis-Papadopoulos CK, Gravvanis GA (2018) Parallel schur complement techniques based on multiprojection methods. SIAM J Sci Comput 40(4):C634–C654
Nataf F, Xiang H, Dolean V, Spillane N (2011) A coarse space construction based on local dirichlet-to-neumann maps. SIAM J Sci Comput 33(4):1623–1642
Nicolaides RA (1987) Deflation of conjugate gradients with applications to boundary value problems. SIAM J Numer Anal 24(2):355–365
Pellegrini F, Roman J (1996) Scotch: a software package for static mapping by dual recursive bipartitioning of process and architecture graphs. In: Liddell H, Colbrook A, Hertzberger B, Sloot P (eds) International Conference on High-performance Computing and Networking. Springer, Berlin, pp 493–498
Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. Society for industrial and applied mathematics. SIAM, Philadelphia
Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with pardiso. Future Gener Comput Syst 20(3):475–487
Smith B, Bjorstad P, Gropp WD, Gropp W (2004) Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge
Tang JM, Nabben R, Vuik C, Erlangga YA (2009) Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods. J Sci Comput 39(3):340–370
Toselli A, Widlund OB (2005) Domain decomposition methods: algorithms and theory, vol 34. Springer, Berlin
Tselepidis N, Filelis-Papadopoulos C, Gravvanis G (2018) Distributed algebraic tearing and interconnecting techniques. Numer Algorithms 82:1–34
Van der Vorst HA (2003) Iterative Krylov methods for large linear systems, vol 13. Cambridge University Press, Cambridge
Xi Y, Li R, Saad Y (2016) An algebraic multilevel preconditioner with low-rank corrections for sparse symmetric matrices. SIAM J Matrix Anal Appl 37(1):235–259
Xiang H, Nataf F (2014) Two-level algebraic domain decomposition preconditioners using jacobi-schwarz smoother and adaptive coarse grid corrections. J Comput Appl Math 261:1–13
Zhang J (2000) On preconditioning schur complement and schur complement preconditioning. Electron Trans Numer Anal 10:115–130
Zhu Y, Sameh AH (2017) PSPIKE+: a family of parallel hybrid sparse linear system solvers. J Comput Appl Math 311:682–703
Acknowledgements
The research work of Byron E. Moutafis, as a Ph.D. candidate, was funded by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) (Grant Code: 1609). The authors acknowledge the Greek Research and Technology Network (GRNET) for the provision of the National HPC facility ARIS under project PR006053-ScaleSciCompIII.
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Moutafis, B.E., Gravvanis, G.A. & Filelis-Papadopoulos, C.K. On the design of two-stage multiprojection methods for distributed memory systems. J Supercomput 76, 9063–9094 (2020). https://doi.org/10.1007/s11227-020-03201-5
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DOI: https://doi.org/10.1007/s11227-020-03201-5