Quadrature-Based Restarted Arnoldi Method for Fast 3-D TEM Forward Modeling of Large-Scale Models

For large-scale geophysical models, the order of the coefficient matrix in 3-D transient electromagnetics (TEMs) forward modeling can reach millions or even tens of millions. Balancing computational efficiency and memory usage presents a challenge worthy of in-depth exploration. In this letter, we utilize an integral representation of the iterative error in the Arnoldi method to construct an efficient quadrature-based restarted forward algorithm. First, the mimetic finite volume (MFV) method on a staggered hexahedral grid is employed to discretize the time-domain Maxwell’s equations, expressing the TEM response after the step-off waveform shutoff as the product of the matrix exponential function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f({\text {A}})$ </tex-math></inline-formula> and vector b. Then, using Cauchy’s integral formula, the expression of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${f}({\text {A}}){b}$ </tex-math></inline-formula> is transformed into an integral form and approximated using the restarted Arnoldi (RA) algorithm. Our method does not require solving linear systems and can leverage GPU parallel technology and optimize the RA algorithm parameters to enhance computational efficiency. Comparative studies with other numerical methods validate the advantages and accuracy of our approach, which numerical example demonstrates can fully achieve large-scale fast 3-D TEM forward modeling.