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Conjugate direction methods for multiple solution of SLAEs
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 26-42 Cet article a éte moissonné depuis la source Math-Net.Ru

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Conjugate gradient and conjugate residual methods for multiple solution of systems of linear algebraic equations (SLAE) with the same matrices but with different successively determined right-hand sides are considered. In order to speed up the iterative processes when solving the second and subsequent SLAEs, deflation algorithms are applied. These algorithms use the direction vectors obtained in the course of solving the first system as the basis vectors. Results of numerical experiments for model examples, illustrating the efficiency of the approaches under consideration, are provided. 
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     author = {Y. L. Gurieva and V. P. Il'in},
     title = {Conjugate direction methods for multiple solution of {SLAEs}},
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Y. L. Gurieva; V. P. Il'in. Conjugate direction methods for multiple solution of SLAEs. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 26-42. http://geodesic.mathdoc.fr/item/ZNSL_2020_496_a1/

[1] V. P. Ilin, Matematicheskoe modelirovanie, v. 1, Nepreryvnye i diskretnye modeli, Izd. SO RAN, Novosibirsk, 2017

[2] R. Nicolaides, “Deflation of conjugate gradients with applications to boundary value problems”, SIAM J. Numer. Annal., 24 (1987), 355–365 | DOI | MR | Zbl

[3] Z. Dostal, “Conjugate gradient method with preconditioning by projector”, Int. J. Computer Math., 23 (1988), 315–323 | DOI | Zbl

[4] L. Mansfield, “On the use of deflation to inprove the convergences of conjugate gradient iteration”, Communs. Appl. Numer. Math., 4 (1998), 151–156 | DOI

[5] L. Mansfield, “Damped Jacobi preconditioning and coarse grid deflation for conjugate gradient iIteration on parallel computers”, SIAM J. Sci. Stat. Comput., 12 (1991), 1314–1323 | DOI | MR | Zbl

[6] A. Gaul, M. H. Gutknecht, J. Liesen, R. Nabben, “A framework for deflated and augmented Krylov subspace methods”, SIAM J. Anal. Appl., 34 (2013), 495–518 | DOI | MR | Zbl

[7] M. H. Gutknecht, “Deflated and augmented Krylov subspace methods: A framework for the deflated BiCG and related solvers”, SIAM J. Matrix Appl., 35 (2014), 1444–1466 | DOI | MR | Zbl

[8] K. Ahuja, E. Sturler, L. Feng, “Recycling BiCG STAB with an application to parametric model order reduction”, SIAM J. Sci. Comput., 37 (2015), 429–446 | DOI | MR

[9] M. P. Neuenhofen, C. Greif, “MSTAB: Stabilized inducted dimension reduction for KRYLOV subspace recycling”, SIAM J. Sci. Comput., 40:2 (2018), 554–571 | DOI | MR

[10] R. Nabben, C. Vuik, “A comparison of deflation and coarse grid correction applied to porous media flow”, SIAM J. Numer. Anal., 42 (2004), 631–647 | DOI | MR

[11] J. M. Tang, R. Nabben, S. Vuik, U. A. Erlangga, “Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods”, J. Sci. Comput., 39 (2009), 340–370 | DOI | MR | Zbl

[12] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edn., SIAM, 2003 | MR | Zbl

[13] Y. Saad, M. Yeung, J. Erhel, F. Guyomarc'h, “A deflated version of conjugate gradient algorithm”, SIAM J. Sci. Comput., 24 (2000), 1909–1926 | DOI | MR

[14] V. Simoncini, D. B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems”, Numer. Linear. Allegra Appl., 14 (2007), 1–39 | DOI | MR

[15] J. I. Aliaga, D. L. Boley, R. W Freund, V. Hernandez, “A Lanczos-type method for multiple starting vectors”, Math. Comp., 69 (2000), 1577–1601 | DOI | MR | Zbl

[16] J. Erhel, F. Guyomarc'h, “An augmented conjugate gradient method for solving consecutive symmetric positive definite linear systems”, SIAM J. Matrix Annal. Appl., 21 (2000), 1279–1299 | DOI | MR | Zbl

[17] L. Yu. Kolotilina, “Pereobuslavlivanie sistem lineinykh algebraicheskikh uravnenii s pomoschyu dvoinogo ischerpyvaniya. I. Teoriya”, Zap. nauchn. semin. POMI, 229, 1995, 95–152

[18] N. Venkovic, P. Mycek, L. Giraud, O. Le Maitre, Comparative study of harmonic and Rayleigh–Ritz procedures with applications for deflated conjugate gradients, Pressed Report Cerfacs, , 2020 hal-02434043

[19] V. P. Ilin, “O problemakh parallelnogo resheniya bolshikh SLAU”, Zap. nauchn. semin. POMI, 439, 2015, 112–127

[20] V. P. Ilin, Metody i tekhnologii konechnykh elementov, Izd. IVM i MG SO RAN, Novosibirsk, 2007

[21] M. A. Olshanskii, E. E. Tyrtyshnikov, Iterative Methods for Linear Systems. Theory and Applications, SIAM, Philadelphia, 2014 | MR | Zbl

[22] M. Arioli, “Generalized Golub–Kahan diagonalization and stopping criteria”, SIAM J. Matrix Anal. Appl., 34:2 (2013), 57–592 | DOI | MR

[23] N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 2002 | MR | Zbl

[24] Yu. V. Vorobev, Metod momentov v prikladnoi matematike, Fizmatlit, M., 1958 | MR

[25] J. Liesen, Z. Strakos, Krylov Subspace Methods, Principles and Analysis, Oxford University Press, 2013 | MR | Zbl

[26] Ya. L. Gureva, V. P. Ilin, “O metodakh grubosetochnoi korrektsii v podprostranstvakh Krylova”, Zap. nauchn. semin. POMI, 463, 2017, 44–57

[27] V. P. Ilin, “Dvukhurovnevye metody naimenshikh kvadratov v podprostranstvakh Krylova”, Zap. nauchn. semin. POMI, 463, 2017, 224–239