A parallel finite‐element method for 3‐D marine controlled‐source electromagnetic forward modeling

Li, Xiong; Li, Yaoguo; Meng, Xiaohong; Koldan, Jelena; Puzyrev, Vladimir; María, José; Puente, Josep de la; Ortigosa, F.

doi:10.1190/1.3659052

Cited by 2 publications

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“…6.4 Tests for Insensitivity to the Maximal Size Ratio Between the Grid Elements Finite-element grids normally have extremely large size ratios between the elements due to local refinements. This is usually a reason for the poor convergence of a solver (Koldan et al, 2011). Having this in mind, we have performed tests to see if our preconditioning scheme can reduce the sensitivity of an iterative solver to a big size difference between the biggest to the smallest element in a mesh.…”

Section: Flat Seabed Modelmentioning

confidence: 99%

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

Koldan¹,

Puzyrev²,

Puente³

et al. 2014

Geophysical Journal International

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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 efficient in the parallel context.

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Section: Flat Seabed Modelmentioning

confidence: 99%

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

Koldan¹,

Puzyrev²,

Puente³

et al. 2014

Geophysical Journal International

View full text Add to dashboard Cite

show abstract

“…FE grids normally have extremely large size ratios between the elements due to local refinements. This is usually a reason for the poor convergence of a solver (Koldan et al, 2011). Having this in mind, I have performed tests to see if this preconditioning scheme can reduce the sensitivity of an iterative solver to a big size difference between the biggest to the smallest element in a mesh.…”

Section: Tests For Insensitivity To the Maximal Size Ratio Between Th...mentioning

confidence: 99%

Numerical solution of 3-D electromagnetic problems in exploration geophysics and its implementation on massively parallel computers

Koldan¹

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The growing significance, technical development and employment of electromagnetic (EM) methods in exploration geophysics have led to the increasing need for reliable and fast techniques of interpretation of 3-D EM data sets acquired in complex geological environments. The first and most important step to creating an inversion method is the development of a solver for the forward problem. In order to create an efficient, reliable and practical 3-D EM inversion, it is necessary to have a 3-D EM modelling code that is highly accurate, robust and very fast. This thesis focuses precisely on this crucial and very demanding step to building a 3-D EM interpretation method. The thesis presents as its main contribution a highly accurate, robust, very fast and extremely scalable numerical method for 3-D EM modelling in geophysics that is based on finite elements (FE) and designed to run on massively parallel computing platforms. Thanks to the fact that the FE approach supports completely unstructured tetrahedral meshes as well as local mesh refinements, the presented solver is able to represent complex geometries of subsurface structures very precisely and thus improve the solution accuracy and avoid misleading artefacts in images. Consequently, it can be successfully used in geological environments of arbitrary geometrical complexities. The parallel implementation of the method, which is based on the domain decomposition and a hybrid MPI-OpenMP scheme, has proved to be highly scalable - the achieved speed-up is close to the linear for more than a thousand processors. Thanks to this, the code is able to deal with extremely large problems, which may have hundreds of millions of degrees of freedom, in a very efficient way. The importance of having this forward-problem solver lies in the fact that it is now possible to create a 3-D EM inversion that can deal with data obtained in extremely complex geological environments in a way that is realistic for practical use in industry. So far, such imaging tool has not been proposed due to a lack of efficient, parallel FE solutions as well as the limitations of efficient solvers based on finite differences. In addition, the thesis discusses physical, mathematical and numerical aspects and challenges of 3-D EM modelling, which have been studied during my research in order to properly design the presented software for EM field simulations on 3-D areas of the Earth. Through this work, a physical problem formulation based on the secondary Coulomb-gauged EM potentials has been validated, proving that it can be successfully used with the standard nodal FE method to give highly accurate numerical solutions. Also, this work has shown that Krylov subspace iterative methods are the best solution for solving linear systems that arise after FE discretisation of the problem under consideration. More precisely, it has been discovered empirically that the best iterative method for this kind of problems is biconjugate gradient stabilised with an elaborate preconditioner. Since most commonly used preconditioners proved to be either unable to improve the convergence of the implemented solvers to the desired extent, or impractical in the parallel context, I have proposed a preconditioning technique for Krylov methods that is based on algebraic multigrid. Tests for various problems with different conductivity structures and characteristics have shown that the new preconditioner greatly improves the convergence of different Krylov subspace methods, which significantly reduces the total execution time of the program and improves the solution quality. Furthermore, the preconditioner is very practical for parallel implementation. Finally, it has been concluded that there are not any restrictions in employing classical parallel programming models, MPI and OpenMP, for parallelisation of the presented FE solver. Moreover, they have proved to be enough to provide an excellent scalability for it.

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A parallel finite‐element method for 3‐D marine controlled‐source electromagnetic forward modeling

Cited by 2 publications

References 0 publications

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

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

Numerical solution of 3-D electromagnetic problems in exploration geophysics and its implementation on massively parallel computers

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