Advances in Iterative Methods and Preconditioners for the Helmholtz Equation

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Abstract

In this paper we survey the development of fast iterative solvers aimed at solving 2D/3D Helmholtz problems. In the first half of the paper, a survey on some recently developed methods is given. The second half of the paper focuses on the development of the shifted Laplacian preconditioner used to accelerate the convergence of Krylov subspace methods applied to the Helmholtz equation. Numerical examples are given for some difficult problems, which had not been solved iteratively before.

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References

  1. 1.
    Abarbanel S, Gottlieb D (1997) A mathematical analysis of the PML method. J Comput Phys 134:357–363 MATHMathSciNetGoogle Scholar
  2. 2.
    Abarbanel S, Gottlieb D (1998) On the construction and analysis of absorbing layers in CEM. Appl Numer Math 27:331–340 MATHMathSciNetGoogle Scholar
  3. 3.
    Alcouffe RE, Brandt A, Dendy JE Jr, Painter JW (1981) The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J Sci Comput 2:430–454 MATHMathSciNetGoogle Scholar
  4. 4.
    Arnoldi WE (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q Appl Math 9:17–29 MathSciNetGoogle Scholar
  5. 5.
    Babuska I, Sauter S (1997) Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?. SIAM J Numer Anal 27:323–352 MathSciNetGoogle Scholar
  6. 6.
    Babuska I, Ihlenburg F, Strouboulis T, Gangaraj SK (1997) Posteriori error estimation for finite element solutions of Helmholtz’s equation. Part I: the quality of local indicators and estimators. Int J Numer Methods Eng 40:3443–3462 MATHMathSciNetGoogle Scholar
  7. 7.
    Babuska I, Ihlenburg F, Strouboulis T, Gangaraj SK (1997) Posteriori error estimation for finite element solutions of Helmholtz’s equation. Part II: estimation of the pollution error. Int J Numer Methods Eng 40:3883–3900 MATHMathSciNetGoogle Scholar
  8. 8.
    Bamberger A, Joly P, Roberts JE (1990) Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem. SIAM J Numer Anal 27:323–352 MATHMathSciNetGoogle Scholar
  9. 9.
    Bayliss A, Gunzburger M, Turkel E (1982) Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J Appl Math 42:430–451 MATHMathSciNetGoogle Scholar
  10. 10.
    Bayliss A, Goldstein CI, Turkel E (1983) An iterative method for Helmholtz equation. J Comput Phys 49:443–457 MATHMathSciNetGoogle Scholar
  11. 11.
    Bayliss A, Goldstein CI, Turkel E (1985) The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics. Comput Math Appl 11:655–665 MATHMathSciNetGoogle Scholar
  12. 12.
    Bayliss A, Goldstein CI, Turkel E (1985) On accuracy conditions for the numerical computation of waves. J Comput Phys 59:396–404 MATHMathSciNetGoogle Scholar
  13. 13.
    Benamou JD, Despres B (1997) Domain decomposition method for the Helmholtz equation and related optimal control problems. J Comput Phys 136:62–88 MathSciNetGoogle Scholar
  14. 14.
    Benzi M, Haws JC, Tuma M (2000) Preconditioning highly indefinite and nonsymmetric matrices. SIAM J Sci Comput 22:1333–1353 MATHMathSciNetGoogle Scholar
  15. 15.
    Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:185–200 MATHMathSciNetGoogle Scholar
  16. 16.
    Berenger JP (1996) Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 127:363–379 MATHMathSciNetGoogle Scholar
  17. 17.
    Berkhout AJ (1982) Seismic migration: imaging of acoustic energy by wave field extrapolation. Elsevier, Amsterdam Google Scholar
  18. 18.
    Bollöffer M (2004) A robust and efficient ILU that incorporates the growth of the inverse triangular factors. SIAM J Sci Comput 25:86–103 Google Scholar
  19. 19.
    Bourgeois A, Bourget M, Lailly P, Poulet M, Ricarte P, Versteeg R (1991) Marmousi, model and data. In: Marmousi experience, pp 5–16 Google Scholar
  20. 20.
    Brackenridge K (1993) Multigrid and cyclic reduction applied to the Helmholtz equation. In: Melson ND, Manteuffel TA, McCormick SF (eds) Proc 6th Copper Mountain conf on multigrid methods, pp 31–41 Google Scholar
  21. 21.
    Brandt A (1977) Multi–level adaptive solutions to boundary–value problems. Math Comput 31:333–390 MATHMathSciNetGoogle Scholar
  22. 22.
    Brandt A (2002) Multigrid techniques: 1984 guide with applications to fluid dynamics. Technical Report GMD-Studie 85, GMD Sankt Augustine, Germany Google Scholar
  23. 23.
    Brandt A, Livshits I (1997) Wave-ray multigrid methods for standing wave equations. Electr Trans Numer Anal 6:162–181 MATHMathSciNetGoogle Scholar
  24. 24.
    Brandt A, Ta’asan S (1986) Multigrid method for nearly singular and slightly indefinite problems. In: Proc EMG’85 Cologne, 1986, pp 99–121 Google Scholar
  25. 25.
    Brezinzky C, Zaglia MR (1995) Look-ahead in bi-cgstab and other product methods for linear systems. BIT 35:169–201 MathSciNetGoogle Scholar
  26. 26.
    Briggs WL (1988) A multigrid tutorial. SIAM, Philadelphia Google Scholar
  27. 27.
    Chow E, Saad Y (1997) ILUS: an incomplete LU factorization for matrices in sparse skyline format. Int J Numer Methods Fluids 25:739–749 MATHMathSciNetGoogle Scholar
  28. 28.
    Clayton R, Engquist B (1977) Absorbing boundary conditions for acoustic and elastic wave equations. Bull Seis Soc Am 67(6):1529–1540 Google Scholar
  29. 29.
    Colloni F, Ghanemi S, Joly P (1998) Domain decomposition methods for harmonic wave propagation: a general presentation. Technical Report, INRIA RR-3473 Google Scholar
  30. 30.
    Colton D, Kress R (1983) Integral equation methods in scattering theory. Willey, New York MATHGoogle Scholar
  31. 31.
    Colton D, Kress R (1998) Inverse matrix and electromagnetic scattering theory. Springer, Berlin Google Scholar
  32. 32.
    D’Azevedo EF, Forsyth FA, Tang WP (1992) Towards a cost effective ILU preconditioner with high level fill. BIT 31:442–463 MathSciNetGoogle Scholar
  33. 33.
    Dendy J Jr (1983) Blackbox multigrid for nonsymmetric problems. Appl Math Comput 13:261–283 MATHMathSciNetGoogle Scholar
  34. 34.
    Deraemaeker A, Babuska I, Bouillard P (1999) Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two, and three dimensions. Int J Numer Methods Eng 46:471–499 MATHGoogle Scholar
  35. 35.
    de Zeeuw PM (1990) Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J Comput Appl Math 33:1–27 MATHMathSciNetGoogle Scholar
  36. 36.
    de Zeeuw PM (1996) Development of semi-coarsening techniques. Appl Numer Math 19:433–465 MATHMathSciNetGoogle Scholar
  37. 37.
    Drespes B (1990) Domain decomposition method and Helmholtz problems. In: Cohen G, Halpern L, Joly P (eds) Mathematical and numerical aspects of wave propagation phenomena. SIAM, Philadelphia, pp 42–51 Google Scholar
  38. 38.
    Elman HC (1986) A stability analysis of incomplete LU factorizations. Math Comput 47:191–217 MATHMathSciNetGoogle Scholar
  39. 39.
    Elman HR, Ernst OG, O’Leary DP (2001) A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations. SIAM J Sci Comput 22:1291–1315 MathSciNetGoogle Scholar
  40. 40.
    Engquist B, Majda A (1977) Absorbing boundary conditions for the numerical simulation of waves. Math Comput 31:629–651 MATHMathSciNetGoogle Scholar
  41. 41.
    Erlangga YA, Vuik C, Oosterlee CW (2004) On a class of preconditioners for solving the Helmholtz equation. Appl Numer Math 50:409–425 MATHMathSciNetGoogle Scholar
  42. 42.
    Erlangga YA, Vuik C, Oosterlee CW (2005) On a robust iterative method for heterogeneous Helmholtz problems for geophysical applications. Int J Numer Anal Model 2:197–208 MathSciNetGoogle Scholar
  43. 43.
    Erlangga YA, Oosterlee CW, Vuik C (2006) A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation. SIAM J Sci Comput 27:1471–1492 MATHMathSciNetGoogle Scholar
  44. 44.
    Erlangga YA, Vuik C, Oosterlee CW (2006) Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl Numer Math 56:648–666 MATHMathSciNetGoogle Scholar
  45. 45.
    Erlangga YA, Vuik C, Oosterlee CW (2006) A semicoarsening-based multigrid preconditioner for the 3D inhomogeneous Helmholtz equation. In: Wesseling P, Oosterlee CW, Hemker P (eds) Proceedings of the 8th European multigrid conference, September 27–30, 2005, Scheveningen, TU Delft, The Netherlands Google Scholar
  46. 46.
    Fan K (1960) Note in M-matrices. Q J Math Oxford Ser 2 11:43–49 MATHGoogle Scholar
  47. 47.
    Farhat C, Macedo A, Lesoinne M (2000) A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numer Math 85:283–308 MATHMathSciNetGoogle Scholar
  48. 48.
    Fish J, Qu Y (2000) Global-basis two-level method for indefinite systems. Int J Numer Methods Eng 49:439–460 MATHMathSciNetGoogle Scholar
  49. 49.
    Fish J, Qu Y (2000) Global-basis two-level method for indefinite systems. Part I: convergence studies. Int J Numer Methods Eng 49:461–478 MathSciNetGoogle Scholar
  50. 50.
    Fletcher R (1975) Conjugate gradient methods for indefinite systems. In: Watson GA (ed) Proc the 1974 Dundee biennial conf on numerical analysis, pp 73–89 Google Scholar
  51. 51.
    Frank J, Vuik C (2001) On the construction of deflation-based preconditioners. SIAM J Sci Comput 23:442–462 MATHMathSciNetGoogle Scholar
  52. 52.
    Freund RW (1992) Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J Sci Stat Comput 13(1):425–448 MATHMathSciNetGoogle Scholar
  53. 53.
    Freund RW (1997) Preconditioning of symmetric but highly indefinite linear systems. In: Sydow A (ed) 15th IMACS world congress on scientific computation modelling and applied mathematics, vol 2. Numerical mathematics, pp 551–556 Google Scholar
  54. 54.
    Freund RW, Nachtigal NM (1991) QMR: A quasi minimum residual method for non-Hermitian linear systems. Numer Math 60:315–339 MATHMathSciNetGoogle Scholar
  55. 55.
    Gander MJ, Nataf F (2000) AILU: a preconditioner based on the analytical factorization of the elliptical operator. Numer Linear Algebra Appl 7:543–567 MathSciNetGoogle Scholar
  56. 56.
    Gander MJ, Nataf F (2001) AILU for Helmholtz problems: a new preconditioner based on the analytic parabolic factorization. J Comput Acoust 9:1499–1509 MathSciNetGoogle Scholar
  57. 57.
    Gander MJ, Nataf F (2005) An incomplete LU preconditioner for problems in acoustics. J Comput Acoust 13:455–476 MathSciNetGoogle Scholar
  58. 58.
    George A, Liu JW (1981) Computer solution of large sparse positive definite systems. Prentice-Hall, Englewood Cliffs MATHGoogle Scholar
  59. 59.
    Ghanemi S (1998) A domain decomposition method for Helmholtz scattering problems. In: Bjørstad, Espedal, Keyes, (eds) The ninth intl conf on domain decomposition methods, pp 105–112 Google Scholar
  60. 60.
    Ghosh-Roy DN, Couchman LS (2002) Inverse problems and inverse scattering of plane waves. Academic, London Google Scholar
  61. 61.
    Goldstein CI (1986) Multigrid preconditioners applied to the iterative methods of singularly perturbed elliptic boundary value and scattering problems. In: Innovative numerical methods in engineering. Springer, Berlin, pp 97–102 Google Scholar
  62. 62.
    Gozani J, Nachshon A, Turkel E (1984) Conjugate gradient coupled with multigrid for an indefinite problem. In: Advances in comput methods for PDEs V, pp 425–427 Google Scholar
  63. 63.
    Greenbaum A (1997) Iterative methods for solving linear systems. SIAM, Philadelphia MATHGoogle Scholar
  64. 64.
    Grote MJ, Huckel T (1997) Parallel preconditioning with sparse approximate inverses. SIAM J Sci Comput 18:838–853 MATHMathSciNetGoogle Scholar
  65. 65.
    Gutknecht MH, Ressel KJ (2000) Look-ahead procedures for Lanczos-type product methods based on three-term recurrences. SIAM J Matrix Anal Appl 21:1051–1078 MATHMathSciNetGoogle Scholar
  66. 66.
    Hackbusch W (1978) A fast iterative method for solving Helmholtz’s equation in a general region. In: Schumman U (ed) Fast elliptic solvers. Advance Publications, London, pp 112–124 Google Scholar
  67. 67.
    Hackbusch W (2003) Multi-grid methods and applications. Springer, Berlin Google Scholar
  68. 68.
    Hadley GR (2006) A complex Jacobi iterative method for the indefinite Helmholtz equation. J Comput Phys 203:358–370 Google Scholar
  69. 69.
    Harari I (2006) A survey of finite element methods for time-harmonic acoustics. Comput Methods Appl Mech Eng 195:1594–1607 MATHMathSciNetGoogle Scholar
  70. 70.
    Harari I, Turkel E (1995) Accurate finite difference methods for time-harmonic wave propagation. J Comput Phys 119:252–270 MATHMathSciNetGoogle Scholar
  71. 71.
    Heikkola E, Rossi T, Toivanen J (2000) A parallel fictitious domain decomposition method for the three-dimensional Helmholtz equation. Technical Report No B 9/2000, Dept Math Info Tech, Univ Jÿvaskÿla Google Scholar
  72. 72.
    Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Nat Bur Stand 49:409–435 MATHMathSciNetGoogle Scholar
  73. 73.
    Ihlenburg F, Babuska I (1995) Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int J Numer Methods Eng 38:3745–3774 MATHMathSciNetGoogle Scholar
  74. 74.
    Ihlenburg F, Babuska I (1995) Finite element solution of the Helmholtz equation with high wave number. Part I: the h-version of the FEM. Comput Math Appl 30(9):9–37 MATHMathSciNetGoogle Scholar
  75. 75.
    Ihlenburg F, Babuska I (1997) Finite element solution of the Helmholtz equation with high wave number. Part II: the hp-version of the FEM. SIAM J Numer Anal 34:315–358 MATHMathSciNetGoogle Scholar
  76. 76.
    Jo C-H, Shin C, Suh JH (1996) An optimal 9-point, finite difference, frequency space, 2-D scalar wave extrapolator. Geophysics 61(2):529–537 Google Scholar
  77. 77.
    Kettler R (1982) Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods. In: Hackbusch W, Trottenberg U (eds) Multigrid methods. Lecture notes in mathematics, vol 960, pp 502–534 Google Scholar
  78. 78.
    Kim S (1994) A parallezable iterative procedure for the Helmholtz equation. Appl Numer Math 14:435–449 MATHMathSciNetGoogle Scholar
  79. 79.
    Kim S (1995) Parallel multidomain iterative algorithms for the Helmholtz wave equation. Appl Numer Math 17:411–429 MATHMathSciNetGoogle Scholar
  80. 80.
    Kim S (1998) Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numer Math 79:231–259 MATHMathSciNetGoogle Scholar
  81. 81.
    Kononov AV, Riyanti CD, de Leeuw SW, Vuik C, Oosterlee CW (2006) Numerical performance of parallel solution of heterogeneous 2d Helmholtz equation. In: Wesseling P, Oosterlee CW, Hemker P (eds) Proceedings of the 8th European multigrid conference, TU Delft Google Scholar
  82. 82.
    Laird AL, Giles MB (2002) Preconditioned iterative solution of the 2D Helmholtz equation. Technical Report NA 02-12, Comp Lab, Oxford Univ Google Scholar
  83. 83.
    Lanczos C (1950) An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J Res Nat Bur Stand 45:255–282 MathSciNetGoogle Scholar
  84. 84.
    Lanczos C (1952) Solution of systems of linear equations by minimized iterations. J Res Nat Bur Stand 49:33–53 MathSciNetGoogle Scholar
  85. 85.
    Larsson E (1999) Domain decomposition method for the Helmholtz equation in a multilayer domain. SIAM J Sci Comput 20:1713–1731 MATHMathSciNetGoogle Scholar
  86. 86.
    Lee B, Manteuffel TA, McCormick SF, Ruge J (2000) First-order system least-squares for the Helmholtz equation. SIAM J Sci Comput 21:1927–1949 MATHMathSciNetGoogle Scholar
  87. 87.
    Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42 MATHMathSciNetGoogle Scholar
  88. 88.
    Lynch RE, Rice JR (1980) A high-order difference method for differential equations. Math Comput 34(150):333–372 MATHMathSciNetGoogle Scholar
  89. 89.
    Made MMM (2001) Incomplete factorization-based preconditionings for solving the Helmholtz equation. Int J Numer Methods Eng 50:1077–1101 MATHGoogle Scholar
  90. 90.
    Manteuffel TA, Parter SV (1990) Preconditioning and boundary conditions. SIAM J Numer Anal 27(3):656–694 MATHMathSciNetGoogle Scholar
  91. 91.
    Meijerink JA, van der Vorst HA (1977) An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math Comput 31(137):148–162 MATHGoogle Scholar
  92. 92.
    Meijerink JA, van der Vorst HA (1981) Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems. J Comput Phys 44:134–155 MATHMathSciNetGoogle Scholar
  93. 93.
    Morgan RB (1995) A restarted GMRES method augmented with eigenvectors. SIAM J Matrix Anal Appl 16:1154–1171 MATHMathSciNetGoogle Scholar
  94. 94.
    Nicolaides RA (1987) Deflation of conjugate gradients with applications to boundary value problems. SIAM J Numer Anal 24:355–365 MATHMathSciNetGoogle Scholar
  95. 95.
    Oosterlee CW (1995) The convergence of parallel multiblock multigrid methods. Appl Numer Math 19:115–128 MATHMathSciNetGoogle Scholar
  96. 96.
    Oosterlee CW, Washio T (1998) An evaluation of parallel multigrid as a solver and as a preconditioner for singularly perturbed problems. SIAM J Sci Comput 19:87–110 MATHMathSciNetGoogle Scholar
  97. 97.
    Otto K, Larsson E (1999) Iterative solution of the Helmholtz equation by a second order method. SIAM J Matrix Anal Appl 21:209–229 MATHMathSciNetGoogle Scholar
  98. 98.
    Plessix RE, Mulder WA (2004) Separation-of-variables as a preconditioner for an iterative Helmholtz solver. Appl Numer Math 44:385–400 MathSciNetGoogle Scholar
  99. 99.
    Pratt RG, Worthington MH (1990) Inverse theory applied to multi-source cross-hole tomography. Part 1: acoustic wave-equation method. Geophys Prosp 38:287–310 Google Scholar
  100. 100.
    Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. Oxford Science Publications, Oxford MATHGoogle Scholar
  101. 101.
    Riyanti CD, Kononov AV, Vuik C, Oosterlee CW (2006) Parallel performance of an iterative solver for heterogeneous Helmholtz problems. In: SIAM conference on parallel processing for scientific computing, San Fransisco, CA Google Scholar
  102. 102.
    Riyanti CD, Kononov A, Erlangga YA, Vuik C, Oosterlee CW, Plessix R-E, Mulder WA (2007) A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation. J Comput Phys 224(1):431–448 MATHMathSciNetGoogle Scholar
  103. 103.
    Saad Y (1993) A flexible inner-outer preconditioned GMRES algorithm. SIAM J Sci Comput 14:461–469 MATHMathSciNetGoogle Scholar
  104. 104.
    Saad Y (1994) ILUT: a dual threshold incomplete LU factorization. Numer Linear Algebra Appl 1:387–402 MATHMathSciNetGoogle Scholar
  105. 105.
    Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia MATHGoogle Scholar
  106. 106.
    Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(12):856–869 MATHMathSciNetGoogle Scholar
  107. 107.
    Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with PARDISO. J Future Gen Comput Syst 20:475–487 Google Scholar
  108. 108.
    Schenk O, Gärtner K (2006) On fast factorization pivoting methods for symmetric indefinite systems. Electron Trans Numer Anal 23:158–179 MATHMathSciNetGoogle Scholar
  109. 109.
    Singer I, Turkel E (1998) High-order finite difference methods for the Helmholtz equation. Comput Methods Appl Mech Eng 163:343–358 MATHMathSciNetGoogle Scholar
  110. 110.
    Singer I, Turkel E (2006) Sixth order accurate finite difference scheme for the Helmholtz equations. J Comput Acoust 14(3):339–351 MathSciNetGoogle Scholar
  111. 111.
    Smith B, Bjorstad P, Gropp W (1996) Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge MATHGoogle Scholar
  112. 112.
    Sonneveld P (1989) CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J Sci Stat Comput 10:36–52 MATHMathSciNetGoogle Scholar
  113. 113.
    Strikwerda JC (1989) Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole, Pacific Groove MATHGoogle Scholar
  114. 114.
    Stüben K, Trottenberg U (1982) Multigrid methods: fundamental algorithms, model problem analysis and applications. In: Hackbusch W, Trottenberg U (eds) Lecture notes in math, vol 960, pp 1–176 Google Scholar
  115. 115.
    Susan-Resiga RF, Atassi HM (1998) A domain decomposition method for the exterior Helmholtz problem. J Comput Phys 147:388–401 MATHMathSciNetGoogle Scholar
  116. 116.
    Szyld DB, Vogel JA (2001) A flexible quasi-minimal residual method with inexact preconditioning. SIAM J Sci Comput 23:363–380 MATHMathSciNetGoogle Scholar
  117. 117.
    Tam CKW, Webb JC (1993) Dispersion-relation-preserving finite difference schemes for computational acoustics. J Comput Phys 107(2):262–281 MATHMathSciNetGoogle Scholar
  118. 118.
    Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics 49:1259–1266 Google Scholar
  119. 119.
    Tezaur R, Macedo A, Farhat C (2001) Iterative solution of large-scale acoustic scattering problems with multiple right hand-sides by a domain decomposition method with Lagrange multipliers. Int J Numer Methods Eng 51:1175–1193 MATHMathSciNetGoogle Scholar
  120. 120.
    Thole CA, Trottenberg U (1986) Basic smoothing procedures for the multigrid treatment of elliptic 3-d operators. Appl Math Comput 19:333–345 MATHMathSciNetGoogle Scholar
  121. 121.
    Tosseli A, Widlund O (2005) Domain decomposition methods. Springer, Berlin Google Scholar
  122. 122.
    Trottenberg U, Oosterlee C, Schüller A (2001) Multigrid. Academic, New York MATHGoogle Scholar
  123. 123.
    Tsynkov S, Turkel E (2001) A Cartesian perfectly matched layer for the Helmholtz equation. In: Tourette L, Harpern L (eds) Absrobing boundaries and layers, domain decomposition methods applications to large scale computation. Springer, Berlin, pp 279–309 Google Scholar
  124. 124.
    Turkel E (2001) Numerical difficulties solving time harmonic equations. In: Multiscale computational methods in chemistry and physics. IOS, Ohmsha, pp 319–337 Google Scholar
  125. 125.
    Turkel E, Erlangga YA (2006) Preconditioning a finite element solver of the Helmholtz equation. In: Wesseling P, Oñate EO, Périaux J (eds), Proceedings ECCOMAS CFD 2006, TU Delft Google Scholar
  126. 126.
    van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13(2):631–644 MATHGoogle Scholar
  127. 127.
    van der Vorst HA (2003) Iterative Krylov methods for large linear systems. Cambridge University Press, New York MATHGoogle Scholar
  128. 128.
    van der Vorst HA, Melissen JBM (1990) A Petrov-Galerkin type method for solving Ax=b, where A is symmetric complex systems. IEEE Trans Magn 26(2):706–708 Google Scholar
  129. 129.
    van der Vorst HA, Vuik C (1993) The superlinear convergence behaviour of GMRES. J Comput Appl Math 48:327–341 MATHMathSciNetGoogle Scholar
  130. 130.
    van der Vorst HA, Vuik C (1994) GMRESR: a family for nested GMRES methods. Numer Linear Algebra Appl 1(4):369–386 MATHMathSciNetGoogle Scholar
  131. 131.
    van Gijzen M, Erlangga YA, Vuik C (2007) Spectral analysis of the shifted Laplace precondtioner. SIAM J Sci Comput 29(5):1942–1958 MathSciNetGoogle Scholar
  132. 132.
    Vandersteegen P, Bienstman P, Baets R (2006) Extensions of the complex Jacobi iteration to simulate photonic wavelength scale components. In: Wesseling P, Oñate E, Périaux J (eds) Proceedings ECCOMAS CFD 2006, TU Delft Google Scholar
  133. 133.
    Vandersteegen P, Maes B, Bienstman P, Baets R (2006) Using the complex Jacobi method to simulate Kerr non-linear photonic components. Opt Quantum Electron 38:35–44 Google Scholar
  134. 134.
    Vanek P, Mandel J, Brezina M (1996) Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 56:179–196 MATHMathSciNetGoogle Scholar
  135. 135.
    Vanek PV, Mandel J, Brezina M (1998) Two-level algebraic multigrid for the Helmholtz problem. Contemp Math 218:349–356 MathSciNetGoogle Scholar
  136. 136.
    Vuik C, Erlangga YA, Oosterlee CW (2003) Shifted Laplace preconditioner for the Helmholtz equations. Technical Report 03-18, Dept Appl Math Anal, Delft Univ Tech, The Netherlands Google Scholar
  137. 137.
    Waisman H, Fish J, Tuminaro RS, Shadid J (2004) The generalized global basis (GGB) methods. Int J Numer Methods Eng 61:1243–1269 MATHMathSciNetGoogle Scholar
  138. 138.
    Washio T, Oosterlee CW (1998) Flexible multiple semicoarsening for three dimensional singularly perturbed problems. SIAM J Sci Comput 19:1646–1666 MATHMathSciNetGoogle Scholar
  139. 139.
    Wesseling P (1992) An introduction to multigrid methods. Willey, London MATHGoogle Scholar
  140. 140.
    Wienands R, Joppich W (2004) Practical Fourier analysis for multigrid methods. Chapman & Hall/CRC, London Google Scholar
  141. 141.
    Wienands R, Oosterlee CW (2001) On three-grid Fourier analysis of multigrid. SIAM J Sci Comput 23:651–671 MATHMathSciNetGoogle Scholar
  142. 142.
    Zhou L, Walker HF (1994) Residual smoothing techniques for iterative methods. SIAM J Sci Comput 15(2):297–312 MATHMathSciNetGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2007

Authors and Affiliations

  1. 1.TU BerlinInstitut für MathematikBerlinGermany

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