Calcul des Probabilités. By Paul Lévy. Pp. 350. Fr. 40. 1925. (Gauthier-Villars.)

SUMMARY The fast and accurate 3-D magnetotelluric (MT) forward modelling is core engine of the interpretation and inversion of MT data. In this study, we develop an improved extrapolation cascadic multigrid method (EXCMG) to solve the large sparse complex linear system arising from the finite-element (FE) discretization on non-uniform orthogonal grids of the Maxwell’s equations using potentials. First, the vector Helmholtz equation and the scalar auxiliary equation are derived from the Maxwell’s equations using Coulomb-gauged potentials. The weighted residual method is adopted to discretize the weak formulation and assemble the FE equation. Secondly, carefully choosing the preconditioned complex stable bi-conjugate gradient method (BiCGStab) as multigrid smoother, we develop an improved EXCMG method on non-uniform grids to solve the resulting large sparse complex non-Hermitian linear systems. Finally, several examples including three standard testing models (COMMEMI3D-1, COMMEMI3D-2 and DTM1.0) and a topographic model are used to validate the accuracy and efficiency of the proposed multigrid solver. Numerical results show that the proposed EXCMG algorithm greatly improves the efficiency of 3-D MT forward modelling, is more efficient than some existing solvers, such as Pardiso, incomplete LU factorization preconditioned biconjugate gradients stabilized method (ILU-BiCGStab) and flexible generalized minimum residual method with auxiliary space Maxwell preconditioner (FGMRES-AMS), and capable to simulate large-scale problems with more than 100 million unknowns. 

OUP accepted manuscript

Practical application of three-dimensional (3D) Magnetotelluric (MT) inversion requires efficient forward modeling of electromagnetic (EM) fields in the Earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly, or even diverge. We develop an efficient MG solver for finite difference (FD) frequency domain EM solution. In this algorithm, the excellent smoothing property of an efficient 4-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are utilized to handle the long-range errors. They work as a whole to speed the convergence of our proposed algorithm remarkably. Since all the nodes for the 4-color cell-block GS are grouped into four colors and in each color the edge components attached to different nodes are totally decoupled, this can be utilized to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver with those from the Bi-Conjugate Gradients Stabilized (BiCGStab) solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will as well be suitable for massively parallel computing.

An efficient multigrid solver based on a 4-color cell-block Gauss-Seidel smoother for 3D Magnetotelluric forward modeling

\n Most of the existing 3-D electromagnetic (EM) modelling solvers based on the integral equation (IE) method exploit fast Fourier transform (FFT) to accelerate the matrix–vector multiplications. This in turn requires a laterally uniform discretization of the modelling domain. However, there is often a need for multiscale modelling and inversion, for instance, to properly account for the effects of non-uniform distant structures and, at the same time, to accurately model the effects from local anomalies. In such scenarios, the usage of laterally uniform grids leads to excessive computational loads, in terms of both memory and time. To alleviate this problem, we developed an efficient 3-D EM modelling tool based on a multinested IE approach. Within this approach, the IE modelling is first performed at a large domain and on a (laterally uniform) coarse grid, and then the results are refined in the region of interest by performing modelling at a smaller domain and on a (laterally uniform) denser grid. At the latter stage, the modelling results obtained at the previous stage are exploited. The lateral uniformity of the grids at each stage allows us to keep using the FFT for the acceleration of matrix–vector multiplications. An important novelty of the paper is the development of a ‘rim domain’ concept that further improves the performance of the multinested IE approach. We verify the developed tool on both idealized and realistic 3-D conductivity models, and demonstrate its efficiency and accuracy.

Advanced three-dimensional electromagnetic modelling using a nested integral equation approach

3D finite difference modeling of controlled-source electromagnetic response in frequency domain based on a modified curl-curl equation

Krylov subspace solvers for frequency-domain electromagnetic forward modeling problems converge remarkably more slowly as the period increases. In this article, we present an efficient four-color cellblock Gauss Seidel (GS) preconditioner for finite-difference (FD) electromagnetic modeling in geophysical applications. Rather than updating the FD electromagnetic (EM) equation edge by edge, as in a traditional GS scheme, we renew six edge components attached to one node simultaneously (i.e., in cellblock manner) effectively enforcing a local divergence free condition for currents. To improve implementation efficiency, we reorder the nodes on the FD grid into four colors so that nodes in each color are uncoupled, allowing the use of highly parallel vectorized algorithms. The four-color cellblock GS preconditioner is implemented in the MATLAB code, in conjunction with a BiCGstab solver. It is compared, in terms of iteration number and computing time, with other three commonly used preconditioners [GS, symmetric successive overrelaxation (SSOR) and incomplete lower and upper triangular matrix decomposition (ILU)] on three models—two synthetic and one modified from the version of real data inversion. The comparison indicates that the proposed algorithm is extremely stable and efficient compared with the other three preconditioners tested, over a range of periods (1-1000 s). Especially at long periods, the improvement of our proposed algorithm is substantial. In addition, a parallel implementation of the cellblock GS preconditioner is straightforward due to the independence of nodes in each color.

Jian Li

Papers

An Efficient Preconditioner for 3-D Finite Difference Modeling of the Electromagnetic Diffusion Process in the Frequency Domain

An efficient multigrid solver based on a 4-color cell-block Gauss-Seidel smoother for 3D Magnetotelluric forward modeling

3D finite difference modeling of controlled-source electromagnetic response in frequency domain based on a modified curl-curl equation

Comparison of stable maximum likelihood estimator with traditional robust estimator in magnetotelluric impedance estimation

Fast numerical simulation of 2D gravity anomaly based on nonuniform fast Fourier transform in mixed space-wavenumber domain