Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media

An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete $LDL^T$ factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls $\|L^{-1}\|$ for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a “black-box” solver to a series of challenging two- and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.

  • [1]  T. Airaksinen, E. Heikkola, A. Pennanen and , and J. Toivanen, An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation, J. Comput. Phys., 226 (2007), pp. 1196–1210. JCTPAH 0021-9991 CrossrefISIGoogle Scholar

  • [2]  Google Scholar

  • [3]  Google Scholar

  • [4]  I. M. Babuska and  and S. A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM Rev., 42 (2000), pp. 451–484. SIREAD 0036-1445 LinkISIGoogle Scholar

  • [5]  R. E. Bank, A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations, SIAM J. Numer. Anal., 18 (1981), pp. 724–743. SJNAAM 0036-1429 LinkISIGoogle Scholar

  • [6]  A. Bayliss, C. I. Goldstein and , and E. Turkel, An iterative method for the Helmholtz equation, J. Comput. Phys., 49 (1983), pp. 443–457. JCTPAH 0021-9991 CrossrefISIGoogle Scholar

  • [7]  A. Bayliss, C. I. Goldstein and , and E. Turkel, On accuracy conditions for the numerical computation of waves, J. Comput. Phys., 59 (1985), pp. 396–404. JCTPAH 0021-9991 CrossrefISIGoogle Scholar

  • [8]  J.-D. Benamou and  and B. Després, A domain decomposition method for the Helmholtz equation and related optimal control problems, J. Comput. Phys., 136 (1997), pp. 62–82. JCTPAH 0021-9991 CrossrefISIGoogle Scholar

  • [9]  M. Benzi, J. C. Haws and , and M. Tuma, Preconditioning highly indefinite and nonsymmetric matrices, SIAM J. Sci. Comput., 22 (2000), pp. 1333–1353. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [10]  M. Bollhöfer and  and Y. Saad, Multilevel preconditioners constructed from inverse-based ILUs, SIAM J. Sci. Comput., 27 (2006), pp. 1627–1650. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [11]  A. Brandt and  and I. Livshits, Wave-ray multigrid method for standing wave equations, Electron. Trans. Numer. Anal., 6 (1997), pp. 162–181. ETNAB7 1097-4067 Google Scholar

  • [12]  J. R. Bunch and  and L. Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems, Math. Comp., 31 (1977), pp. 163–179. MCMPAF 0025-5718 CrossrefISIGoogle Scholar

  • [13]  X.-C. Cai, M. A. Casarin, F. W. Elliott and , and O. B. Widlund, Overlapping Schwarz algorithms for solving Helmholtz's equation, Contemp. Math., 218 (1998), pp. 391–399. CTMAEH 0271-4132 CrossrefGoogle Scholar

  • [14]  X.-C. Cai and  and O. B. Widlund, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 243–258. SIJCD4 0196-5204 LinkISIGoogle Scholar

  • [15]  A. K. Cline, C. B. Moler, G. W. Stewart and , and J. H. Wilkinson, An estimate for the condition number of a matrix, SIAM J. Numer. Anal., 16 (1979), pp. 368–375. SJNAAM 0036-1429 LinkISIGoogle Scholar

  • [16]  F. Collino, S. Ghanemi and , and P. Joly, Domain decomposition method for harmonic wave propagation: A general presentation, Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 171–211. CMMECC 0045-7825 CrossrefISIGoogle Scholar

  • [17]  I. S. Duff and  and J. Koster, The design and use of algorithms for permuting large entries to the diagonal of sparse matrices, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 889–901. SJMAEL 0895-4798 LinkISIGoogle Scholar

  • [18]  I. S. Duff and  and S. Pralet, Strategies for scaling and pivoting for sparse symmetric indefinite problems, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 313–340. SJMAEL 0895-4798 LinkISIGoogle Scholar

  • [19]  H. C. Elman, O. G. Ernst and , and D. P. O'Leary, A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations, SIAM J. Sci. Comput., 23 (2001), pp. 1291–1315. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [20]  Y. A. Erlangga, C. W. Oosterlee and , and C. Vuik, A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM J. Sci. Comput., 27 (2006), pp. 1471–1492. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [21]  Y. A. Erlangga, C. Vuik and , and C. W. Oosterlee, On a class of preconditioners for solving the Helmholtz equation, Appl. Numer. Math., 50 (2004), pp. 409–425. ANMAEL 0168-9274 CrossrefISIGoogle Scholar

  • [22]  C. Farhat and  and J. Li, An iterative domain decomposition method for the solution of a class of indefinite problems in computational structural dynamics, Appl. Numer. Math., 54 (2005), pp. 150–166. ANMAEL 0168-9274 CrossrefISIGoogle Scholar

  • [23]  C. Farhat, A. Macedo and , and M. Lesoinne, A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems, Numer. Math., 85 (2000), pp. 283–308. NUMMA7 0029-599X CrossrefISIGoogle Scholar

  • [24]  D. G. Feingold and  and R. S. Varga, Block diagonally dominant matrices and generalization of the gershgorin circle theorem, Pacific J. Math., 12 (1962), pp. 1241–1250. PJMAAI 0030-8730 CrossrefISIGoogle Scholar

  • [25]  R. Freund and  and F. Jarre, A QMR-based interior-point algorithm for solving linear programs, Math. Program. Ser. B, 76 (1997), pp. 183–210. CrossrefISIGoogle Scholar

  • [26]  R. Freund and  and N. Nachtigal, Software for simplified Lanczos and QMR algorithms, Appl. Numer. Math., 19 (1995), pp. 319–341. ANMAEL 0168-9274 CrossrefISIGoogle Scholar

  • [27]  M. J. Gander, F. Magoulès and , and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., 24 (2002), pp. 38–60. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [28]  A. George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10 (1973), pp. 345–363. SJNAAM 0036-1429 LinkISIGoogle Scholar

  • [29]  Google Scholar

  • [30]  M. Hagemann and  and O. Schenk, Weighted matchings for preconditioning of symmetric indefinite linear systems, SIAM J. Sci. Comput., 28 (2006), pp. 403–420. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [31]  G. Karypis and  and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), pp. 359–392. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [32]  S. Kim, Parallel multidomain iterative algorithms for the Helmholtz wave equation, Appl. Numer. Math., 17 (1995), pp. 411–429. ANMAEL 0168-9274 CrossrefISIGoogle Scholar

  • [33]  S. Kim and  and S. Kim, Multigrid simulations for high-frequency solutions of the Helmholtz problem in heterogeneous media, SIAM J. Sci. Comput., 24 (2002), pp. 684–701. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [34]  H. W. Kuhn, The Hungarian method for solving the assignment problem, Naval Res. Logist. Quart., 2 (1955), pp. 83–97. NRLOEP 0894-069X CrossrefGoogle Scholar

  • [35]  Google Scholar

  • [36]  B. Lee, T. A. Manteuffel, S. F. McCormick and , and J. Ruge, First-order system least-squares for the Helmholtz equation, SIAM J. Sci. Comput., 21 (2000), pp. 1927–1949. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [37]  X. S. Li and  and J. W. Demmel, SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Trans. Math. Software, 29 (2003), pp. 110–140. ACMSCU 0098-3500 CrossrefISIGoogle Scholar

  • [38]  M. M. M. Made, Incomplete factorization based preconditionings for solving the Helmholtz equation, Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1077–1101. IJNMBH 0029-5981 CrossrefISIGoogle Scholar

  • [39]  Google Scholar

  • [40]  J. Munkres, Algorithms for the assignment and transportation problems, J. SIAM, 5 (1957), pp. 32–38. LinkISIGoogle Scholar

  • [41]  M. Olschowka and  and A. Neumaier, A new pivoting strategy for Gaussian elimination, Linear Algebra Appl., 240 (1996), pp. 131–151. LAAPAW 0024-3795 CrossrefISIGoogle Scholar

  • [42]  C. D. Riyanti, Y. A. Erlangga, R.-E. Plessix, W. A. Mulder, C. Vuik and , and C. Oosterlee, A new iterative solver for the time-harmonic wave equation, Geophysics, 71 (2006), pp. E57–E63. GPYSA7 0016-8033 CrossrefISIGoogle Scholar

  • [43]  C. D. Riyanti, A. Kononov, Y. A. Erlangga, C. Vuik, C. W. Oosterlee, R.-E. Plessix and , and W. A. Mulder, A parallel multigrid-based preconditioner for the 3d heterogeneous high-frequency Helmholtz equation, J. Comput. Phys., 224 (2007), pp. 431–448. JCTPAH 0021-9991 CrossrefISIGoogle Scholar

  • [44]  Google Scholar

  • [45]  Google Scholar

  • [46]  A. H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), pp. 959–962. MCMPAF 0025-5718 CrossrefISIGoogle Scholar

  • [47]  O. Schenk, M. Bollhöfer and , and R. A. Römer, On large-scale diagonalization techniques for the Anderson model of localization, SIAM J. Sci. Comput., 28 (2006), pp. 963–983. SJOCE3 1064-8275 LinkISIGoogle Scholar

  • [48]  O. Schenk, M. Bollhöfer and , and R. A. Römer, On large-scale diagonalization techniques for the Anderson model of localization, SIAM Rev., 50 (2008), pp. 91–112. SIREAD 0036-1445 LinkISIGoogle Scholar

  • [49]  O. Schenk and  and K. Gärtner, On fast factorization pivoting methods for symmetric indefinite systems, Electron. Trans. Numer. Anal., 23 (2006), pp. 158–179. ETNAB7 1097-4067 ISIGoogle Scholar

  • [50]  O. Schenk, S. Röllin and , and A. Gupta, The effects of unsymmetric matrix permutations and scalings in semiconductor device and circuit simulation, IEEE Trans. Computer-Aided Design Integrated Circuits Syst., 23 (2004), pp. 400–411. CrossrefISIGoogle Scholar

  • [51]  H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631–644. SIJCD4 0196-5204 LinkISIGoogle Scholar

  • [52]  M. B. van Gijzen, Y. A. Erlangga and , and C. Vuik, Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian, SIAM J. Sci. Comput., 29 (2007), pp. 1942–1958. SJOCE3 1064-8275 LinkISIGoogle Scholar