A Time-Segmented SAI-Krylov Subspace Approach for Large-Scale Transient Electromagnetic Forward Modeling

After nearly two decades of development, transient electromagnetic (TEM) 3D forward modeling technology has significantly improved both numerical precision and computational efficiency, primarily through advancements in mesh generation and the optimization of linear equation solvers. However, the dominant approach still relies on direct solvers, which require substantial memory and complicate the modeling of electromagnetic responses in large-scale models. This paper proposes a new method for solving large-scale TEM responses, building on previous studies. The TEM response is expressed as a matrix exponential function with an analytic initial field for a step-off source, which can be efficiently solved using the Shift-and-Invert Krylov (SAI-Krylov) subspace method. The Arnoldi algorithm is used to construct the orthogonal basis for the Krylov subspace, and the preconditioned conjugate gradient (PCG) method is applied to solve large-scale linear equations. The paper further explores how dividing the off-time and optimizing parameters for each time interval can enhance computational efficiency. The numerical results show that this parameter optimization strategy reduces the iteration count of the PCG method, improving efficiency by a factor of 5 compared to conventional iterative methods. Additionally, the proposed method outperforms direct solvers for large-scale model calculations. Conventional approaches require numerous matrix factorizations and thousands of back-substitutions, whereas the proposed method only solves about 300 linear equations. The accuracy of the approach is validated using 1D and 3D models, and the propagation characteristics of the TEM field are studied in large-scale models.