Algebraic multigrid preconditioning within parallel finite-element solvers for 3-D electromagnetic modelling problems in geophysics

We present an elaborate preconditioning scheme for Krylov subspace methods which has been developed to improve the performance and reduce the execution time of parallel node-based finite-element solvers for three-dimensional electromagnetic numerical modelling in exploration geophysics. This new preconditioner is based on algebraic multigrid that uses different basic relaxation methods, such as Jacobi, symmetric successive over-relaxation and Gauss-Seidel, as smoothers and the wave-front algorithm to create groups, which are used for a coarse-level generation. We have implemented and tested this new preconditioner within our parallel nodal finite-element solver for three-dimensional forward problems in electromagnetic induction geophysics. We have performed series of experiments for several models with different conductivity structures and characteristics to test the performance of our algebraic multigrid preconditioning technique when combined with biconjugate gradient stabilised method. The results have shown that, the more challenging the problem is in terms of conductivity contrasts, ratio between the sizes of grid elements and/or frequency, the more benefit is obtained by using this preconditioner. Compared to other preconditioning schemes, such as diagonal, symmetric successive over-relaxation and truncated approximate inverse, the algebraic multigrid preconditioner greatly improves the convergence of the iterative solver for all tested models. Also, when it comes to cases in which other preconditioners succeed to converge to a desired precision, algebraic multigrid is able to considerably reduce the total execution time of the forward-problem code -up to an order of magnitude. Furthermore, the tests have confirmed that our algebraic multigrid scheme ensures grid-independent rate of convergence, as well as improvement in convergence regardless of how big local mesh refinements are. In addition, algebraic multigrid is designed to be a black-box preconditioner, which makes it easy to use and combine with different iterative methods. Finally, it has proved to be very practical and eficient in the parallel context.

[1]  Gregory A. Newman,et al.  Electromagnetic induction in a generalized 3D anisotropic earth, Part 2: The LIN preconditioner , 2003 .

[2]  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..

[3]  R.D. Falgout,et al.  An Introduction to Algebraic Multigrid Computing , 2006, Computing in Science & Engineering.

[4]  D FalgoutRobert.  An Introduction to Algebraic Multigrid , 2006 .

[5]  Dan Li,et al.  Parallel numerical solution of the time-harmonic Maxwell equations in mixed form , 2012, Numer. Linear Algebra Appl..

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

[7]  Max A. Meju,et al.  Useful characteristics of shallow and deep marine CSEM responses inferred from 3D finite-difference modeling , 2009 .

[8]  Mark E. Everett,et al.  Theoretical Developments in Electromagnetic Induction Geophysics with Selected Applications in the Near Surface , 2011, Surveys in Geophysics.

[9]  J. Dendy.  Black box multigrid , 1982 .

[10]  Michael Commer,et al.  A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources , 2004 .

[11]  Rainald Löhner,et al.  A parallel implicit incompressible flow solver using unstructured meshes , 1993 .

[12]  Elizabeth R. Jessup,et al.  A Technique for Accelerating the Convergence of Restarted GMRES , 2005, SIAM J. Matrix Anal. Appl..

[13]  Uri M. Ascher,et al.  Multigrid Preconditioning for Krylov Methods for Time-Harmonic Maxwell's Equations in Three Dimensions , 2002, SIAM J. Sci. Comput..

[14]  Guillaume Houzeaux,et al.  A massively parallel fractional step solver for incompressible flows , 2009, J. Comput. Phys..

[15]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[16]  Gregory A. Newman,et al.  Three-dimensional induction logging problems, Part 2: A finite-difference solution , 2002 .

[17]  W. A. Mulder,et al.  A multigrid solver for 3D electromagnetic diffusion , 2006 .

[18]  Algebraic MultiGrid Methods for Nodal and Edge based Discretizations of Maxwell ’ s Equations , 2002 .

[19]  Chester J. Weiss,et al.  Mapping thin resistors and hydrocarbons with marine EM methods, Part II -Modeling and analysis in 3D , 2006 .

[20]  Guillaume Houzeaux,et al.  A parallel finite-element method for three-dimensional controlled-source electromagnetic forward modelling , 2013 .

[21]  Dmitry B. Avdeev,et al.  Three-Dimensional Electromagnetic Modelling and Inversion from Theory to Application , 2005 .

[22]  Guillaume Houzeaux,et al.  Parallel uniform mesh multiplication applied to a Navier–Stokes solver , 2013 .

[23]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[24]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[25]  David S. Burnett,et al.  Finite Element Analysis: From Concepts to Applications , 1987 .

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

[27]  Oszkar Biro,et al.  Finite-element analysis of controlled-source electromagnetic induction using Coulomb-gauged potentials , 2001 .

[28]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[29]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[30]  Christoph Schwarzbach,et al.  Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics—a marine CSEM example , 2011 .

[31]  Francisco Ortigosa,et al.  A parallel finite‐element method for 3‐D marine controlled‐source electromagnetic forward modeling , 2011 .

[32]  René-Édouard Plessix,et al.  An approach for 3D multisource, multifrequency CSEM modeling , 2007 .

[33]  François Baumgartner,et al.  CR1Dmod: A Matlab program to model 1D complex resistivity effects in electrical and electromagnetic surveys , 2006, Comput. Geosci..

[34]  J. David Moulton,et al.  Robust and adaptive multigrid methods: comparing structured and algebraic approaches , 2012, Numer. Linear Algebra Appl..

[35]  Chester J. Weiss,et al.  Mapping thin resistors and hydrocarbons with marine EM methods: Insights from 1D modeling , 2006 .

[36]  V. E. Henson,et al.  BoomerAMG: a parallel algebraic multigrid solver and preconditioner , 2002 .

[37]  G. Newman,et al.  Three-dimensional magnetotelluric inversion using non-linear conjugate gradients , 2000 .