Academic literature on the topic 'Shift-and-invert Krylov'

Abstract For model reduction techniques, there have been relatively few studies performed regarding the forward modeling of anisotropic media in comparison to transient electromagnetic (TEM) forward modeling of isotropic media. The transient electromagnetic method (TEM) responses after the current has been turned off can be represented as a homogeneous ordinary differential equation (ODE) with an initial value, and the ODE can be solved using a matrix exponential function. However, the order of the matrix exponential function is large and solving it directly is challenging, thus this study employs the Shift-and-Invert (SAI-Krylov) subspace algorithm. The SAI-Krylov subspace technique is classified as a single-pole approach compared to the multi-pole rational Krylov subspace approach. It only takes one LU factorization of the coefficient matrix, along with hundreds of backward substitutions. The research in this paper shows that the anisotropic medium has little effect on the optimal shift ${\gamma _{opt}}$ and subspace order m. Furthermore, as compared to the mimetic finite volume method (SAI-MFV) of the SAI-Krylov subspace technique, the method proposed in this paper (SAI-FEM) can further improve the computing efficiency by roughly 13%. In contrast to the standard implicit time step iterative technique, the SAI-FEM method does not require discretization in time, and the TEM response at any moment within the off-time period can be easily computed. Next, the accuracy of the SAI-FEM algorithm was verified by 1D solutions for an anisotropic layer model and a 3D anisotropic model. Finally, the electromagnetic characteristics of the anisotropic anomalous body of the center loop device and separated device of the airborne transient electromagnetic method were analyzed, and it was found that horizontal conductivity has a considerable influence on the TEM response of the anisotropic medium.

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.