3D transient electromagnetic modeling using a shift-and-invert Krylov subspace method

Abstract

We present a fast and efficient algorithm for simulation of a three-dimensional (3D) transient electromagnetic (TEM) response using a modified shift-and-invert Krylov subspace method. The mimetic finite volume method with a staggered grid is carried out for spatial discretization of the time-domain Maxwell's equations. The transient electromagnetic response then can be expressed as a matrix exponential function with an analytic initial magnetic field for a step-off loop source. The shift-and-invert Krylov subspace method can be used to solve the matrix exponential function. However, it requires solving dozens of large sparse linear equations at every time point to reconstruct the Krylov subspace, which makes the conventional shift-and-invert Krylov subspace method time-consuming. By analyzing the characteristics of the optimal shift and shift-and-invert Krylov subspace dimension in detail, we proposed a fast substitute approach to obtain the optimal shift and subspace order by using single optimal shift and constant subspace order with a useful stopping criterion, and developed an efficient modified shift-and-invert Krylov subspace method. Only one LU factorization of a shift coefficient matrix and hundreds of times backward substitutions are required to obtain the results of the TEM modeling data. Time savings are considerable, and this approach makes it possible to compute the response at any time point in the given time interval within the given residual easily and accurately. This is illustrated by using synthetic examples both in layered models and in a 3D complicated model.