Comparison of iterative and direct solvers for 3D CSEM modeling

Grayver, Alexander; Streich, Rita

doi:10.1190/segam2012-0727.1

Cited by 7 publications

(7 citation statements)

References 21 publications

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“…In all maps the relative error in the air is orders of magnitude larger than in the water or formation. Similar observations have been earlier reported by Grayver & Streich (2012). Fortunately, large errors in the air do not create a problem in most practical CSEM applications.…”

Section: Choice Of Blr Thresholdsupporting

confidence: 89%

Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver

Shantsev

Jaysaval

Ryhove³

et al. 2017

Geophysical Journal International

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S U M M A R YWe put forward the idea of using a Block Low-Rank (BLR) multifrontal direct solver to efficiently solve the linear systems of equations arising from a finite-difference discretization of the frequency-domain Maxwell equations for 3-D electromagnetic (EM) problems. The solver uses a low-rank representation for the off-diagonal blocks of the intermediate dense matrices arising in the multifrontal method to reduce the computational load. A numerical threshold, the so-called BLR threshold, controlling the accuracy of low-rank representations was optimized by balancing errors in the computed EM fields against savings in floating point operations (flops). Simulations were carried out over large-scale 3-D resistivity models representing typical scenarios for marine controlled-source EM surveys, and in particular the SEG SEAM model which contains an irregular salt body. The flop count, size of factor matrices and elapsed run time for matrix factorization are reduced dramatically by using BLR representations and can go down to, respectively, 10, 30 and 40 per cent of their full-rank values for our largest system with N = 20.6 million unknowns. The reductions are almost independent of the number of MPI tasks and threads at least up to 90 × 10 = 900 cores. The BLR savings increase for larger systems, which reduces the factorization flop complexity from O(N 2 ) for the full-rank solver to O(N m ) with m = 1.4-1.6. The BLR savings are significantly larger for deep-water environments that exclude the highly resistive air layer from the computational domain. A study in a scenario where simulations are required at multiple source locations shows that the BLR solver can become competitive in comparison to iterative solvers as an engine for 3-D controlled-source electromagnetic Gauss-Newton inversion that requires forward modelling for a few thousand right-hand sides.

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Section: Choice Of Blr Thresholdsupporting

confidence: 89%

Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver

Shantsev

Jaysaval

Ryhove³

et al. 2017

Geophysical Journal International

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“…Computation time with an increasing number of right‐hand side vectors (i.e. sources) for a preconditioned BiCG iterative solver with a simultaneous solving strategy (Grayver and Streich ) and a direct solver (PARDISO) to solve the EM fields due to multiple sources for the inclined sheet model (Figure ). Numbers on each line represent the number of cores used.…”

Section: Resultsmentioning

confidence: 99%

“…After matrix factorisation, the number of right‐hand side vectors barely affects the total solution time in a direct solver, as shown in Figure , whereas an iterative solver typically calculates multiple right‐hand side vectors repeatedly. Figure also shows a simultaneous solving strategy (Grayver and Streich ), which was introduced to optimise the performance of preconditioned BiCG by utilising a set of cores to concurrently solve each divided sub‐problem. As mentioned in the above‐cited paper, the iterative solver combined with this strategy scales less favourably because the increasing number of concurrent processes decreases the amount of shared resources accessible to each core.…”

Section: Resultsmentioning

confidence: 99%

Three‐dimensional modelling of controlled‐source electromagnetic surveys using an edge finite‐element method with a direct solver

Chung

Son

Lee

et al. 2014

Geophysical Prospecting

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A fully three‐dimensional finite‐element algorithm has been developed for simulating controlled‐source electromagnetic surveys. To exploit the advantages of geometric flexibility, frequency‐domain Maxwell's equations of the secondary electric field were discretised using edge‐based finite elements while the primary field was calculated analytically for a horizontally layered‐earth model. The resulting system of equations for the secondary field was solved using a parallel version of direct solvers. The accuracy of the algorithm was successfully verified by comparisons with integral‐equations and iterative solutions, and the applicability to models containing large conductivity contrasts was verified against published data. The advantages of geometry‐conforming meshes have been demonstrated by comparing different mesh systems to simulate an inclined sheet model. A comparison of the performance between direct and iterative solvers demonstrated the superior efficiency of direct solvers, particularly for multisource problems.

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“…Direct solvers are generally more expensive than iterative solvers in terms of computation time and memory (Grayver and Streich, 2012). However, they also have advantages.…”

Section: Theorymentioning

confidence: 97%

A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids

2014

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The application of unstructured grids can improve the solution of total field electromagnetic problems because these grids allow efficient local refinement of the mesh at the locations of high field gradients. Unstructured grids also provide the flexibility required for representing arbitrary topography and subsurface interfaces. We investigated the generalization of the standard Yee's staggered scheme to unstructured tetrahedral-Voronoï grids using a finite-volume approach. We discretized the Helmholtz equation for the electric field in the frequency domain and solved the problem to find the projection of the total electric field along the edges of the tetrahedral elements. To compute the electric and magnetic fields at the observation points, an interpolation technique was employed, which uses the edge vector interpolation functions of the tetrahedral elements. To verify the presented scheme, two examples were evaluated, which revealed the computation of the total and secondary fields due to electric and magnetic sources in half-spaces that contain anomalous bodies. The results had good agreement with those from the literature. For the second example, accuracy studies were conducted to better understand the relative importance of different refinements in the grid and also the effect of the general quality of the grid. The results revealed the utmost importance of the quality of the tetrahedral grid and refinement at the observation locations. To validate the versatility of the approach, we synthesized helicopter-borne data for a model of the Ovoid ore body at Voisey's Bay, Labrador, Canada, which had good agreement with real data.

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Comparison of iterative and direct solvers for 3D CSEM modeling

Cited by 7 publications

References 21 publications

Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver

Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver

Three‐dimensional modelling of controlled‐source electromagnetic surveys using an edge finite‐element method with a direct solver

A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids

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