Geodesic
Performance analysis of iterative methods of combined linear algebraic equations solution
Matematičeskoe modelirovanie i čislennye metody (2014), pp. 37-52.

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When sampling partial differential equations one has to solve a system of linear algebraic equations. To select the optimal in the sense of the computational efficiency of iterative method for solving such equations, in addition to the rate of convergence we should take into account such characteristics of the system and method, as the condition number, the smoothing factor, the indicator "costs on". The last two characteristics are calculated by the coefficients of harmonics amplification that give evidence of the smoothing properties of the iterative method and its "costs on", i. e. how worse the method suppresses frequency components of the error as compared with the highfrequency ones. The suggested method of determining harmonic gain factors is based on of the discrete Fourier transform. As an example, an analysis of the effectiveness of the BiCGStab method with ILU and multigrid preconditioning when solving difference analogues of the Helmholtz and Poisson equations is described.
Keywords: Sparse linear systems, preconditioning, smoothers, multigrid methods.
Mots-clés : discrete fourier transform 
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     author = {I. K. Marchevsky and V. V. Puzikova},
     title = {Performance analysis of iterative methods of combined linear algebraic equations solution},
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     pages = {37--52},
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     year = {2014},
     language = {ru},
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I. K. Marchevsky; V. V. Puzikova. Performance analysis of iterative methods of combined linear algebraic equations solution. Matematičeskoe modelirovanie i čislennye metody (2014), pp. 37-52. http://geodesic.mathdoc.fr/item/MMCM_2014_a2/

[1] Zarubin V.S., Kuvyrkin G.N., Mathematical Modeling and Computational Methods, 2014, no. 1, 5–17

[2] PETSc <ext-link ext-link-type='uri' href='http://www.mcs.anl.gov/petsc'>http://www.mcs.anl.gov/petsc</ext-link>

[3] Scalable linear solvers: HYPRE <ext-link ext-link-type='uri' href='http://computation.llnl.gov/casc/linear_solvers/sls_hypre.html'>http://computation.llnl.gov/casc/linear_solvers/sls_hypre.html</ext-link>

[4] Intel Math Kernel Library 11.0 <ext-link ext-link-type='uri' href='http://software.intel.com/en-us/intel-mkl'>http://software.intel.com/en-us/intel-mkl</ext-link>

[5] Wesseling P., An introduction to multigrid methods, John Willey & Sons Ltd., Chichester, 1991, 284 pp. <ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=1156079'>1156079</ext-link>

[6] Ol'shansky M.A., Lectures and Exercises on Multigrid Methods, Lomonosov MSU Publ., Moscow, 2003, 163 pp.

[7] Fedorenko R.P., Introduction to Computational Physics, Moscow Institute of Physics and Technology Publ., Moscow, 1994, 528 pp.

[8] Galanin M.P., Savenkov E.B., Methods of Numerical Analysis of Mathematical Models, BMSTU Publ., Moscow, 2010, 590 pp.

[9] Saad Y., Iterative Methods for Sparse Linear Systems, PWS Publ., New-York, 1996, 547 pp. <ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:1031.65047'>1031.65047</ext-link>

[10] Sergienko A.B., Digital Signal Processing, Piter Publ., St. Petersburg, 2002, 608 pp.

[11] Van der Vorst H.A., “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for solution of non-symmetric linear systems”, SIAM J. Sci. Stat. Comp., 1992, no. 2, 631–644 <ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=1149111'>1149111</ext-link><ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:0761.65023'>0761.65023</ext-link>

[12] Il'in V.P., Siberian Journal of Industrial Mathematics, 9:4 (2008), 47–60

[13] Puzikova V.V., Herald of the Bauman Moscow State Technical University, 2011, 124–133

[14] Van Kan J., Vuik C., Wesseling P., “Fast pressure calculation for 2D and 3D time dependent incompressible flow”, Numer. Linear Algebra Appl., 2000, no. 7, 429–447 pp. <ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=1784561'>1784561</ext-link>

[15] OpenFOAM <ext-link ext-link-type='uri' href='http://www.openfoam.com'>http://www.openfoam.com</ext-link>