3D large-scale transient electromagnetic modeling using a Shift-and-Invert Krylov subspace method

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In this paper, the combination of the Shift-and-Invert Krylov (SAI-Krylov) subspace method and the preconditioned conjugate gradient method (PCG) solves the problem of 3D large-scale time-domain transient electromagnetic (TEM) forward modeling. Firstly, the Galerkin method is used to obtain the spatial discretization governing equation of the electric field from the governing equation of the electric field. The electric field can be expressed in the form of a matrix exponential function using the model reduction method, according to the homogeneous ODE. The TEM response of small and medium-scale models can be solved quickly by using the SAI-Krylov technique, selecting the optimal shift γopt, and combining with the direct method solver. The orthogonal basis of the SAI-Krylov subspace is difficult to build by the direct method due to the high coefficient matrix order of the large-scale TEM forward modeling problem, but it can be solved successfully using the PCG method. Since the optimal shift γopt in the SAI-Krylov method applies to the direct method solver, the computation efficiency is quite low if directly applied to the PCG method. In conclusion, it is necessary to re-select the optimal shift γopt when solving the orthogonal basis of SAI-Krylov subspace using the PCG method. In this paper, the optimal shift γopt is selected based on the relationship between the condition number of the coefficient matrix and shift γ to reduce the conditional number of the coefficient matrix and minimize the number of iterations in the calculation process. Furthermore, the tolerance ε and relaxation factors ω in the PCG method are chosen based on the 5% error threshold. By reselecting the shift γ, the calculation time is reduced by 2/3. Finally, the validity and applicability of the proposed method are verified by the homogeneous half-space model with different resistivity, the 3D vertical contact zone model, and the large-scale complex sulfide ore body model.