Multigrid-based 'shifted-Laplacian' preconditioning for the time-harmonic elastic wave equation

We investigate the numerical performance of an iterative solver for a frequency-domain finite-difference discretization of the isotropic elastic wave equation. The solver is based on the 'shifted-Laplacian' preconditioner, originally designed for the acoustic wave equation. This preconditioner represents a discretization of a heavily damped wave equation and can be efficiently inverted by a multigrid iteration. However, the application of multigrid to the elastic case is not straightforward because standard methods, such as point-Jacobi, fail to smooth the S-wave wavenumber components of the error when high P-to-S velocity ratios are present. We consider line smoothers as an alternative and apply local-mode analysis to evaluate the performance of the various components of the multigrid preconditioner. Numerical examples in 2-D demonstrate the efficacy of our method.

[1]  Laurent Demanet,et al.  Nested Domain Decomposition with Polarized Traces for the 2D Helmholtz Equation , 2015, SIAM J. Sci. Comput..

[2]  Jianlin Xia,et al.  On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver , 2011 .

[3]  Jianlin Xia,et al.  Massively parallel structured multifrontal solver for time-harmonic elastic waves in 3-D anisotropic media , 2012 .

[4]  Ray S. Tuminaro,et al.  Augmented AMG‐shifted Laplacian preconditioners for indefinite Helmholtz problems , 2015, Numer. Linear Algebra Appl..

[5]  Reinhard Nabben,et al.  Multilevel Projection-Based Nested Krylov Iteration for Boundary Value Problems , 2008, SIAM J. Sci. Comput..

[6]  René-Édouard Plessix,et al.  How to choose a subset of frequencies in frequency-domain finite-difference migration , 2004 .

[7]  Cornelis Vuik,et al.  A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems , 2005, SIAM J. Sci. Comput..

[8]  Cornelis Vuik,et al.  On the Construction of Deflation-Based Preconditioners , 2001, SIAM J. Sci. Comput..

[9]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[10]  Cornelis Vuik,et al.  On the convergence of shifted Laplace preconditioner combined with multilevel deflation , 2013, Numer. Linear Algebra Appl..

[11]  Cornelis Vuik,et al.  Closing the performance gap for 3-D migration with an iterative frequency-domain solver and an explicit time-domain scheme on parallel architectures , 2014 .

[12]  I. Štekl,et al.  Accurate viscoelastic modeling by frequency‐domain finite differences using rotated operators , 1998 .

[13]  Cornelis Vuik,et al.  Closing the performance gap between an iterative frequency-domain solver and an explicit time-domain scheme for 3D migration on parallel architectures , 2014 .

[14]  Michael B. Giles,et al.  Preconditioned iterative solution of the 2D Helmholtz equation , 2002 .

[15]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[16]  A. Abubakar,et al.  Iterative solution of 3D acoustic wave equation with perfectly matched layer boundary condition and multigrid preconditioner , 2013 .

[17]  Changsoo Shin,et al.  Free-Surface Boundary Condition in Finite-Difference Elastic Wave Modeling , 2004 .

[18]  Changsoo Shin,et al.  Weighted-Averaging Finite-Element Method for 2D Elastic Wave Equations in the Frequency Domain , 2003 .

[19]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[20]  Christiaan C. Stolk,et al.  A rapidly converging domain decomposition method for the Helmholtz equation , 2012, J. Comput. Phys..

[21]  Cornelis W. Oosterlee,et al.  On Three-Grid Fourier Analysis for Multigrid , 2001, SIAM J. Sci. Comput..

[22]  Gary Martin,et al.  Marmousi2 An elastic upgrade for Marmousi , 2006 .

[23]  K. R. Kelly,et al.  SYNTHETIC SEISMOGRAMS: A FINITE ‐DIFFERENCE APPROACH , 1976 .

[24]  Alan George,et al.  Computer Solution of Large Sparse Positive Definite , 1981 .

[25]  Cornelis Vuik,et al.  On a Class of Preconditioners for Solving the Helmholtz Equation , 2003 .

[26]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[27]  R. Pratt.  Frequency-domain elastic wave modeling by finite differences : a tool for crosshole seismic imaging , 1990 .

[28]  William L. Briggs,et al.  A multigrid tutorial, Second Edition , 2000 .

[29]  J. Virieux.  P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method , 1986 .

[30]  Dianne P. O'Leary,et al.  A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations , 2001, SIAM J. Sci. Comput..

[31]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[32]  Wim Vanroose,et al.  Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer , 2011, Numer. Linear Algebra Appl..

[33]  Jianlin Xia,et al.  Superfast Multifrontal Method for Large Structured Linear Systems of Equations , 2009, SIAM J. Matrix Anal. Appl..

[34]  A. Bayliss,et al.  An Iterative method for the Helmholtz equation , 1983 .

[35]  Changsoo Shin,et al.  Improved frequency-domain elastic wave modeling using weighted-averaging difference operators , 2000 .

[36]  Aria Abubakar,et al.  An effective perfectly matched layer design for acoustic fourth-order frequency-domain finite-difference scheme , 2012 .

[37]  W. Mulder.  A new multigrid approach to convection problems , 1989 .

[38]  Martin B. van Gijzen,et al.  IDR(s): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations , 2008, SIAM J. Sci. Comput..

[39]  P. M. De Zeeuw,et al.  Matrix-dependent prolongations and restrictions in a blackbox multigrid solver , 1990 .

[40]  Achi Brandt,et al.  Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, Revised Edition , 2011 .

[41]  William H. Press,et al.  Numerical Recipes 3rd Edition: The Art of Scientific Computing , 2007 .

[42]  R. Nicolaides.  Deflation of conjugate gradients with applications to boundary value problems , 1987 .