On block triangular preconditioned iteration methods for solving the Helmholtz equation
To further improve the efficiency of solving the Helmholtz equation in heterogeneous media with large wavenumber, the Krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two-by-two linear system derived from the discret...
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| Published in | Applied mathematics and computation Vol. 369; p. 124695 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
15.03.2020
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| ISSN | 0096-3003 |
| DOI | 10.1016/j.amc.2019.124695 |
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| Abstract | To further improve the efficiency of solving the Helmholtz equation in heterogeneous media with large wavenumber, the Krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two-by-two linear system derived from the discrete Helmholtz equation. We further develop the eigenvalue properties of the preconditioned matrices to discuss the convergence of the corresponding preconditioned iteration methods. The superiority of such preconditioned iteration methods is prominent according to the numerical results when comparing with other classical iteration methods. We also investigate how the wavenumber influences the performance of the corresponding methods and it is shown that the iteration number of our proposed methods linearly increase with the wavenumber, roughly. Furthermore, the computational wave-fields which conform the real physical law are exhibited to show the correctness of our proposed numerical modeling algorithm. |
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| AbstractList | To further improve the efficiency of solving the Helmholtz equation in heterogeneous media with large wavenumber, the Krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two-by-two linear system derived from the discrete Helmholtz equation. We further develop the eigenvalue properties of the preconditioned matrices to discuss the convergence of the corresponding preconditioned iteration methods. The superiority of such preconditioned iteration methods is prominent according to the numerical results when comparing with other classical iteration methods. We also investigate how the wavenumber influences the performance of the corresponding methods and it is shown that the iteration number of our proposed methods linearly increase with the wavenumber, roughly. Furthermore, the computational wave-fields which conform the real physical law are exhibited to show the correctness of our proposed numerical modeling algorithm. |
| ArticleNumber | 124695 |
| Author | Gao, Rong Qiu, Chujun Lang, Chao |
| Author_xml | – sequence: 1 givenname: Chao orcidid: 0000-0002-8406-6239 surname: Lang fullname: Lang, Chao organization: School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China – sequence: 2 givenname: Rong orcidid: 0000-0003-2267-3153 surname: Gao fullname: Gao, Rong email: gaor14@tsinghua.org.cn, gaor92@sina.com organization: School of Economics and Management, Hebei University of Technology, Tianjin 300401, China – sequence: 3 givenname: Chujun surname: Qiu fullname: Qiu, Chujun organization: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China |
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| Cites_doi | 10.1190/1.1444597 10.1002/fld.1433 10.1016/0893-9659(88)90097-3 10.1088/1742-2132/11/5/055009 10.1785/012003155 10.1137/S1064827501385426 10.1137/0714011 10.1002/nla.1882 10.1190/1.1442147 10.1785/0120060209 10.1016/0021-9991(83)90139-0 10.1006/jcph.1994.1159 10.1007/s11425-013-4695-9 10.1142/S0218396X05002803 10.1190/1.1443849 10.1785/0120020125 10.1002/nla.1881 10.1007/s11430-013-4805-0 10.1016/j.jappgeo.2015.03.007 10.1016/j.jcp.2007.03.033 10.1137/0907058 10.1785/0120080346 10.1142/S0218396X01001510 10.1190/1.2738849 |
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| Keywords | Block two-by-two matrix Helmholtz equation Incomplete factorization Block triangular preconditioning |
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| References | Liu, Li, Wang, Liu, Zhang, Zhang (bib0020) 2014; 57 Plessix (bib0022) 2007; 72 Sleijpen, Fokkema (bib0028) 1993; 1 Yang, Song, Lu (bib0032) 2007; 97 Bai (bib0002) 2013; 56 Gander, Nataf (bib0010) 2001; 9 Yang, Wang, Chen, Song (bib0034) 2009; 99 Hohmann (bib0014) 1988; 1 A.L. Laird, M. Giles, Preconditioned iterative solution of the 2D Helmholtz equation, 2002. Riyanti, Kononov, Erlangga, Vuik, Oosterlee, Plessix, Mulder (bib0024) 2007; 224 Bayliss, Goldstein, Turkel (bib0003) 1983; 49 Ernst, Gander (bib0008) 2012 George (bib0012) 1977; 14 Berenger (bib0004) 1994; 114 Erlangga, Vuik, Oosterlee (bib0007) 2004; 50 Yang, Teng, Zhang, Liu (bib0033) 2003; 93 Kim, Kim (bib0016) 2002; 24 Pratt (bib0023) 1999; 64 Liu, Li, Wang, Xu, Li (bib0021) 2015; 116 Yang, Lu, Wu, Peng (bib0031) 2004; 94 Saad, Schultz (bib0026) 1986; 7 Gander, Halpern, Magoules (bib0009) 2007; 55 Liu, Li, Wang, Liu (bib0019) 2014; 11 Bai (bib0001) 2013 Sheikh, Lahaye, Vuik (bib0027) 2013; 20 Cools, Vanroose (bib0006) 2013; 20 Lang, Ren (bib0018) 2013 Igel, Mora, Riollet (bib0015) 1995; 60 Virieux (bib0029) 1986; 51 Yang, Liu, Zhang, Teng (bib0030) 2002; 148 Cocquet, Gander (bib0005) 2016 Saad (bib0025) 1996 Gander, Nataf (bib0011) 2005; 13 Hagstrom, Tewarson, Jazcilevich (bib0013) 1988; 1 Berenger (10.1016/j.amc.2019.124695_bib0004) 1994; 114 Liu (10.1016/j.amc.2019.124695_bib0019) 2014; 11 Bai (10.1016/j.amc.2019.124695_bib0002) 2013; 56 Cocquet (10.1016/j.amc.2019.124695_bib0005) 2016 George (10.1016/j.amc.2019.124695_bib0012) 1977; 14 Sleijpen (10.1016/j.amc.2019.124695_bib0028) 1993; 1 Bai (10.1016/j.amc.2019.124695_bib0001) 2013 Saad (10.1016/j.amc.2019.124695_bib0025) 1996 Saad (10.1016/j.amc.2019.124695_bib0026) 1986; 7 Gander (10.1016/j.amc.2019.124695_bib0009) 2007; 55 Liu (10.1016/j.amc.2019.124695_bib0020) 2014; 57 Igel (10.1016/j.amc.2019.124695_bib0015) 1995; 60 Gander (10.1016/j.amc.2019.124695_bib0010) 2001; 9 Yang (10.1016/j.amc.2019.124695_bib0032) 2007; 97 Erlangga (10.1016/j.amc.2019.124695_bib0007) 2004; 50 Gander (10.1016/j.amc.2019.124695_bib0011) 2005; 13 Liu (10.1016/j.amc.2019.124695_bib0021) 2015; 116 Sheikh (10.1016/j.amc.2019.124695_bib0027) 2013; 20 Hohmann (10.1016/j.amc.2019.124695_bib0014) 1988; 1 Bayliss (10.1016/j.amc.2019.124695_bib0003) 1983; 49 Hagstrom (10.1016/j.amc.2019.124695_bib0013) 1988; 1 Virieux (10.1016/j.amc.2019.124695_bib0029) 1986; 51 Riyanti (10.1016/j.amc.2019.124695_bib0024) 2007; 224 Ernst (10.1016/j.amc.2019.124695_bib0008) 2012 10.1016/j.amc.2019.124695_bib0017 Yang (10.1016/j.amc.2019.124695_bib0031) 2004; 94 Cools (10.1016/j.amc.2019.124695_bib0006) 2013; 20 Yang (10.1016/j.amc.2019.124695_bib0030) 2002; 148 Kim (10.1016/j.amc.2019.124695_bib0016) 2002; 24 Plessix (10.1016/j.amc.2019.124695_bib0022) 2007; 72 Pratt (10.1016/j.amc.2019.124695_bib0023) 1999; 64 Lang (10.1016/j.amc.2019.124695_bib0018) 2013 Yang (10.1016/j.amc.2019.124695_bib0034) 2009; 99 Yang (10.1016/j.amc.2019.124695_bib0033) 2003; 93 |
| References_xml | – volume: 9 start-page: 1499 year: 2001 end-page: 1506 ident: bib0010 article-title: AILU For Helmholtz problems: a new preconditioner based on the analytic parabolic factorization publication-title: J. Comput. Acoust. – volume: 93 start-page: 882 year: 2003 end-page: 890 ident: bib0033 article-title: A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media publication-title: Bull. Seismol. Soc. Am. – volume: 1 start-page: 313 year: 1988 end-page: 363 ident: bib0014 article-title: Numerical modeling for electromagnetic methods of geophysics publication-title: Electromag. Methods Appl. Geophys. – volume: 13 start-page: 455 year: 2005 end-page: 476 ident: bib0011 article-title: An incomplete LU preconditioner for problems in acoustics publication-title: J. Comput. Acoust. – volume: 57 start-page: 751 year: 2014 end-page: 758 ident: bib0020 article-title: A new kind of optimal second-order symplectic scheme for seismic wave simulations publication-title: Sci. China Earth Sci. – volume: 1 start-page: 2000 year: 1993 ident: bib0028 article-title: Bicgstab (l) for linear equations involving unsymmetric matrices with complex spectrum publication-title: Electron. Trans. Numer. Anal. – volume: 55 start-page: 163 year: 2007 end-page: 175 ident: bib0009 article-title: An optimized Schwarz method with two-sided robin transmission conditions for the Helmholtz equation publication-title: Int. J. Numer. Methods Fluids – volume: 56 start-page: 2523 year: 2013 end-page: 2538 ident: bib0002 article-title: Rotated block triangular preconditioning based on PMHSS publication-title: Sci. China Math. – volume: 224 start-page: 431 year: 2007 end-page: 448 ident: bib0024 article-title: A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency helmholtz equation publication-title: J. Comput. Phys. – volume: 50 start-page: 409 year: 2004 end-page: 425 ident: bib0007 article-title: On a class of preconditioners for solving the Helmholtz equation publication-title: Numer. Linear Algebra Appl. – volume: 24 start-page: 684 year: 2002 end-page: 701 ident: bib0016 article-title: Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media publication-title: SIAM J. Scient. Comput. – year: 1996 ident: bib0025 article-title: Iterative Methods for Sparse Linear Systems – volume: 99 start-page: 3340 year: 2009 end-page: 3354 ident: bib0034 article-title: An explicit method based on the implicit runge–Kutta algorithm for solving wave equations publication-title: Bull. Seismol. Soc. Am. – start-page: 137 year: 2016 end-page: 145 ident: bib0005 article-title: On the Minimal Shift in the Shifted Laplacian Preconditioner for Multigrid to Work publication-title: Domain Decomposition Methods in Science and Engineering XXII – start-page: 87 year: 2013 end-page: 98 ident: bib0018 article-title: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices publication-title: J. Eng. Math. – volume: 64 start-page: 888 year: 1999 end-page: 901 ident: bib0023 article-title: Seismic waveform inversion in the frequency domain, part 1: theory and verification in a physical scale model publication-title: Geophysics – volume: 51 start-page: 889 year: 1986 end-page: 901 ident: bib0029 article-title: P-SV Wave propagation in heterogeneous media: velocity-stress finite-difference method publication-title: Geophysics – volume: 60 start-page: 1203 year: 1995 end-page: 1216 ident: bib0015 article-title: Anisotropic wave propagation through finite-difference grids publication-title: Geophysics – volume: 94 start-page: 1982 year: 2004 end-page: 1992 ident: bib0031 article-title: An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations publication-title: Bull. Seismol. Soc. Am. – volume: 1 start-page: 299 year: 1988 end-page: 302 ident: bib0013 article-title: Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems publication-title: Appl. Math. Lett. – volume: 7 start-page: 856 year: 1986 end-page: 869 ident: bib0026 article-title: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems publication-title: SIAM J. Scient. Stat. Comput. – volume: 97 start-page: 1557 year: 2007 end-page: 1569 ident: bib0032 article-title: Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3D anisotropic media publication-title: Bull. Seismol. Soc. Am. – volume: 49 start-page: 443 year: 1983 end-page: 457 ident: bib0003 article-title: An iterative method for the Helmholtz equation publication-title: J. Comput. Phys. – volume: 20 start-page: 645 year: 2013 end-page: 662 ident: bib0027 article-title: On the convergence of shifted laplace preconditioner combined with multilevel deflation publication-title: Numer. Linear Algebra Appl. – volume: 148 start-page: 320 year: 2002 end-page: 328 ident: bib0030 article-title: Finite-difference modelling in two-dimensional anisotropic media using a flux-corrected transport technique publication-title: Geophys. J. Int. – volume: 14 start-page: 161 year: 1977 end-page: 179 ident: bib0012 article-title: Numerical experiments using dissection methods to solve n by n grid problems publication-title: SIAM J. Numer. Anal. – volume: 72 start-page: SM185 year: 2007 end-page: SM194 ident: bib0022 article-title: A helmholtz iterative solver for 3D seismic-imaging problems publication-title: Geophysics – volume: 114 start-page: 185 year: 1994 end-page: 200 ident: bib0004 article-title: A perfectly matched layer for the absorption of electromagnetic waves publication-title: J. Comput. Phys. – volume: 116 start-page: 110 year: 2015 end-page: 120 ident: bib0021 article-title: A modified symplectic scheme for seismic wave modeling publication-title: J. Appl. Geophy. – start-page: 41 year: 2013 end-page: 60 ident: bib0001 article-title: On preconditioned iteration methods for complex linear systems publication-title: J. Eng. Math. – reference: A.L. Laird, M. Giles, Preconditioned iterative solution of the 2D Helmholtz equation, 2002. – volume: 11 start-page: 055009 year: 2014 ident: bib0019 article-title: A mixed-grid finite element method with PML absorbing boundary conditions for seismic wave modelling publication-title: J. Geophys. Eng. – start-page: 325 year: 2012 end-page: 363 ident: bib0008 article-title: Why it is difficult to solve Helmholtz problems with classical iterative methods publication-title: Numerical Analysis of Multiscale Problems – volume: 20 start-page: 575 year: 2013 end-page: 597 ident: bib0006 article-title: Local fourier analysis of the complex shifted Laplacian preconditioner for helmholtz problems publication-title: Numer. Linear Algeb. Appl. – volume: 64 start-page: 888 issue: 3 year: 1999 ident: 10.1016/j.amc.2019.124695_bib0023 article-title: Seismic waveform inversion in the frequency domain, part 1: theory and verification in a physical scale model publication-title: Geophysics doi: 10.1190/1.1444597 – volume: 55 start-page: 163 issue: 2 year: 2007 ident: 10.1016/j.amc.2019.124695_bib0009 article-title: An optimized Schwarz method with two-sided robin transmission conditions for the Helmholtz equation publication-title: Int. J. Numer. Methods Fluids doi: 10.1002/fld.1433 – volume: 1 start-page: 299 issue: 3 year: 1988 ident: 10.1016/j.amc.2019.124695_bib0013 article-title: Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems publication-title: Appl. Math. Lett. doi: 10.1016/0893-9659(88)90097-3 – volume: 11 start-page: 055009 issue: 5 year: 2014 ident: 10.1016/j.amc.2019.124695_bib0019 article-title: A mixed-grid finite element method with PML absorbing boundary conditions for seismic wave modelling publication-title: J. Geophys. Eng. doi: 10.1088/1742-2132/11/5/055009 – volume: 94 start-page: 1982 issue: 5 year: 2004 ident: 10.1016/j.amc.2019.124695_bib0031 article-title: An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations publication-title: Bull. Seismol. Soc. Am. doi: 10.1785/012003155 – volume: 24 start-page: 684 issue: 2 year: 2002 ident: 10.1016/j.amc.2019.124695_bib0016 article-title: Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media publication-title: SIAM J. Scient. Comput. doi: 10.1137/S1064827501385426 – start-page: 87 year: 2013 ident: 10.1016/j.amc.2019.124695_bib0018 article-title: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices publication-title: J. Eng. Math. – volume: 14 start-page: 161 issue: 2 year: 1977 ident: 10.1016/j.amc.2019.124695_bib0012 article-title: Numerical experiments using dissection methods to solve n by n grid problems publication-title: SIAM J. Numer. Anal. doi: 10.1137/0714011 – volume: 20 start-page: 645 issue: 4 year: 2013 ident: 10.1016/j.amc.2019.124695_bib0027 article-title: On the convergence of shifted laplace preconditioner combined with multilevel deflation publication-title: Numer. Linear Algebra Appl. doi: 10.1002/nla.1882 – volume: 51 start-page: 889 issue: 4 year: 1986 ident: 10.1016/j.amc.2019.124695_bib0029 article-title: P-SV Wave propagation in heterogeneous media: velocity-stress finite-difference method publication-title: Geophysics doi: 10.1190/1.1442147 – volume: 148 start-page: 320 issue: 2 year: 2002 ident: 10.1016/j.amc.2019.124695_bib0030 article-title: Finite-difference modelling in two-dimensional anisotropic media using a flux-corrected transport technique publication-title: Geophys. J. Int. – start-page: 41 year: 2013 ident: 10.1016/j.amc.2019.124695_bib0001 article-title: On preconditioned iteration methods for complex linear systems publication-title: J. Eng. Math. – volume: 50 start-page: 409 issue: 3 year: 2004 ident: 10.1016/j.amc.2019.124695_bib0007 article-title: On a class of preconditioners for solving the Helmholtz equation publication-title: Numer. Linear Algebra Appl. – volume: 97 start-page: 1557 issue: 5 year: 2007 ident: 10.1016/j.amc.2019.124695_bib0032 article-title: Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3D anisotropic media publication-title: Bull. Seismol. Soc. Am. doi: 10.1785/0120060209 – year: 1996 ident: 10.1016/j.amc.2019.124695_bib0025 – volume: 49 start-page: 443 issue: 3 year: 1983 ident: 10.1016/j.amc.2019.124695_bib0003 article-title: An iterative method for the Helmholtz equation publication-title: J. Comput. Phys. doi: 10.1016/0021-9991(83)90139-0 – ident: 10.1016/j.amc.2019.124695_bib0017 – volume: 114 start-page: 185 issue: 2 year: 1994 ident: 10.1016/j.amc.2019.124695_bib0004 article-title: A perfectly matched layer for the absorption of electromagnetic waves publication-title: J. Comput. Phys. doi: 10.1006/jcph.1994.1159 – start-page: 325 year: 2012 ident: 10.1016/j.amc.2019.124695_bib0008 article-title: Why it is difficult to solve Helmholtz problems with classical iterative methods – volume: 56 start-page: 2523 issue: 12 year: 2013 ident: 10.1016/j.amc.2019.124695_bib0002 article-title: Rotated block triangular preconditioning based on PMHSS publication-title: Sci. China Math. doi: 10.1007/s11425-013-4695-9 – volume: 13 start-page: 455 issue: 03 year: 2005 ident: 10.1016/j.amc.2019.124695_bib0011 article-title: An incomplete LU preconditioner for problems in acoustics publication-title: J. Comput. Acoust. doi: 10.1142/S0218396X05002803 – volume: 60 start-page: 1203 issue: 4 year: 1995 ident: 10.1016/j.amc.2019.124695_bib0015 article-title: Anisotropic wave propagation through finite-difference grids publication-title: Geophysics doi: 10.1190/1.1443849 – volume: 93 start-page: 882 issue: 2 year: 2003 ident: 10.1016/j.amc.2019.124695_bib0033 article-title: A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media publication-title: Bull. Seismol. Soc. Am. doi: 10.1785/0120020125 – volume: 20 start-page: 575 issue: 4 year: 2013 ident: 10.1016/j.amc.2019.124695_bib0006 article-title: Local fourier analysis of the complex shifted Laplacian preconditioner for helmholtz problems publication-title: Numer. Linear Algeb. Appl. doi: 10.1002/nla.1881 – start-page: 137 year: 2016 ident: 10.1016/j.amc.2019.124695_bib0005 article-title: On the Minimal Shift in the Shifted Laplacian Preconditioner for Multigrid to Work – volume: 57 start-page: 751 issue: 4 year: 2014 ident: 10.1016/j.amc.2019.124695_bib0020 article-title: A new kind of optimal second-order symplectic scheme for seismic wave simulations publication-title: Sci. China Earth Sci. doi: 10.1007/s11430-013-4805-0 – volume: 116 start-page: 110 year: 2015 ident: 10.1016/j.amc.2019.124695_bib0021 article-title: A modified symplectic scheme for seismic wave modeling publication-title: J. Appl. Geophy. doi: 10.1016/j.jappgeo.2015.03.007 – volume: 224 start-page: 431 issue: 1 year: 2007 ident: 10.1016/j.amc.2019.124695_bib0024 article-title: A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency helmholtz equation publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2007.03.033 – volume: 1 start-page: 313 year: 1988 ident: 10.1016/j.amc.2019.124695_bib0014 article-title: Numerical modeling for electromagnetic methods of geophysics publication-title: Electromag. Methods Appl. Geophys. – volume: 7 start-page: 856 issue: 3 year: 1986 ident: 10.1016/j.amc.2019.124695_bib0026 article-title: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems publication-title: SIAM J. Scient. Stat. Comput. doi: 10.1137/0907058 – volume: 1 start-page: 2000 issue: 11 year: 1993 ident: 10.1016/j.amc.2019.124695_bib0028 article-title: Bicgstab (l) for linear equations involving unsymmetric matrices with complex spectrum publication-title: Electron. Trans. Numer. Anal. – volume: 99 start-page: 3340 issue: 6 year: 2009 ident: 10.1016/j.amc.2019.124695_bib0034 article-title: An explicit method based on the implicit runge–Kutta algorithm for solving wave equations publication-title: Bull. Seismol. Soc. Am. doi: 10.1785/0120080346 – volume: 9 start-page: 1499 issue: 04 year: 2001 ident: 10.1016/j.amc.2019.124695_bib0010 article-title: AILU For Helmholtz problems: a new preconditioner based on the analytic parabolic factorization publication-title: J. Comput. Acoust. doi: 10.1142/S0218396X01001510 – volume: 72 start-page: SM185 issue: 5 year: 2007 ident: 10.1016/j.amc.2019.124695_bib0022 article-title: A helmholtz iterative solver for 3D seismic-imaging problems publication-title: Geophysics doi: 10.1190/1.2738849 |
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| SubjectTerms | Block triangular preconditioning Block two-by-two matrix Helmholtz equation Incomplete factorization |
| Title | On block triangular preconditioned iteration methods for solving the Helmholtz equation |
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