Spectral analysis of complex shifted-laplace preconditioners for the Helmholtz equation

Spectral analysis of complex shifted-laplace preconditioners for the Helmholtz equation C. Vuik, Y.A. Erlangga, M.B. van Gijzen, and C.W. Oosterlee Delft Institute of Applied Mathematics c.vuik@tudelft.nl http://ta.twi.tudelft.nl/users/vuik/ Seminar für Analysis und Numerik Mathematisches Intitut, Universität Basel December 7, 2007, Basel, Switzerland C. Vuik, December 7, 2007 1 p.1/28

Contents 1. Introduction 2. Spectrum of shifted Laplacian preconditioners 3. Shift with an SPD real part 4. General shift 5. Numerical experiments 6. Conclusions Wim Mulder, René Edouard Plessix, Paul Urbach, Alex Kononov and Dwi Riyanti Financially supported by the Dutch Ministry of Economic Affairs: project BTS01044 C. Vuik, December 7, 2007 2 p.2/28

1. Introduction The Helmholtz problem is defined as follows where: xx u yy u z 1 k 2 (x, y)u = f, in Ω, Boundary conditions on Γ = Ω, z 1 = α 1 + iβ 1 and k(x, y) is the wavenumber for solid boundaries: Dirichlet/Neumann for fictitious boundaries: Sommerfeld du dn iku = 0 Perfectly Matched Layer (PML) Absorbing Boundary Layer (ABL) C. Vuik, December 7, 2007 3 p.3/28

Discretization In general: Finite Difference/Finite Element Methods. Particular to the present case: 5-point Finite Difference stencil, O(h 2 ). Linear system Ax = b, A C N N, b, x C N, C. Vuik, December 7, 2007 4 p.4/28

Discretization In general: Finite Difference/Finite Element Methods. Particular to the present case: 5-point Finite Difference stencil, O(h 2 ). Linear system Ax = b, A C N N, b, x C N, A is a sparse, highly indefinite matrix for practical values of k. Special property A = A T. For high resolution a very fine grid is required: 30 60 grid-points per wavelength (or 5 10 k) A is extremely large! C. Vuik, December 7, 2007 4 p.4/28

Characteristic properties of the problem A C N N is sparse wavenumber k and grid size N are very large wavenumber k varies discontinuously real parts of the eigenvalues of A are positive and negative C. Vuik, December 7, 2007 5 p.5/28

Application: geophysical survey Marmousi model (hard) C. Vuik, December 7, 2007 6 p.6/28

Application: geophysical survey 2000 1800 old method new method iterations 1600 1400 1200 1000 800 600 400 200 0 0 5 10 15 20 25 30 Frequency f Marmousi model (hard) C. Vuik, December 7, 2007 6 p.6/28

2. Spectrum of shifted Laplacian preconditioners Operator based preconditioner P is based on a discrete version of xx u yy u (α 2 + iβ 2 )k 2 (x, y)u = f, in Ω. appropriate boundary conditions Matrix P 1 is approximated by an inner iteration process. α 2 = 0 β 2 = 0 Laplacian Bayliss and Turkel, 1983 α 2 = 1 β 2 = 0 Definite Helmholtz Laird, 2000 α 2 = 0 β 2 = 1 Complex Erlangga, Vuik and α 2 = 1 β 2 = 0.5 Optimal Oosterlee, 2004, 2006 C. Vuik, December 7, 2007 7 p.7/28

Spectrum of shifted Laplacian preconditioners After discretization we obtain the (un)damped Helmholtz operator L z 1 M, where L and M are SPD matrices and z 1 = α 1 + iβ 1. The preconditioner is then given by L z 2 M, where z 2 = α 2 + iβ 2 is chosen such that systems with the preconditioner are easy to solve, the outer Krylov process is accelerated significantly. C. Vuik, December 7, 2007 8 p.8/28

Spectrum of shifted Laplacian preconditioners References: Manteuffel, Parter, 1990; 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. σ j = λ j z 1 λ j z 2 holds. C. Vuik, December 7, 2007 9 p.9/28

Spectrum of shifted Laplacian preconditioners Theorem 2 If β 2 = 0, 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. C. Vuik, December 7, 2007 10 p.10/28

Spectrum of shifted Laplacian preconditioners Theorem 2 If β 2 = 0, 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 0, 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 > 0 the origin is not enclosed in the circle. C. Vuik, December 7, 2007 10 p.10/28

Spectrum of shifted Laplacian preconditioners Using Sommerfeld boundary conditions, it impossible to write the matrix as L z 1 M where, L and M are SPD. Generalized matrix L + ic z 1 M, where L, M, and C are SPD. Matrix C contains Sommerfeld boundary conditions (or other conditions: PML, ABL). Use as preconditioner L + ic z 2 M. C. Vuik, December 7, 2007 11 p.11/28

Spectrum of shifted Laplacian preconditioners Suppose then (L + ic)v = λ C Mv (L + ic z 1 M)v = σ C (L + ic z 2 M)v. Theorem 4 Let β 2 0 then the eigenvalues σ C are in or on the circle with center c = z 1 z 2 z 2 z 2 and radius R = z 2 z 1 z 2 z 2. C. Vuik, December 7, 2007 12 p.12/28

3. Shift with an SPD real part Motivation: the preconditioned system is easy to solve. 1 z1 = 1 0.5i, z2 = 1 1.5 z1 = 1, z2 = i 0.8 1 Im 0.6 0.4 0.2 Im 0.5 0 0 1 0.5 0 0.5 1 Re 0.5 0.5 0 0.5 1 1.5 Re 1.5 z1 = 1 0.5i, z2 = 1 0.5i i 2 z1 = 1 0.5i, z2 = 1 0.5i i 1 1 Im 0.5 Im 0 0 1 0.5 0.5 0 0.5 1 1.5 Re 2 2 1 0 1 2 Re C. Vuik, December 7, 2007 13 p.13/28

Optimization of the shift Which choices for z2 are optimal? 0 0.5 1 1.5 Damping: 0.1 2 1 0 2 alpha2 (scaled) 0 0.95 0.5 0.85 0.9 1 0.95 1.5 Damping: 0.5 0.85 0.9 2 1 0 2 alpha2 (scaled) 0 0.5 1 1.5 Damping: 1 1 1 0.95 0.7 0.9 0.65 0.6 0.75 0.8 0.85 0.2 0.25 0.3 0.35 0.4 0.7 0.45 0.5 0.6 0.55 0.65 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 2 1 0 2 alpha2 (scaled) C. Vuik, December 7, 2007 14 p.14/28 1 0.95 0.8 0.7 0.75 0.9 0.85 0.65 0.45 0.5 0.55 0.6 0.7 0.75 0.8 0.4 0.65 beta2 (scaled) beta2 (scaled) beta2 (scaled) 0.45 0.5 0.55 0.6 0.7

Optimal choices for z 2? 70 60 beta1 = 0 beta1 = 0.1 beta1 = 0.5 beta1 = 1 Frequency: 8 50 Number of iterations 40 30 20 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 imaginary shift (negative, scaled) C. Vuik, December 7, 2007 15 p.15/28

Optimal choices for z 2? Damping Optimal β 2 "optimal" iterations Minimum iterations β 1 = 0-1 56 54 β 1 = 0.1-1.005 42 41 β 1 = 0.5-1.118 20 20 β 1 = 1-1.4142 13 13 C. Vuik, December 7, 2007 16 p.16/28

Optimal choices for z 2? Damping Optimal β 2 "optimal" iterations Minimum iterations β 1 = 0-1 56 54 β 1 = 0.1-1.005 42 41 β 1 = 0.5-1.118 20 20 β 1 = 1-1.4142 13 13 Number of iterations h 100/2 100/4 100/8 100/16 100/32 f 2 4 8 16 32 β 1 = 0 14 25 56 116 215 β 1 = 0.1 13 22 42 63 80 β 1 = 0.5 11 16 20 23 23 β 1 = 1 9 11 13 13 23 C. Vuik, December 7, 2007 16 p.16/28

Superlinear convergence of GMRES 10 0 f = 8. beta1 = 0, no radiation condition 10 1 10 2 Scaled residual norm 10 3 10 4 10 5 10 6 10 7 0 20 40 60 80 100 120 Number of iterations C. Vuik, December 7, 2007 17 p.17/28

Superlinear convergence of GMRES 10 0 f = 8. beta1 = 0.05, no radiation condition 10 1 10 2 Scaled residual norm 10 3 10 4 10 5 10 6 10 7 0 20 40 60 80 100 120 Number of iterations C. Vuik, December 7, 2007 17 p.17/28

Superlinear convergence of GMRES 10 0 f = 8. beta1 = 0.1, no radiation condition 10 1 10 2 Scaled residual norm 10 3 10 4 10 5 10 6 10 7 0 20 40 60 80 100 120 Number of iterations C. Vuik, December 7, 2007 17 p.17/28

4. General Shifted Laplacian preconditioner No restriction on α 2 For the outer loop α 2 = 1 and β 2 = 0 is optimal. Convergence in 1 iteration. But, the inner loop does not converge with multi-grid (original problem). However, it appears that multi-grid works well for α 2 = 1 and β 2 = 1 and the convergence of the outer loop is much faster than for the choice α 2 = 0 and β 2 = 1. C. Vuik, December 7, 2007 18 p.18/28

Eigenvalues for Complex preconditioner k = 100 and α 2 = 1 Spectrum is independent of the grid size β 2 = 1 β 2 = 0.5 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C. Vuik, December 7, 2007 19 p.19/28

Eigenvalues for β 1 = 0.025 (damping) and α 2 = 1, β 2 = 0.5 Spectrum is independent of the grid size and the choice of k. 0.5 k = 100 k = 400 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C. Vuik, December 7, 2007 20 p.20/28

5. Numerical experiments Multi-grid components geometric multi-grid ω-jac smoother matrix dependent interpolation, restriction operator full weighting Galerkin coarse grid approximation F(1,1)-cycle P 1 is approximated by one multi-grid iteration in 3D semi-coarsening is used C. Vuik, December 7, 2007 21 p.21/28

Spectrum with inner iteration 0.6i (a) 0.5i (b) 0 0.5i 0 1 1.6 0.5i 0 1 C. Vuik, December 7, 2007 22 p.22/28

Sigsbee model 0 4500 1000 2000 4000 3000 3500 depth (m) 4000 5000 3000 6000 2500 7000 8000 2000 9000 0 0.5 1 1.5 2 x direction (m) x 10 4 1500 C. Vuik, December 7, 2007 23 p.23/28

Sigsbee model dx = dz = 22.86 m; D = 24369 9144 m 2 ; grid points 1067 401. Bi-CGSTAB 5 Hz 10 Hz CPU (sec) Iter CPU (sec) Iter NO preco 3128 16549 1816 9673 With preco 86 48 92 58 Note: Without preconditioner, number of iterations > 10 4, With shifted Laplacian preconditioner, only 58 iterations. C. Vuik, December 7, 2007 24 p.24/28

3D wedge problem (0,0.25,1) (0,0.8,1) bk f 2 k x 3 ak (0,0.6,0) (1,0.7,0) f 1 x 2 (0,0.4,0) (1,0.2,0) x 1 Unit source C. Vuik, December 7, 2007 25 p.25/28

Numerical results for 3D wedge problem 80 70 (a,b) = (1.2,1.5) (a,b) = (1.2,2.0) 60 iterations 50 40 30 20 10 0 0 10 20 30 40 50 60 Frequency f C. Vuik, December 7, 2007 26 p.26/28

Numerical results for 3D wedge problem 80 70 (a,b) = (1.2,1.5) (a,b) = (1.2,2.0) 60 iterations 50 40 30 20 10 0 0 10 20 30 40 50 60 Frequency f C. Vuik, December 7, 2007 26 p.26/28

6. Conclusions The shifted Laplacian operator leads to robust preconditioners for the 2D and 3D Helmholtz equations with various boundary conditions. For real shifts the eigenvalues of the preconditioned operator are on a straight line. For complex shifts the eigenvalues of the preconditioned operator are on a circle. 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. C. Vuik, December 7, 2007 27 p.27/28

Further information/research http://ta.twi.tudelft.nl/nw/users/vuik/pub_it_helmholtz.html Y.A. Erlangga, C. Vuik and C.W. Oosterlee On a class of preconditioners for solving the Helmholtz equation Appl. Num. Math., 50, pp. 409-425, 2004 Y.A. Erlangga, C.W. Oosterlee and C. Vuik A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems SIAM J. Sci. Comput.,27, pp. 1471-1492, 2006 M.B. van Gijzen, Y.A. Erlangga and C. Vuik Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian SIAM J. Sci. Comput., 2007, 29, pp. 1942-1958, 2007 C. Vuik, December 7, 2007 28 p.28/28