c 2006 Society for Industrial and Applied Mathematics

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1 SIAM J. SCI. COMPUT. Vol. 27, No. 4, pp c 26 Society for Industrial and Applied Matematics A NOVEL MULTIGRID BASED PRECONDITIONER FOR HETEROGENEOUS HELMHOLTZ PROBLEMS Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK Abstract. An iterative solution metod, in te form of a preconditioner for a Krylov subspace metod, is presented for te Helmoltz equation. Te preconditioner is based on a Helmoltz-type differential operator wit a complex term. A multigrid iteration is used for approximately inverting te preconditioner. Te coice of multigrid components for te corresponding preconditioning matrix wit a complex diagonal is validated wit Fourier analysis. Multigrid analysis results are verified by numerical experiments. Hig wavenumber Helmoltz problems in eterogeneous media are solved indicating te performance of te preconditioner. Key words. Helmoltz equation, nonconstant ig wavenumber, complex multigrid preconditioner, Fourier analysis AMS subject classifications. 65N55, 65F, 65N22, 78A45, 76Q5 DOI..37/ Introduction. In tis paper we present a novel preconditioner for ig wavenumber Helmoltz problems in eterogeneous media. Te preconditioner is based on te Helmoltz operator, were an imaginary term is added. Tis preconditioner can be andled by multigrid. Tis is somewat surprising as multigrid, witout enancements, as convergence troubles for te original Helmoltz operator at ig wavenumbers. A part of tis paper is terefore reserved for te analysis of te multigrid metod for Helmoltz problems wit a complex zerot order term. Tis is done, for constant wavenumbers, by means of Fourier analysis. Te preconditioned system leads to a favorably clustered spectrum for a Krylov subspace convergence acceleration. As te preconditioner is not based on a regular splitting of te original Helmoltz problem, it must be used in te setting of Krylov subspace metods. Te particular example presented can be viewed as a generalization of te work by Bayliss, Goldstein, and Turkel [3] from te 98s, were te Laplacian was used as a preconditioner for Helmoltz problems. Tis work as been generalized by Laird and Giles [7], proposing a Helmoltz preconditioner wit a positive sign in front of te Helmoltz term. In [3] we ave proposed a preconditioner wit a purely imaginary sift added to te Laplacian. Te metod ere is an improvement of tat metod. In tis paper we benefit from Fourier analysis in several ways. First of all, for idealized (omogeneous boundary conditions, constant coefficients) versions of te preconditioned system it is possible to visualize its spectrum for different values of te wavenumber, as Fourier analysis provides all eigenvalues. Second, for analyzing multigrid algoritms quantitatively, Fourier smooting, two-, and tree-grid analysis [6, 7, 23, 24, 3] are te tools of coice. Received by te editors September 6, 24; accepted for publication (in revised form) May 9, 25; publised electronically January 27, 26. Te researc is financially supported by Dutc Ministry of Economic Affairs project BTS44. ttp:// Faculty of Electrical Engineering, Matematics and Computer Science, Delft University of Tecnology, Delft, Te Neterlands (y.a.erlangga@mat.tudelft.nl, c.w.oosterlee@mat.tudelft.nl, c.vuik@mat.tudelft.nl). 47

2 472 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK Table Number of grid points employed, related to te wavenumber, so tat k =.625. k /64 /8 /28 /6 /24 /32 /8 /96 Te outline of tis paper is as follows. In section 2 te Helmoltz problem is introduced and te convergence difficulties of multigrid for tis equation are detailed. Te new preconditioner is introduced in section 3, were multigrid components for Helmoltz problems wit a complex term (smooting, operator-dependent prolongation) are presented. Fourier analysis to obtain quantitative performance estimates of components and metods is performed in section 4. Numerical experiments on two-dimensional ig wavenumber eterogeneous Helmoltz problems are presented in section Helmoltz equation, standard multigrid. Consider te Helmoltz equation for a wave problem in a eterogeneous medium () Aφ := xx φ yy φ ( αi)k 2 (x, y)φ = g(x, y) in Ω R 2. Here, φ = φ(x, y) represents te solution, usually a pressure field, and g represents te source term. Te medium is barely attenuative if α<<, wit α indicating te fraction of damping in te medium (i =, te imaginary unit). In geopysical applications, wic are our main interest, tis damping can be set up to 5% (α =.5). Te wavenumber k = ω f /c is space-dependent because of a spatially dependent speed of sound c(x, y) in a eterogeneous medium. Wit ω f := 2πf te angular frequency (f is te frequency), wavelengt l is defined by l = c/f. Te number of wavelengts in a domain of size L equals L/l. n w, te number of points per wavelengt, is typically cosen to be 2 points. Wavenumber k can be large. Te dimensionless wavenumber k on a nondimensional [, ] 2 domain is defined by k =2πfL/c. A dimensionless discretization step reads = l/(n w L), and terefore for te angular frequency one finds ω f =2π/(n w )=2πL/l. Wit domain size L =, an accuracy requirement for second order discretizations is tat k π/5(.63) for n w = points per wavelengt, and k.53 wit n w = 2 points per wavelengt. In Table, te number of grid points used for several wavenumbers k is displayed. For eac combination we ave k =.625. Tese mes refinements assume a linear connection between k and. However, in order to avoid a reduction of accuracy for a second order sceme due to te so-called pollution effect [26, 4] k 2 3 sould be set constant. As for an iterative solution metod, keeping k constant is more severe; we stay wit k as in Table. In tis paper we empasize te iterative solution rater tan te accuracy of te discretization. Typically, boundary conditions at te boundary Γ = Ω are in te form of first- or second-order absorbing boundary conditions or of a perfectly matced layer (PML). We use approximate radiation (or nonreflecting) boundary conditions at an artificial boundary. Te well-known second-order radiation boundary condition [2], to avoid unpysical reflections at boundaries, reads (2) A Γ φ := φ ν ikφ i 2 φ = on Γ, 2k τ2 wit ν te outward normal direction to te boundary and τ pointing in te tangential direction. At te cornerpoints te suggestions in [2] to avoid corner reflections

3 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 473 ave been adopted. Wereas tese conditions are commonly applied for problems in omogeneous media, tey are less obvious for inomogeneous media wit discontinuous wavenumbers at te boundaries. Te reason is tat tese discontinuities may act as unpysical scatterers at te boundaries. Tis can be avoided by appropriately increasing damping, modeled by te imaginary part in (2). Anoter natural approac wit an inomogeneous medium at te boundaries may be te use of te PML. If a discretization is applied to () and (2), a linear system of te form (3) Aφ = g, A C N N, φ, g C N, is obtained, were N is te number of unknowns in te computational domain Ω. Matrix A as complex components due to te discrete boundary operator (2) and te damping term in (). A is in general symmetric wit eigenvalues in te left and rigt alf-plane, non-hermitian, and, because of te accuracy requirements, also large for ig wavenumbers. However, A is also sparse; its sparsity pattern depends on te discretization metod used. We consider ere, in stencil notation, te well-known O( 2 ) 5-point discretization stencil: (4) A = 2 4 (k) 2 ( αi) We use matrix and stencil notation simultaneously: Matrix A (3) relates to te discretization of (), (2), and discrete operator A (4) relates to te discretization of (). Te discrete solution is represented by φ and φ, respectively. Te eigenvalues (for constant k-problems wit omogeneous Diriclet boundary conditions). (5) λ l,m = λ l,m k 2 ( αi) 2 2 (2 cos lπ cos mπ) k2 ( αi) (l, m =, 2,..., N ) are not equal to zero as long as k 2 ( αi) is not equal to any of te eigenvalues of l,m te corresponding discrete Laplacian λ. Oterwise, te matrix is singular, and its null-space is spanned by te eigenfunctions (6) v l,m = sin lπx sin mπy, wit l, m for wic λ l,m =. 2.. Multigrid convergence for te Helmoltz equation. Textbook multigrid metods are typically set up so tat a smooting metod reduces ig frequency components of an error between te numerical approximation and te exact discrete solution, and a coarse grid correction andles te low frequency error components. Wereas suc metods are easily defined for elliptic Poisson-like equations, tis is not te case for te Helmoltz equation witout any damping in (), α =. Depending on te particular value of k 2, tis equation gives rise to bot smooting and coarse grid correction difficulties. Te matrix as eigenvalues in only te rigt alf-plane as long as k 2, is less tan te smallest eigenvalue of te Laplacian, λ. For k2, > λ, te matrix does not ave only positive eigenvalues. Pointwise Jacobi iteration wit underrelaxation does not converge in tat case, but since its smooting properties are

4 474 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK satisfactory, te multigrid convergence will deteriorate only gradually for increasing k 2. By te time k 2 l,m approaces te 6t eigenvalue λ (k 2 5), te standard multigrid metod diverges. Te Jacobi relaxation now diverges for smoot eigenfrequencies v l,m l,m wit λ <k 2. Consequently, te multigrid metod will still converge as long as te coarsest level used is fine enoug to represent tese smoot eigenfrequencies sufficiently. So, te coarsest level cosen limits te convergence. Wen k 2 gets larger more variables need to be represented on te coarsest level for standard multigrid convergence. Eventually, tis does not result in an O(N) iterative metod. In addition to tis feature, te Helmoltz equation also brings a multigrid coarse grid correction difficulty. Eigenvalues close to te origin may undergo a sign cange after discretization on a coarser grid. If a sign cange occurs, te coarse grid solution does not give a convergence acceleration to te finer grid problem but gives a severe convergence degradation (or even divergence) instead. In [] tis penomenon is analyzed and a remedy for te coarse grid correction related to tese problematic eigenvalues is proposed. Te efficient treatment in [] is tat te multigrid metod is combined wit Krylov subspace iteration metods. GMRES is proposed as a smooter and as a cure for te problematic coarse grid correction. Standard multigrid will also fail for k 2 -values very close to eigenvalues. In tat case subspace correction tecniques sould be employed [9]. An advanced multigrid based solution metod for te Helmoltz equation is te wave-ray multigrid metod [8]. Te metod as been adapted for a first-order system least-squares version of te Helmoltz equation in [8]. Wave-ray multigrid as been developed for Helmoltz problems wit constant or smootly varying wavenumbers. A toroug overview for te numerical solution of te Helmoltz equation is presented in [25]. 3. Sifted Laplacian preconditioner. To solve (3), iterative metods based on te Krylov subspace are of interest. In particular, we coose preconditioned Bi- CGSTAB. In [3], Bi-CGSTAB is preferred over oter Krylov subspace metods as te convergence for Helmoltz problems is reported typically faster tan tat of GMRES. We ave also tested advanced versions suc as Bi-CGSTAB(2) [22] and GMRESR [28], but Bi-CGSTAB remains te metod of coice, especially for te Helmoltz equation witout damping (α = ). A preconditioner M C N N for A is developed suc tat te preconditioned system (7) AM ψ = g, ψ = Mφ, as better spectral properties tan te original system. Te preconditioner M proposed ere is based on te following operator: (8) M xx yy (β β 2 i)k 2 (x, y), β,β 2 R, wit (β,β 2 ) parameters tat can be cosen freely and wit i te imaginary unit. Boundary conditions are set identical to tose for te original Helmoltz problem (2). A large imaginary value for te wavenumber pysically corresponds to adding some form of damping for preconditioning. In te time domain te eat equation is sometimes used as te preconditioner for te Laplacian. Also te Jacobi iteration for te Laplacian can be interpreted as a time stepping procedure for te eat equation. For te wave equation suc an iteration is a less common approac. A large imaginary Helmoltz term can be seen as a time-dependent term tat is transformed to Fourier space.

5 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 475 If one is interested in only te interior problem witout any damping term and real-valued boundary conditions, te original problem would be real-valued. Introducing a complex sift ten introduces complex aritmetic into a real problem. Besides te extra work, it necessitates possibly different iterative solvers. Here, owever, we are interested in geopysics applications wit outgoing waves. Te basic coice in tis paper is (β,β 2 )=(, ). Tuning of multigrid components is especially necessary for β 2 <, for example, for β 2 =.5, to be presented below. In [3] we ave proposed a positive purely imaginary sift (β,β 2 )=(, ) to te Laplacian for a satisfactory convergence. Preconditioner (8) is an improvement of tis preconditioner wit β =. We perform Fourier analysis to visualize te effect of te coice of (β,β 2 )in te preconditioner on te clustering of te eigenvalues of te preconditioned system. For tis we consider operator (7) wit omogeneous Diriclet boundary conditions, wavenumber k constant, and a discrete version of Helmoltz operator (), A, and of preconditioner (8), M. Tis particular coice of te boundary conditions especially simplifies te analysis. For radiation boundary conditions te Helmoltz operator is nonnormal. Hence eigenvalue analysis alone would not be sufficient for analyzing preconditioned Krylov subspace metods. Here, we perform te analysis as a first indication of wat we can expect from te solver to be developed and concentrate on te eigenvalues. For bot A and M we coose te 5-point stencil, as in (4). Te components (6) are eigenfunctions of tese discrete operators wit constant coefficients. Wit tese eigenfunctions A M is diagonalizable and te eigenvalues are easily determined. In te first tests we do not include damping in A, α = in (), (4). Figure presents spectra of A M for (β,β 2 )=(, ) (Laplacian preconditioner), (β,β 2 )=(, ) (Laird preconditioner [7]), (β,β 2 )=(, ) (preconditioner from [3]), (β,β 2 )=(, ) (basic parameter coice), (β,β 2 )=(,.5), and (β,β 2 )=(,.3) (more advanced parameters). Te results are for k =4(k 2 = 6) and = /64. Similar eigenvalue distributions are observed for finer grids. From te spectra presented wit te new preconditioner, te lower pictures of Figure are favorable as teir real parts vary between and. Te Laplacian preconditioner in Figure (a) exibits large isolated eigenvalues; for te Laird preconditioner te eigenvalues in Figure (b) are distributed between andonte real axis. Te preconditioners wit complex Helmoltz terms give rise to a curved spectrum. Wereas te real part of te spectrum in Figure (c) still includes a part of te negative real axis, tis is not te case for te (β,β 2 )-preconditioners wit β =. Te difference between Figures (d), (e), and (f) is tat, wit a smaller value of β 2, fewer outliers close to te origin are observed. Tis is favorable for te convergence of te preconditioned Krylov metod. Te approximate inversion of te preconditioner itself by multigrid, owever, will be sown to be arder for smaller values of β 2. In Figure 2 te spectra for k = (k 2 = 4 ) are presented on a grid wit =/6 for β = and β 2 varying between and.3. Te spectra are very similar to tose in Figure. More eigenvalues lie, owever, in te vicinity of te origin due to te iger wavenumber and te correspondingly finer grid. Figure 3 presents te distribution of eigenvalues for te case tat 5% damping (α =.5) is set in A. Parameters in te preconditioner are (β,β 2 )=(,.5). Again te 5-point stencil as in (4) is used for discretization. Figure 3(a) presents te spectrum for k = 4, = /64, and Figure 3(b) presents te spectrum for k =, = /6. An interesting observation is tat now te eigenvalues move away from te origin into te rigt alf-plane. Tis

6 476 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK i i (a) i (c) (b) i i 8.5i.5i.5i (f).2.5i.5i (e) (d).4i.5i Fig.. Spectral pictures of A M wit α = and different values of (β, β2 ) in (8). (a) (β, β2 ) = (, ), (b) (, ), (c) (, ), (d) (, ), (e) (,.5), and (f) (,.3)..5 i.5 i (a).5i (c).5i.5 i (b).5i Fig. 2. Spectral pictures of A M for k =, = /6, and α = ; (a) (β, β2 ) = (, ), (b) (β, β2 ) = (,.5), and (c) (β, β2 ) = (,.3). is beneficial for iterative solution metods. From te spectra in Figure 3 it is expected tat te Bi-CGSTAB (and GMRES) convergence in te case of damping will be considerably faster tan for te undamped case. 4. Multigrid for te preconditioner. 4.. Multigrid components. Geometric multigrid converges satisfactorily for te Helmoltz operator (8) for certain coices of β and β2 (assumed in [5], see also [6]). In tis section, we detail te multigrid components tat can be specified for approximately inverting a discrete version of M in (8). We consider a 5-point discretization and denote te equation for te preconditioner by M φ = ψ. Standard multigrid coarsening, i.e., doubling te mes size in every direction, is cosen. For smooting te pointwise Jacobi relaxation wit underrelaxation (ω-jac) is cosen. Tis smooter is well parallelizable, wic is an important aspect for our researc (w.r.t. a generalization to tree dimensions). In principle, one can coose te underrelaxation parameter ω C, but te Fourier analysis indicates tat tere is no real benefit for te problems considered. So, we coose ω R.

7 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 477.5i.5i (a) (b).5i.5i Fig. 3. Spectral pictures of AM wit 5 % damping in A and (β,β 2 )=(,.5); (a) k = 4,=/64 and (b) k =,=/6. nw n ne C D w c e q r sw s se A p B Fig. 4. Left: Nine point stencil wit numbering. Rigt: Coarse grid cell and four fine cells, (Coarse grid indices designated by capital letters and fine grid indices designated by lower case letters). Te coarse grid correction components are also based on establised operators. For te discrete coarse grid operators M 2,M 4,...,te Galerkin coarse grid operator is used: M 2 := R 2 M P 2, M 4 := R 4 2M 2 P 2 4, etc. In te Fourier analysis to follow, tis discretization will be compared to a direct coarse grid discretization of (). Te Galerkin coarse grid discretization is a natural coice for eterogeneous problems. Also wit boundary conditions containing first and second derivatives, it is convenient to coose te Galerkin coarse grid discretization as it defines te appropriate coarse grid boundary stencils automatically. Te transfer operators used in building te coarse grid operators are te same as tose used for transferring coarse and fine grid quantities to fine and coarse grids, respectively. Te prolongation operator considered is an operator-dependent interpolation based on de Zeeuw s transfer operators [3]. Originally, tis prolongation was set up for general (possibly unsymmetric) real-valued matrices wit a splitting of matrix M into a symmetric and an antisymmetric part, M s = 2 (M + M T ), M t = M M s in [3]. However, since te discretization ere leads to a complex symmetric matrix, te prolongation is adapted and briefly explained for suc matrices wit nine diagonals. Te numbering in a stencil for te explanation of te prolongation is as sown in Figure 4 (left side). Te rigt side of Figure 4 sows one coarse and four fine grid cells wit indices for te explanation of te interpolation weigts. Capital letters denote coarse grid points and lower case letters denote fine grid points. Operator element m w p, for example, denotes te west element of operator M at point p on te fine grid. Te corrections from te coarse to te fine grid are obtained by interpolation among nearest coarse grid neigbors. Te operator-dependent interpolation weigts,

8 478 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK w, to determine te fine grid correction quantities e are derived wit te following formulas. For fine grid points p in Figure 4, e,p = w A e H,A + w B e H,B. w A = min(, max(,w w )); w B = min(, max(,w e )), were (9) () () wit (2) (3) (4) d w = max( m sw p d e = max( m se p w w = + m w p + m nw p, m sw p, m nw p ), p ), + m s p + m ne p, m se p, m ne d w d e, w e =. d w + d e d w + d e For fine grid points q in Figure 4, e,q = w A e H,A + w C e H,C. w A = min(, max(,w s )); w C = min(, max(,w n )), d n = max( m nw q + m n q + m ne q, m nw q, m ne q ), d s = max( m sw q + m s q + m se q, m sw q, m se q ), w s = d s, w n = d n. d s + d n d s + d n On te remaining points te prolongation is defined as follows: (5) (6) On fine grid points tat are also coarse points, e (A) =e 2 (A). On points r, e (r) is determined so tat M P2e 2 = atr. Te interpolation weigts are te same as in [3] but are specially tailored to te symmetric complex Helmoltz equation, i.e., te unsymmetric components in [3] ave been removed.. denotes te modulus, in tis case, leading to real-valued interpolation weigts. As for symmetric problems wit jumping coefficients, te prolongation operator by de Zeeuw [3] is very similar to te original operator-dependent prolongation in []. In [], for d w, for example, te lumped sum of tree elements, m sw p + m w p + m nw p, is cosen. For satisfactory convergence it is, owever, important to consider te modulus of te operator elements, as in (9), (), (2), and (3), in te definition of te interpolation weigts. Tis prolongation is also valid at boundaries. Te full weigting operator is employed as te restriction operator. So, we do not coose te adjoint of te prolongation operator, wic is commonly used but is not absolutely necessary, as already stated in [] (an example were te restriction is not te adjoint of te prolongation operator as been given in []). We coose te combination of a full weigting restriction and te operator-dependent interpolation, as it brings a robust convergence for a variety of Helmoltz problems wit constant and nonconstant coefficients. For constant coefficients and mildly varying wavenumbers, bilinear interpolation also gives very satisfactory convergence results, but for strongly varying coefficients, as in te Marmousi problem discussed in section 5.3, a robust and efficient convergence on different grid sizes and for many frequencies is observed for te combination of te transfer operators cosen Fourier analysis. Fourier smooting and two-grid analysis, two classical multigrid analysis tools, ave been used for quantitative estimates of te smooting properties and of te oter multigrid components in a two-grid metod [5, 6, 7, 23, 24]. Consider a discretization of (7), (8), M φ = ψ, were φ represents te exact discrete solution. Te error w l = φl φ after te lt iteration is transformed by a

9 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 479 two-grid cycle as (7) w l+ = T 2 w, l T 2 = S ν2 K2 S ν, K2 = I P 2(M 2 ) R 2 M. M, M 2 correspond to discretizations of (8) on te -, 2-grid, S is te smooting operator on te fine grid, and I is te identity operator. ν l (l =, 2) represents te number of pre- and postsmooting steps, and R 2 and P 2 denote te restriction and prolongation operator, respectively. In te analysis we assume an equidistant grid wit N points in eac direction. Te O( 2 )-discrete complex Helmoltz operator from (8) wit constant wavenumber and Diriclet boundary conditions belongs to te class of symmetric stencils. For tese stencils it is possible to apply Fourier analysis on te basis of discrete sine-eigenfunctions v l,m,l,m=,..., N (6), instead of te local Fourier analysis wit exponential functions. For problems wit symmetric stencils and omogeneous Diriclet boundary conditions, tis analysis can predict -dependent convergence factors. From te discussion of multigrid metods for te original Helmoltz equation, it seems necessary to gain insigt into te -dependency of te multigrid metods developed also for te complex Helmoltz operator. (Te definition of te operator-dependent prolongation and te Galerkin coarse grid stencils in section 4. also leads to symmetric operators tat can be analyzed witin tis framework.) For te pointwise Jacobi smooter, te v l,m (6) are also eigenfunctions of te smooting operator. Tis is not true for te two-grid iteration operator T 2. However, 4-dimensional linearly independent spaces, te armonics, (8) E l,m = [ v l,m,v N l, N m, v N l,m, v l, N m ] for l, m =,..., N 2 are invariant under tese operators. One can sow [23, 24] tat and T 2 M : span [v l,m ] span [vl,m ], (M 2) : span [v l,m 2 ] span [vl,m S : span [v l,m ] span [vl,m ], R 2 : El,m span [v l,m 2 ], P 2 : span [vl,m 2 ] El,m, : E l,m wit respect to E l,m (9) E l,m T 2 (l, m =,..., N 2 leads to a block-diagonal matrix, T 2 [ ] = T 2 (l, m) l,m=,..., N 2 l,m N 2 2 ], 2 ). Terefore, te representation of T 2 T, =: T 2. Here te blocks (l, m) are 4 4 matrices if l, m < N 2, and are 2 2( ) matrices if eiter l = N 2 or m = N 2 (l = N 2 and m = N 2 ). Te two-grid convergence factor is defined as ) ρ 2g := max ρ( T 2 (2) (l, m). Tus, te spectral radii of at most 4 4 matrices (l, m) ave to be determined, and teir maximum wit respect to l and m as to be found. Te definition of te smooting factor μ is closely related. Te smooting factor measures te reduction of ig frequency error components by an iterative metod. T 2

10 48 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK It is based on a coarse grid correction operator tat anniilates te low frequency error components completely and keeps te ig frequency components uncanged. K 2 is replaced by a projection operator Q2 mapping onto te space of ig frequencies, i.e., a block-diagonal matrix wit Q 2 at most 4 4-diagonal blocks defined by diag(,,, ). So, μ is computed as ρ 2g (2) wit K in T replaced by Q. Recently, tree-grid Fourier analysis was proposed in [3]. An issue tat can be analyzed in some more detail wit a tird grid is te coarse grid correction. If a large difference occurs between te two-grid and te tree-grid convergence factors, ρ 2g and ρ 3g, tis is an indication for a problematic coarse grid correction. For te complex Helmoltz preconditioner it is important to analyze te coarse grid correction carefully. Te error transformation by a tree-grid cycle is given by (2) T 4 w l+ = T 4 w l wit = S ν2 K4 S ν and K 4 = I P2(I 2 (T2 4 ) γ )(M 2 ) R 2 M. Here T2 4 4, defined by (7), reads T2 = Sν2 2 (I 2 P4 2(M 4) R4 2)Sν 2.M 4 corresponds to 4-grid discretization of (8); S 2 is te smooting operator, and I 2 is te identity on te 2-grid; and R2 4 2 and P4 are transfer operators between te different grids. Te 2-equation is solved approximately in a tree-grid cycle (2) by performing γ two-grid iterations T2 4 wit zero initial approximation; see also [23, 3]. Te tree-grid analysis is a recursive application of te two-grid analysis. Four frequencies are coupled not only in te transition from te - to te 2-grid but also in te transition from te 2- to te 4-grid. Tus te tree-grid error transformation operator couples 6 Fourier frequencies. As a consequence, T 4 is unitarily equivalent 4 4 to a block-diagonal matrix T wit at most 6 6 blocks, T (l, m). Te block matrices are composed of te Fourier symbols from te two-grid analysis, wic is due to te recursive application of te two-grid analysis. One may compute te tree-grid factor ρ 3g as te supremum of te spectral radii from te 6 6 block matrices, T 4 (l, m). For more details about te tree-grid analysis, we refer to [3]. Tree-grid Fourier analysis software, based on te exponential functions, is freely available; see ttp:// Fourier analysis and multigrid results. We first compare te numerical multigrid convergence wit asymptotic convergence factors μ, ρ 2g,ρ 3g from Fourier analysis. For tis, we consider ere solely te preconditioner M (8). (Te beavior of te complete solution metod will be considered in te next section.) Wavenumber k is taken as a constant ere and a square domain wit an equidistant grid is used. Te second-order boundary conditions (2) are set in te numerical experiments to mimic reality. An interesting aspect is tat almost identical convergence factors are obtained, bot from te analysis and from te actual experiments, for constant values of k. Tey are set as in Table. Te results are validated from k =4uptok = 6, te igest wavenumber tested is (k 2 =3.6 5 ). During testing te following abbreviations are used: ω-jac is te Jacobi smooter wit underrelaxation, Galerkin is te Galerkin coarse grid discretization, and direct is a direct coarse grid discretization of te PDE. Direct as not been implemented in te numerical code, but it can be used in te analysis framework. Multigrid coarsening is continued until fewer tan points are processed on te coarsest grid. Te number of levels is - and terefore also k-dependent, as

11 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 48 finest coarsest Fig. 5. An F-cycle for five grids. Table 2 Comparison of asymptotic convergence from Fourier analysis wit numerical multigrid convergence, (β,β 2 )=(, ). μ is te smooting factor; ρ 2g, ρ 3g are te two- and tree-grid convergence factors from Fourier analysis; and ρ is te numerical multigrid convergence factor. Te smooter is ω-jac wit ω =.8. (ν,ν 2 ) μ ρ 2g ρ 3g,γ=2 ρ, F-cycle (,) (,) k is kept constant on te finest grid, and varies between 5 and 9 grids. Te F-cycle (see Figure 5) is always used in te numerical tests; te V-cycle s performance was generally too poor; and te W-cycle is considered too expensive on te very fine grids processed at ig wavenumbers. Te F-cycle often sows te robustness of te W-cycle at te efficiency of te V-cycle. In te tree-grid analysis, γ = 2, te W-cycle analysis is used. Remark. Te Fourier analysis applied directly to te Helmoltz equation () wit α = and te specified mes sizes gives a satisfactory smooting factor, but te twoand tree-grid analysis convergence factors and also te actual multigrid results sow a strong divergence, as expected. Te case (β,β 2 )=(, ). We start wit (β,β 2 )=(, ), as in [3]. Tis case is not of te igest interest as a preconditioner, as te Bi-CGSTAB convergence for te corresponding preconditioned system is worse tan wit β = (sown in te next section). Tis case (β,β 2 )=(, ) serves as a reference for te comparison between Fourier analysis and numerical convergence. Te underrelaxation parameter ω is set to ω =.8, as tis is te optimal coice for te Laplacian [24]. Te agreement between te smooting two- and tree-grid Fourier analysis results wit one and two smooting iterations and te numerical convergence is excellent, presented in Table 2. Te results obtained are very similar to te convergence factors for te Laplacian wit ω-jac. Remark. For te case (β,β 2 )=(, ), one can adopt te well-known multigrid components: direct PDE coarse grid discretization and red-black Gauss Seidel relaxation. Tis gives ρ 3g =.6 for γ = and ρ 3g =.8 for γ = 2 wit two smooting iterations, very similar to te Laplacian situation. Red-black Gauss Seidel relaxation is, owever, not as robust as te ω-jac relaxation for te β = cases. Furtermore, te cost in CPU time on a Linux PC of one red-black Gauss Seidel iteration is about twice tat of a Jacobi iteration. Te case (β,β 2 )=(, ). Te second test is for (β,β 2 )=(, ). In tis test we employ ω-jac smooting wit ω =.7 in an F(,)-cycle (ν = ν 2 = ). It is necessary to adapt te relaxation parameter ω for satisfactory numerical convergence. Te

12 482 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK Table 3 Comparison of convergence (β,β 2 )=(, ) wit Fourier analysis convergence (γ =), ω-jac, ω =.7, and F (, )-cycle. Coarse grid discretizations are compared. (Te direct discretization as not been implemented). Coarse discr. μ ρ 2g ρ 3g,γ=2 ρ, F(,) Galerkin Direct Table 4 Fourier analysis convergence factors compared to multigrid convergence (β,β 2 )=(,.5). Te smooter is ω-jac wit ω =.5. (Te direct discretization as not been implemented). Coarse discr. μ ρ 2g ρ 3g,γ=2 ρ, F(,) Galerkin Direct performance of te ω-jac smooter is not sensitive wit respect to coosing somewat smaller values of ω. We compare te Galerkin discretization wit te direct coarse grid PDE discretization. Analysis results wit two smooting iterations are sown in Table 3, and tey are compared to te numerical F(,) multigrid convergence. Convergence factors well below.5 are obtained wit te F(,)-cycle and ω- JAC relaxation wit ω =.7. Te Fourier analysis results wit te Galerkin coarse grid discretization are very similar to tose obtained wit a direct coarse grid PDE discretization. Te case (β,β 2 )=(,.5). Te preconditioner of coice in tis paper is based on te parameters (β,β 2 )=(,.5). For tis parameter set it is possible to define a converging multigrid iteration by means of an F(,)-cycle, ω-jac relaxation wit ω =.5, and a Galerkin coarse grid discretization. Te underrelaxation parameter needs to be adapted for a robust convergence for a variety of eterogeneous Helmoltz problems. For values β 2 <.5 it is very difficult to define a satisfactory converging multigrid F(,)-cycle wit te components at and. Tey are terefore not considered. Table 4 compares te Galerkin wit te direct PDE coarse grid discretization. Also ere, te operator-dependent interpolation and full weigting restriction are cosen, and two smooting iterations are applied. Te smooting factors and twoand tree-grid factors are very similar, wic is an indication for te proper coice of coarse grid correction components for te problems under investigation. Te numerical convergence wit te F(,)-cycle is again very similar to te Fourier results. In te following tree remarks, we explain te satisfactory convergence of a standard multigrid metod for te complex Helmoltz equation and β = wit some euristic arguments. Te remarks sow tat, wen β 2 is cosen small but suc tat te multigrid metod still converges, te coarse grid stencils do not represent te fine grid problem well. In particular, te main diagonal operator elements in (22) on different coarse grids are smaller tan te off-diagonal elements. Tis will be te reason wy te V-cycle does not perform as well as te F-cycle and wy damped Jacobi relaxation is a more robust smooter tan red-black Gauss Seidel relaxation for te case β 2 =.5. Remark: Smooting. Te Fourier symbol of ω-jac for te complex Helmoltz equation reads

13 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 483 Table 5 Smooting factors μ for ω-jac on different coarse grids and various (β,β 2 )-values. (β,β 2 ) ω in ω-jac /64 /32 /6 /8 (, ) (, ) (, ) (,.5) ω ( S = 4 (β 4 (β β 2 i)(k) 2 β 2 i)(k) 2 2 cos lπ 2 cos mπ ), l, m =,..., N. We consider te case k =4, =/64 and take ω as in te previous experiments. Table 5 presents smooting factors on four consecutive grids for (β,β 2 ) = (, ) (te original Helmoltz equation) and for (β,β 2 )=(, ), (, ), and (,.5). For simplicity, a direct PDE discretization on te coarse grids as been used. From Table 5, one confirms tat for = /6,ω-JAC diverges for te original Helmoltz operator (also found wit oter relaxation parameters). Tis is in accordance wit te remarks in [9, ] tat smooting problems do not occur on te very fine or te very coarse grids but do occur on te intermediate grids. Furtermore, it can be observed tat te (β,β 2 )=(, )-preconditioner resembles a Laplacian-type situation, wit excellent smooting factors on all grids. Te preconditioners wit β =,β 2 give smooting factors less tan one on every grid. Te (,)-preconditioner exibits better smooting factors tan te set (β,β 2 )=(,.5), wic represents a limit case for wic smooting factors are still below one. Remark: Simplified coarse grid analysis. Some insigt into te coarse grid correction can be gained from te so-called simplified coarse grid analysis or firstdifferential-approximation analysis [7, 9, 24]. As in [] we apply tis analysis for a one-dimensional (D) Helmoltz operator. Assuming tat transfer operators do not ave any effect on te lowest frequencies, te quantity λ l /λ2l 2 (l small) gives some insigt into te relation between te discrete fine and coarse grid operators. Tis quantity sould be close to zero and is an indication of te suitability of a coarse grid operator in a multigrid metod. For te original D Helmoltz equation and α = (no damping), tis quantity reads [] λ l /λ 2l sin 4 (lπ/2) 2 = sin 2 (lπ/2) cos 2 (lπ/2) (k/2), l =,...,N. 2 It may give rise to a problematic coarse grid correction in te range were sin 2 (lπ/2) cos 2 (lπ/2) (k/2) 2 and l is associated wit a smoot mode. For a D version of te complex Helmoltz operator, tis quantity reads λ l /λ 2l sin 4 (lπ/2) 2 = sin 2 (lπ/2) cos 2 (lπ/2) (k/2) 2 (β β 2 i) = sin4 (lπ/2) ( sin 2 (lπ/2) cos 2 (lπ/2) (k/2) 2 (β + β 2 i) ) ( sin 2 ) (lπ/2) cos 2 (lπ/2) (k/2) 2 2, β +(k/2)2 β2 2 l =,...,N.

14 484 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK Tis expression for te complex Helmoltz operator is close to zero for te (β,β 2 )- sets under consideration: te denominator does not reac zero, and te numerator contains te term sin 4 lπ/2 wic is very small for smoot eigenmodes. Remark: -ellipticity. Wen a Galerkin coarse grid discretization is used, it is difficult to gain insigt into te coarse grid correction, as te coarse grid stencil elements are constructed wit nontrivial formulas. Terefore, we discuss ere for te case (β,β 2 )=(,.5) two coarse grid discretizations. Wit =/64,k = 4, α = in (4), we obtain by direct PDE discretization similar coarse grid stencils as te fine grid stencil wit grid sizes 2 or 4, respectively. In tat case, only te central stencil element contains an imaginary contribution. Wen te Galerkin coarse grid operator is employed, te imaginary part is distributed over all entries. Wit operator-dependent interpolation and full weigting restriction we find (22) A 2 = A 4 = i i i i i i i i i, i i i i i i i i i Te -ellipticity measures are.28 and.8, indicating te suitability of te stencils for pointwise smooting [7, 24]. For te direct PDE discretization, te -ellipticity measures are.3 and.45 for te 2- and 4-discretizations, respectively. Te fact tat tese qualitative measures are not close to zero means tat pointwise smooters can be constructed for tese stencils. From tese complicated coarse grid stencils it is, owever, difficult to judge between te different smooters, relaxation parameters, etc., but te tree-grid Fourier analysis elps to some extent. We obtain very satisfactory multigrid convergence wit simple multigrid components, altoug te coarse grid discretization (22) seems awkward. At least it does not spoil te -independent multigrid convergence. One merely needs to coose te underrelaxation parameter in te smooter wit some care Multigrid for te preconditioner. One multigrid iteration is taken for approximating te inverse of te operator in (8). After some experimentation it was found tat it is sufficient to employ a multigrid iteration wit a convergence factor ρ.6 for te preconditioner. To some extend tis can also be observed qualitatively from spectral pictures obtained by Fourier analysis (again, constant k, Diriclet boundary conditions). Starting wit a regular splitting of M, (23) C φ l+ =(C M )φ l + ψ, or φ l+. =(I C M )φ l + C ψ. Tis splitting is considered to represent a multigrid iteration, wit iteration matrix (I C M ) and C an approximation of M. T 2 in (7) represents te twogrid version of a multigrid iteration matrix. Terefore, we equate T 2 = I C M. 2 Matrix T in (9) is a block matrix related to T 2: 2 T = U T 2U, were U is a unitary matrix wit four consecutive rows defined by te ortogonal eigenvectors related to (6). U transforms te two-grid iteration matrix into te block-diagonal 2 matrix T. Clearly, 2 T = I U C M U, and U C M U = U C U U M U =: C M

15 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 485.6i (a).5i (b).5i.6.5i Fig. 6. Spectral pictures of te preconditioned system wit one two-grid iteration used for preconditioning (β,β 2 )=(, ), k =4, =/64: (a) one ω-jac relaxation and (b) two ω-jac relaxations, ω =.7. (Te eigenvalues wit te exact inversion lie at te circles.) is in block-diagonal form. We ave for te block-diagonal form ( Ã C C C T 2 M M =(I ) M. So, te expression is te approximation of M ) from (7) reads (24) Ã C = Ã(I 2 T ) M. As all te symbols of te operators in te rigt-and side of (24) can be formed easily wit Fourier two-grid analysis, te corresponding eigenvalues can be visualized for various multigrid cycles. Tese spectra can be compared to tose in Figure, were operator M from (8) is inverted exactly. Figure 6, for example, presents te spectrum of te (β,β 2 )=(, )-preconditioned system were a two-grid iteration is used for preconditioning for wavenumber k = 4( = /64). Figure 6(a) sows te spectrum for one ω-jac (ω =.7) smooting iteration for wic ρ 2g.7, wereas Figure 6(b) sows te two-grid spectral picture wit two ω-jac smooting iterations, ν +ν 2 =2, operator-dependent interpolation, full weigting restriction, and Galerkin coarse grid discretization (ρ 2g =.45). Figure 6(b) sows a spectrum tat coincides well wit te spectrum related to te exact inversion in Figure (d), wereas in Figure 6(a) eigenvalues are also outside te circle obtained wit te exact inversion. Figure 7 presents te spectra wit a two-grid iteration for te (β,β 2 )=(,.5)- preconditioner and Galerkin coarsening, wit ω-jac relaxation (ω =.5). Figure 7(a) is for ν = ; Figure 7(b) is for ν = 2. Also for tis approximate inversion of te preconditioner te spectrum obtained in Figure 7(b) compares well wit te exact inversion in Figure (e), indicating tat one multigrid iteration wit two ω-jac smooting steps may be sufficient for approximating M. Indeed, a numerical comparison between inverting te preconditioner by several multigrid iterations versus by only one multigrid iteration did not lead to substantially different numbers of Krylov subspace iterations for solving te Helmoltz problem. 5. Applications. In tis section te overall solution metod, preconditioned Bi- CGSTAB for te indefinite eterogeneous Helmoltz problems () wit te complex Helmoltz (β,β 2 )-preconditioner, is evaluated. One multigrid F(,)-cycle is used for approximately inverting te preconditioner equation wit te complex Helmoltz operator. Tree problems of increasing difficulty are discussed. 5.. Constant wavenumber. For constant wavenumbers k te Bi-CGSTAB convergence for te Helmoltz equation wit te tree preconditioners is presented.

16 486 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK.6 i (a).6i (b).4i.2.5i Fig. 7. Spectral pictures of preconditioned system wit one two-grid iteration used for preconditioning (β,β 2 )=(,.5), k =4, =/64: (a) one ω-jac relaxation and (b) two ω-jac relaxations, ω =.5. (Te eigenvalues wit exact inversion lie at te circles.) Fig. 8. Numerical solution at k =5for te model problem wit k constant. We consider a square domain Ω = (, ) 2. A point source is located at te center of te domain. Te solution satisfies te second-order conditions (2). In tese experiments te finest grid size for eac wavenumber is as sown in Table. Te numerical solution corresponding to k = 5 is presented in Figure 8. Unpysical reflections at te boundaries are not present due to te boundary treatment. A zero initial guess as been used during te computations. Te Bi-CGSTAB iteration is terminated as soon as te initial residual is reduced by 7 orders of magnitude. Note tat eac Bi-CGSTAB iteration involves two preconditioning steps. For all tree preconditioners, (β,β 2 )=(, ), (, ), and (,.5), te metod cosen to approximately invert te preconditioner consists of one multigrid F(,)- cycle wit ω-jac, operator-dependent interpolation plus full weigting as te transfer operators, and a Galerkin coarse grid discretization. Te only difference is te value of te underrelaxation parameter in ω-jac, wic is ω =.8 for (β,β 2 )=(, ), ω =.7for (β,β 2 )=(, ), and ω =.5for (β,β 2 )=(,.5). Te results for different values of k and (β,β 2 )=(, ) are presented in te upper part of Table 6. In te middle part of Table 6, te Bi-CGSTAB convergence wit te (β,β 2 )=(, )- preconditioner is presented. In te lower lines of Table 6 te (β,β 2 ) = (,.5)- preconditioner is employed. Next to te results for te Helmoltz equation witout

17 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 487 Table 6 Number of preconditioned Bi-CGSTAB iterations and CPU time in seconds (in parenteses) to reduce te initial residual by 7 orders. Damping parameter α is varied in te Helmoltz problem. k (β,β 2 ) α from () α = 57 (.44) 73 (.92) 2 (4.3) 26 (7.7) 88 (28.5) (, ) 2.5% damping 48 (.38) 6 (.77) 84 (3.3) 93 (5.6) 2 (8.5) 5% damping 45 (.35) 55 (.7) 69 (2.7) 75 (4.7) 97 (4.9) α = 36 (.3) 39 (.5) 54 (2.2) 74 (4.5) 9 (3.9) (,) 2.5% damping 33 (.27) 37 (.48) 44 (.8) 5 (3.2) 6 (9.6) 5% damping 28 (.24) 3 (.39) 36 (.5) 4 (2.6) 49 (7.5) α = 26 (.2) 3 (.4) 44 (.8) 52 (3.3) 73 (.8) (,.5) 2.5% damping 24 (.2) 26 (.35) 33 (.4) 39 (2.5) 47 (7.3) 5% damping 2 (.8) 23 (.32) 28 (.2) 32 (2.) 37 (5.8) Table 7 Hig wavenumbers, number of Bi-CGSTAB iterations, and CPU time in seconds (in parenteses) needed to reduce te initial residual by 7 orders wit and witout damping in te Helmoltz problem. k (β,β 2 ) α in () α = 4 (3.8) 29 (55) 352 (89) (,) 2.5 % damping 74 (2.2) 25 (227) 45 (372) 5 % damping 56 (5.5) 95 (74) 8 (25) α = 92 (25.4) 25 (425) 298 (726) (,.5) 2.5 % damping 57 (5.2) 9 (64) 2 (252) 5 % damping 44 (.9) 64 (5) 66 (65) any damping (α = ), we also sow te convergence wit 2.5% (α =.25) and 5% (α =.5) damping. Te number of Bi-CGSTAB iterations are presented as well as te CPU time on a Pentium 4 PC wit 2.4 Gz and 2 Gb RAM. From te results in Table 6 we conclude tat te preferred metods among te coices are te preconditioners wit β =. Tis was already expected from te spectra in Figure. Fastest convergence is obtained for (β,β 2 )=(,.5). Te components of te multigrid iteration for tis preconditioner ave been validated wit te elp of te Fourier analysis. Table 6 sows tat te Bi-CGSTAB convergence wit some damping in te Helmoltz problem is considerably faster tan for α =. Tis was already expected from te spectra in Figure 3. Furtermore, te number of iterations in te case of damping grows only slowly for increasing wavenumbers, especially for te (β,β 2 )=(,.5)-preconditioner. Te difference between te two preconditioners wit β = is more pronounced if we compute iger wavenumbers. Te Bi-CGSTAB convergence and CPU time for te iger wavenumbers, witout and wit damping in te Helmoltz problem, are presented in Table 7. Also for te iger wavenumbers damping in te Helmoltz problem by means of α improves te convergence significantly. Very satisfactory convergence is found for ig wavenumbers on fine grids Te wedge model. A problem of intermediate difficulty is te wedge model. It is used to evaluate te preconditioner s beavior for a simple eterogeneous medium. Te problem is adopted from [9]. Te domain is defined to be a rectangle of dimension 6 m 2. Te second-order boundary conditions (2) are set, and a point source is located at te center of te upper surface (wic is assigned

18 488 Y. A. ERLANGGA, C. W. OOSTERLEE, AND C. VUIK 2 m/s 4 dept(m) 5 m/s 8 x axis (m) 3 m/s 6 (a) (b) (c) Fig. 9. Wedge problem: (a) Problem geometry wit velocity profile indicated, (b) real part of numerical solution at 3 Hz, and (c) real part of numerical solution at 5 Hz. to be y = ) wit frequency, f = kc/(2π)l, varying from to 6 Hz (were c is te speed of sound). Te corresponding values of te local dimensionless wavenumbers k vary between 2 (smallest for Hz) and 24 (biggest for 6 Hz). For te problem at Hz approximately 8 points per wavelengt are used. Figure 9(a) presents te domain, te wedge, and te variation of c in te medium. Te variation of c is due to te different local properties of te medium. Te real part of te numerical solution for te wedge problem at 3 Hz and 5 Hz is plotted in Figures 9(b) and 9(c). In te preconditioner (8) wavenumber k(x, y) is cosen as in te original problem. Also te boundary conditions in te preconditioner are as for te original problem. Te number of Bi-CGSTAB iterations wit one multigrid iteration for te preconditioner wit (β,β 2 )=(, ), (, ), and (,.5) are displayed in Table 8 for frequencies ranging from to 6 Hz on corresponding grid sizes. Results wit and witout damping in te Helmoltz problem are presented. Te only difference in te multigrid metods for te preconditioner is te value of te relaxation parameter: for (β,β 2 )=(, ) ω =.8, for (β,β 2 )=(, ) ω =.7, and for (β,β 2 )=(,.5) ω =.5. A zero initial guess as been used as a starting approximation. Te convergence results for (β,β 2 ) = (,.5) are best, also witout any damping in te original problem. Te convergence wit te (,.5)-preconditioner is about.5 times faster tan wit te (,)-preconditioner and about 3 times faster tan wit te (,)- preconditioner. Te Bi-CGSTAB convergence for te wedge problem for α = and different frequencies are also visualized for (β,β 2 )=(,.5) in Figure Te Marmousi problem. Tis example is a part of te full Marmousi problem wic mimics subsurface geology [4]; see also [9]. Te domain is rectangular wit a dimension of 6 6 m 2. A point source is placed at te center of te upper surface. Te values for te speed of sound c are irregularly structured trougout te

19 MULTIGRID BASED PRECONDITIONER FOR HELMHOLTZ PROBLEMS 489 Table 8 Bi-CGSTAB convergence for te wedge problem wit and witout damping and te tree multigrid based (β,β 2 )-preconditioners compared. Te number of Bi-CGSTAB iterations and te CPU time in seconds (in parenteses) are sown. (β,β 2 ) f (Hz) Grid Damping (,) (,) (,.5).% 52 (.2) 3 (.67) 9 (.42) % 48 (.) 27 (.62) 7 (.39) 5.% 42 (.9) 25 (.57) 6 (.38).% 9 (8.8) 45 (4.5) 27 (2.8) % 75 (7.2) 39 (4.) 23 (2.4) 5.% 65 (6.3) 35 (3.5) 2 (2.).% 28 (3.6) 64 (5.8) 37 (9.4) % 94 (22.8) 49 (2.3) 29 (7.5) 5.% 86 (2.) 42 (.7) 25 (6.6).% 6 (66.) 8 (33.5) 49 (2.8) % 6 (48.) 6 (25.4) 35 (5.2) 5.% 9 (37.9) 46 (9.8) 28 (2.4).% 25 (34.5) 98 (65.5) 58 (38.7) % 35 (89.) 67 (45.5) 37 (24.8) 5.% 99 (66.5) 54 (37.) 32 (22.).% 232 (247.3) 8 (27.6) 66 (7.9) % 47 (59.) 74 (8.) 42 (47.) 5.% (9.6) 58 (64.5) 32 (36.7) 2 f = Hz f = 2 Hz f = 3 Hz f = 4 Hz f = 5 Hz f = 6 Hz 3 Relative residual Iteration Fig.. Bi-CGSTAB convergence plot for (β,β 2 )=(,.5) for te wedge problem at different frequencies, α =. domain; see Figure (a). Te minimum number of points per wavelengt equals 7. Te frequency is varied between and 3 Hz. Preconditioning consists of one multigrid iteration for te complex Helmoltz equation wit te multigrid components prescribed. Te underrelaxation parameter in ω-jac is varied as usual depending on (β,β 2 ). In te preconditioner again te wavenumbers k(x, y) are as in te original problem. Also te boundary conditions are as in te original problem. Table 9 presents te number of Bi-CGSTAB iterations to solve te indefinite Helmoltz Marmousi problem wit te CPU times required sown