The paper considers preconditioned iterative methods in Krylov subspaces for solving large systems of linear algebraic equations with sparse coefficient matrices arising in solving multidimensional boundary-value problems by finite volume or finite element methods of different orders on unstructured grids. Block versions of the weighted Cimmino methods, based on various orthogonal and/or variational approaches and realizing preconditioning functions for two-level multi-preconditioned semi-conjugate residual algorithms with periodic restarts, are proposed. At the inner iterations between restarts, additional acceleration is achieved by applying deflation methods, providing low-rank approximations of the original matrix and playing the part of an additional preconditioner. At the outer level of the Krylov process, in order to compensate the convergence deceleration caused by restricting the number of the orthogonalized direction vectors, restarted approximations are corrected by using the least squares method. Scalable parallelization of the methods considered, based on domain decomposition, where the commonly used block Jacobi–Schwarz iterative processes is replaced by the block Cimmino–Schwarz algorithm, is discussed. Hybrid programming technologies for implementing different stages of the computational process on heterogeneous multi-processor systems with distributed and hierarchical shared memory are described.
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References
S. Kaczmarz, “Angenaherte Auflosung von Systemen lineare Gleichungen,” Bull. Int. Acad. Polon. Sci. Lett., 35, 355–357 (1937).
G. Cimmino, “Calcolo approssimati de soluzioni dei sistemi di equazioni linear,” Ric. Sci., 11.9, 326–333 (1938).
V. P. Il’in, Methods and Technologies of Finite Elements [in Russian], IVMIMG SO RAN, Novosibirsk (2007).
W. Hackbush, Multigrid Methods and Applicatios, Springer-Verlag, Berlin (1985).
V. P. Il’in, On the Kaczmarz iterative method and its generalizations,” Sib. Zh. Industr. Mat., 9, No. 3, 39–40 (2006).
Y. Saad, Iterative Methods for Sparse Linear Systems, PWS (1996).
G. H. Golub and Y. C. F. Loan, Matrix Computations, John‘s Hopkins Univ. Press (1996).
M. T. Chu, R. E. Funderlic, and G. H. Golub, “A rank-one reduction formula and its application to matrix factorization,” SIAM Rev., 37, 512–530 (1999).
L. Hubert, J. Meulman, and W. Heise, “Two purposes for matrix factorization: A historical appraisal,” SIAM Rev., 42, No. 1, 68–82 (2000).
G. Appleby and D. S. Smolarski, “A linear acceleration row action method for projecting onto subspaces,” Eectr. Trans. Numer. Anal. Kent State Univ., 20, 253–275 (2005).
H. Scolnik, N. Echebest, M. T. Guardarrdarucci, and M. C. Vacachino, “A class of optimized row projection methods for solving large nonsymmetric linear systems,” Appl. Numer. Math., 41, 499–513 (2002).
F. P. Gantmakher, The Theory of Matrices [in Russian], Nauka, Moscow (1968).
V. P. Il’in, “Problems of parallel solution of large systems of linear algebraic equations,” Zap. Nauchn. Semin. POMI, 439, 112–127 (2015).
V. P. Il’in, “Two-level least-squares methods in Krylov subspaces,” Zap. Nauchn. Semin. POMI, 463, 224–239 (2017).
V. P. Il’in, “Methods of semiconjugate directions,” Russ. J. Numer. Anal. Math. Model., 23, No. 4, 369–387 (2008).
R. A. Nicolaydes, “Deflation of conjugate gradients with application to boundary-value problems,” SIAM J. Numer. Anal., 24, 335–365 (1987).
Y. Gurieva and V. P. Il’in, “On parallel computational technologies of augmented domain decomposition methods,” in: Parallel Computing Technologies, 13th International Conference PaCT, Petrozavodsk, Russia (2015), pp. 33–46.
J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu, “Convergence estimates for product iterative methods with applications to domain decomposition,” Math. Comp., 57, No. 195, 1–21 (1991).
L. Yu. Kolotilina, “Two-fold deflation preconditioning of linear algebraic systems. I. Theory,” Zap. Nauchn. Semin. POMI, 229, 95–152 (1995).
J. Liesen, M. Rozlo
nik, and Z. Strakoš, “Least squares resudials and minimal residual methods,” SIAM J. Sci. Comput., 23, No. 5, 1503–1525 (2002).
nik, and Z. Strakoš, “Least squares resudials and minimal residual methods,” SIAM J. Sci. Comput., 23, No. 5, 1503–1525 (2002).Author information
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 472, 2018, pp. 103–119.
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Il’in, V.P. Projection Methods in Krylov Subspaces. J Math Sci 240, 772–782 (2019). https://doi.org/10.1007/s10958-019-04395-7
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DOI: https://doi.org/10.1007/s10958-019-04395-7