A decade of fast and robust Helmholtz solvers

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A decade of fast and robust Helmholtz solvers Werkgemeenschap Scientific Computing Spring meeting Kees Vuik May 11th, 212 1 Delft University of Technology

Contents Introduction Preconditioning (22-28) Numerical experiments Second-level preconditioning (28-212) Fourier Analysis of two-level method Numerical experiments Conclusions May 11th, 212 2

The Helmholtz equation The Helmholtz equation without damping u(x,y) k 2 (x,y)u(x,y) = g(x,y) in Ω u(x, y) is the pressure field, k(x, y) is the wave number, g(x, y) is the point source function and Ω is the domain. Absorbing boundary conditions are used on Γ. u n ιu = n is the unit normal vector pointing outwards on the boundary. Perfectly Matched Layer (PML) and Absorbing Boundary Layer (ABL) May 11th, 212 3

Application: geophysical survey hard Marmousi Model May 11th, 212 4

Problem description Second order Finite Difference stencil: 1 1 4 k 2 h 2 1 1 Linear system Au = g: properties Sparse & complex valued Symmetric & Indefinite for large k For high resolution a very fine grid is required: 3 6 gridpoints per wavelength (or 5 1 k) A is extremely large! Is traditionally solved by a Krylov subspace method, which exploits the sparsity. May 11th, 212 5

Survey of solution methods Special Krylov methods COCG van der Vorst and Melissen, 199 QMR Freund and Nachtigal, 1991 General purpose Krylov methods CGNR Paige and Saunders, 1975 Short recurrences Bi-CGSTAB van der Vorst, 1992 IDR(s) Van Gijzen and Sonneveld, 28 Minimal residual GMRES Saad and Schultz, 1986 GCR Eisenstat, Elman and Schultz, 1983 GMRESR van der Vorst and Vuik, 1994 May 11th, 212 6

Preconditioning Equivalent linear system M1 1 1 AM2 x = b, where M = M 1 M 2 is the preconditioning matrix and Requirements for a preconditioner x = M 2 x, b = M1 b. better spectral properties of M 1 A cheap to perform M 1 r. Spectrum of A is {µ i k 2 }, with k is constant and µ i are the eigenvalues of the Laplace operator. Note µ 1 k 2 may be negative. May 11th, 212 7

Preconditioning ILU Meijerink and van der Vorst, 1977 ILU(tol) Saad, 23 SPAI Grote and Huckle, 1997 Multigrid Lahaye, 21 Elman, Ernst and O Leary, 21 AILU Gander and Nataf, 21 analytic parabolic factorization ILU-SV Plessix and Mulder, 23 separation of variables May 11th, 212 8

Preconditioning Laplace operator Bayliss and Turkel, 1983 Definite Helmholtz Laird, 2 Shifted Laplace Y.A. Erlangga, C. Vuik and C.W.Oosterlee, 23 Shifted Laplace preconditioner M (β 1 iβ 2 )k 2, β 1, β 2 R, and β 1. Condition β 1 is used to ensure that M is a (semi) definite operator. β 1, β 2 = : Bayliss and Turkel β 1 = 1, β 2 = : Laird β 1 = 1, β 2 =.5 : Y.A. Erlangga, C. Vuik and C.W.Oosterlee May 11th, 212 9

Application: geophysical survey hard Marmousi Model 2 18 old method new method iterations 16 14 12 1 8 6 4 2 May 11th, 212 1 5 1 15 2 25 3 Frequency f

Numerical experiments Example with constant k in Ω Iterative solver: Bi-CGSTAB Preconditioner: Shifted-Laplace operator, discretized using the same method as the Helmholtz operator. May 11th, 212 11 k ILU(.1) M M 1 M i 5 9 13 13 13 1 25 29 28 22 15 47 114 45 26 2 82 354 85 34 3 139 > 1 15 52

Eigenvalues for Complex preco k = 1.2 2 smallest eigenvalues 75 grid points 15 grid points.2.2.2.4.4.6.6.8.8 1 1.2.5 1 1.2.5 1 May 11th, 212 12

Eigenvalues for Complex preco k = 1.2 5 smallest eigenvalues 75 grid points 15 grid points.2.2.2.4.4.6.6.8.8 1 1.2.5 1 1.2.5 1 May 11th, 212 13

Eigenvalues for Complex preco k = 1.2 spectrum is independent of the grid size 75 grid points 15 grid points.2.2.2.4.4.6.6.8.8 1 1.2.2.2.4.6.8 1 1.2 1.2.2.2.4.6.8 1 1.2 May 11th, 212 14

Inner iteration Possible solvers for solution of Mz = r: ILU approximation of M inner iteration with ILU as preconditioner Multigrid Multigrid components - geometric multigrid - Gauss-Seidel with red-black ordering - matrix dependent interpolation, full weighting restriction - Galerkin coarse grid approximation May 11th, 212 15

Inner iteration geometric multigrid ω-jac smoother bilinear interpolation, restriction operator full weighting Galerkin coarse grid approximation F(1,1)-cycle M 1 is approximated by one multigrid iteration May 11th, 212 16

Numerical results for a wedge problem k 2 1 2 4 5 1 grid 32 2 64 2 128 2 192 2 384 2 No-Prec 21(.56) 128(12) 517(316) ILU(A,) 55(.36) 348(9) 1484(131) 2344(498) ILU(A,1) 26(.14) 126(4) 577(62) 894(27) ILU(M,) 57(.29) 213(8) 1289(122) 272(451) ILU(M,1) 28(.28) 116(4) 443(48) 763(191) 221(1875) MG(V(1,1)) 13(.21) 38(3) 94(28) 115(82) 252(85) May 11th, 212 17

Spectrum of shifted Laplacian preco References: Manteuffel, Parter, 199; Yserentant, 1988 Since L and M are SPD we have the following eigenpairs Lv j = λ j Mv j, where, λ j R + The eigenvalues σ j of the preconditioned matrix satisfy Theorem 1 Provided that z 2 λ j, the relation (L z 1 M)v j = σ j (L z 2 M)v j. May 11th, 212 18 σ j = λ j z 1 λ j z 2 holds.

Spectrum of shifted Laplacian preco Theorem 2 If β 2 =, the eigenvalues σ r + iσ i are located on the straight line in the complex plane given by β 1 σ r (α 1 α 2 )σ i = β 1. Theorem 3 If β 2, the eigenvalues σ r + iσ i are on the circle in the complex plane with center c and radius R: c = z 1 z 2 z 2 z 2, R = z 2 z 1 z 2 z 2. Note that if β 1 β 2 > the origin is not enclosed in the circle. May 11th, 212 19

Spectrum with inner iteration.6i (a).5i (b).5i 1 1.6.5i 1 May 11th, 212 2

Sigsbee model 45 1 2 4 3 35 depth (m) 4 5 3 6 25 7 8 2 9.5 1 1.5 2 x direction (m) x 1 4 15 May 11th, 212 21

Sigsbee model dx = dz = 22.86 m; D = 24369 9144 m 2 ; grid points 167 41. Bi-CGSTAB 5 Hz 1 Hz CPU (sec) Iter CPU (sec) Iter NO preco 3128 16549 1816 9673 With preco 86 48 92 58 May 11th, 212 22

Second Level Preconditioning ILU and variants From Laplace to complex Shifted Laplace Preconditioner (25) Shifted Laplace Preconditioner (SLP) M := u (β 1 ιβ 2 )k 2 u Results shows: (β 1, β 2 ) = (1,.5) is the shift of choice What is the effect of SLP? May 11th, 212 23

Shifted Laplace Preconditioner Introduces damping, Multi-grid approximation is possible The modulus of all eigenvalues of the preconditioned operator is bounded by 1 Small eigenvalues move to zero, as k increases. Spectrum of M 1 (1,.5)A for k = 3 and k = 12.5.4.3.2.1.1.2.3.4.5.4.3.2.1.1.2.3.4.5.2.4.6.8 1.5.2.4.6.8 1 May 11th, 212 24

Some Results at a Glance Number of GMRES iterations. Shifts in the preconditioner are (1,.5) Grid k = 1 k = 2 k = 3 k = 4 k = 5 k = 1 n = 32 1 17 28 44 7 14 n = 64 1 17 28 36 45 163 n = 96 1 17 27 35 43 97 n = 128 1 17 27 35 43 85 n = 16 1 17 27 35 43 82 n = 32 1 17 27 35 42 8 Number of iterations depends linearly on k. May 11th, 212 25

Deflation improves the convergence Number of GMRES iterations. Shifts in the preconditioner are (1,.5) Grid k = 1 k = 2 k = 3 k = 4 k = 5 k = 1 n = 32 5/1 8/17 14/28 26/44 42/7 13/14 n = 64 4/1 6/17 8/28 12/36 18/45 173/163 n = 96 3/1 5/17 7/27 9/35 12/43 36/97 n = 128 3/1 4/17 6/27 7/35 9/43 36/85 n = 16 3/1 4/17 5/27 6/35 8/43 25/82 n = 32 3/1 4/17 4/27 5/35 5/42 1/8 Erlangga and Nabben, 28, seems to be independent of k. with / without deflation. May 11th, 212 26

Erlangga and Nabben algorithm Setting up: For k = 1, set A (1) = A, M (1) = M, construct Z (1,2), λ (k) max = 1, k. From above,  (1) = A (1) M (1) 1 and P (1) λ max = I ˆQ (1)  (1) + ˆQ (1) with ˆQ (1) = Z (1,2)  (2) 1 Z (1,2)T For k = 2,...,m, construct Z (k 1,k) and compute A (k) = Z (k 1,k)T A (k 1) Z (k 1,k), M (k) = Z (k 1,k)T M (k 1) Z (k 1,k) and P (k) λ max = I Z (k,k+1)  (k+1) 1Z ( ) (k,k+1)t  (k) I with  (k) = A (k) M (k) 1 May 11th, 212 27

Inside Iterations Solve: A (2) M (2) 1 v (2) R = (v R) (2) with Krylov v (2) A = A(2) v (2) ; s (2) = M (2) 1 v (2) A ; t (2) = s (2) λ (2) maxv (2) ; Restriction: (v R ) (3) = Z (2,3)T t (2) If k = m = A(m) 1 (v R )(m) else v (m) R Solve: A (3) M (3) 1 v (3) R = (v R) (3) with Krylov... Interpolation: v (2) I = Z (2,3) v (3) R q (2) = v (2) v (2) I w (2) = M (2) 1 q (2) p (2) = A (2) w (2) May 11th, 212 28

Deflation: or two-grid method For any deflation subspace matrix Z R n r, with deflation vectors Z = [z 1,..., z r ], rankz = r P = I AQ, with Q = ZE 1 Z T and E = Z T AZ Solve PAu = Pg preconditioned by M 1 or M 1 PA = M 1 Pg For e.g. say, spec(a) = {λ 1, λ 2, λ 3,..., λ n } and if Z is the matrix with columns the r eigenvectors then spec(pa) = {,...,, λ r+1,...λ n } May 11th, 212 29

Deflation We use multi-grid inter-grid transfer operator (Prolongation) as deflation matrix. Setting Z = I 2h h and ZT = I h 2h then P = I AQ, with Q = I 2h h E 1 I h 2h and E = I h 2hAI 2h h where P can be interpreted as a coarse grid correction and Q as the coarse grid operator May 11th, 212 3

Fourier Analysis Dirichlet boundary conditions for analysis. With above deflation, spec(pm 1 A) = f(β 1, β 2, k, h) is a complex valued function. Setting kh =.625, Spectrum of PM 1 A with shifts (1,.5) is clustered around 1 with a few outliers. Spectrum remains almost the same, when the imaginary shift is varied from.5 to 1. May 11th, 212 31

Fourier Analysis Analysis shows spectrum clustered around 1 with few outliers. k = 3 k = 12.5.5.1.15.2.5.25.3 1.35.4 1.5.45.5.5.5 1 1.5 2 2.5 2 2 2 4 6 8 1 May 11th, 212 32

Fourier Analysis Analysis shows that an increase in the imaginary shift does not change the spectrum. (β 1, β 2 ) = (1,.5) (β 1, β 2 ) = (1, 1).5.4.3.2.1.1.2.3.4.5 1.5.5 1 1.5 2 2.5.5.4.3.2.1.1.2.3.4.5 1.5.5 1 1.5 2 2.5 May 11th, 212 33

Numerical results Sommerfeld boundary conditions are used for test problem. What is the effect of an increase in the imaginary shift in SLP? Constant wavenumber problem Wedge problem 8 7 With deflation Without deflation 15 With deflation Without deflation 6 No. of iterations 5 4 3 2 No. of iteratrions 1 5 1.2.4.6.8 1 β 2.2.4.6.8 1 β 2 May 11th, 212 34

Numerical results Number of GMRES iterations with/without deflation. Shifts in the preconditioner are (1,.5) Grid k = 1 k = 2 k = 3 k = 4 k = 5 k = 1 n = 32 5/1 8/17 14/28 26/44 42/7 13/14 n = 64 4/1 6/17 8/28 12/36 18/45 173/163 n = 96 3/1 5/17 7/27 9/35 12/43 36/97 n = 128 3/1 4/17 6/27 7/35 9/43 36/85 n = 16 3/1 4/17 5/27 6/35 8/43 25/82 n = 32 3/1 4/17 4/27 5/35 5/42 1/8 May 11th, 212 35

Numerical results Number of GMRES iterations with/without deflation to solve a Wedge problem. Shifts in the preconditioner are (1,.5) Grid freq = 1 freq = 2 freq = 3 freq = 4 freq = 5 74 124 7/33 2/6 79/95 267/156 49/292 148 248 5/33 9/57 17/83 42/112 15/144 232 386 5/33 7/57 1/81 25/18 18/129 3 5 4/33 6/57 8/81 12/15 18/129 374 624 4/33 5/57 7/8 9/14 13/128 May 11th, 212 36

Numerical results 25 Def. Prec. GMRES Prec. GMRES 2 No. of iterations 15 1 5 1 2 3 4 5 6 7 8 Wave number k Number of GMRES iterations for the 1D Helmholtz equation 1 k 8 May 11th, 212 37

Numerical results 25 Def. Prec. GMRES 2 No. of iterations 15 1 5.5 1 1.5 2 Wave number k x 1 4 Number of GMRES iterations for the 1D Helmholtz equation 1 k 2 May 11th, 212 38

Numerical results Number of GMRES outer-iterations in multilevel algorithm. (β 1, β 2 ) = (1,.5) kh =.3125 or 2 gp/wl and MG Vcycle(1,1) for SLP Grid k = 1 k = 2 k = 4 k = 8 k = 16 MLMGV(4,2,1) 9 11 16 27 1+ MLMGV(6,2,1) 9 1 14 21 47 MLMGV(8,2,1) 9 1 13 2 38 MLMGV(8,3,2) 9 1 13 19 37 May 11th, 212 39

Results Petsc solve-time in Seconds; a Two-level solver. Solver 2 4 8 12 16 2 SLP.1(23).24(54) 2.62(113) 11.6(168) 33.59(222) 83.67(274) Def/SLP.3(1).14(14).82(23) 2.92(37) 8.98(61) 23.13(87) SLP : GCR preconditioned with SLP M(1,1). Def/SLP: Deflated and preconditioned GCR. Grid resolution is such that there are 1 grid points per wavelength. May 11th, 212 4

Results Petsc time and Krylov outer-iterations in multilevel algorithm. (β 1, β 2 ) = (1, 1) kh =.625 or 1 gp/wl MLMGV is Multilevel with VCycle(1,1), MLMGF is Multilevel with FCycle(1,1) - k = 1 k = 2 k = 4 k = 8 k = 12 MLMGV(8,2,1) 16(1.3) 27(2.8) 58(7.3) 116(38.7) 177(76.8) MLMGF(8,2,1) 1(.9) 11(4.3) 16(12.9) 28(39.2) 41(6.5) May 11th, 212 41

Fourier Analysis Spectrum of A, M 1 A and PM 1 A (from left to right) in bar-graph. 9 K=3,n=48 2.5 K=3,n=48 2.5 K=3,n=48 8 7 2 2 6 5 1.5 1.5 λ 4 λ 1 λ 1 3 2.5.5 1 5 1 15 2 25 Number of eigenvalues λ 5 1 15 2 25 Number of eigenvalues λ 5 1 15 2 25 Number of eigenvalues λ 14 x 14 K=12,n=256 2.5 K = 12, n = 256 9 K = 12, n = 256 12 8 2 7 1 6 8 1.5 5 λ 6 λ 1 λ 4 4 3 2.5 2 1 1 2 3 4 5 6 7 8 9 1 Number of eigenvalues λ 1 2 3 4 5 6 7 8 9 Number of eigenvalues λ 1 2 3 4 5 6 7 8 9 1 Number of eigenvalues λ May 11th, 212 42

Conclusions The proposed preconditioner (shifted Laplacian + multi-grid) is independent of the grid size and linearly dependent of k. With physical damping the proposed preconditioner is also independent of k. Without deflation, when imaginary shift is increased in SLP, spectrum remains bounded above 1, but lower part moves to zero. Flexibility to increase imaginary shift, when deflation is combined with SLP. Further research Multilevel scheme, applying similarly for coarse problem in deflation. Questions: gain in CPU time? why not scalable?... May 11th, 212 43

References Y.A. Erlangga and R. Nabben. On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian. ETNA, 28. M.B. van Gijzen, Y.A. Erlangga and C. Vuik. Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J.of Sc. Comp. 27. J.M. Tang. Two level preconditioned Conjugate Gradient methods with applications to bubbly flow problems. PhD Thesis, DIAM TU Delft 28. A.H. Sheikh, D. Lahaye and C. Vuik. On the convergence of shifted Laplace preconditioner combined with multi-grid deflation. submitted H. Knibbe and C.W. Oosterlee and C. Vuik GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multi-grid method. Journal of Computational and Applied Mathematics, 236, pp. 281-293, 211 http://ta.twi.tudelft.nl/nw/users/vuik/pub_it_helmholtz.html May 11th, 212 44

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