Efficient Multigrid Algorithms for Three-Dimensional Electromagnetic Forward Modeling

Multigrid (MG) methods solve large linear equations on fine grids by projecting them onto progressively coarser grids, on which the problem can be solved more cheaply. They have become among the most effective and prospective solvers for large linear systems. However, due to the abundant null solution space and the inclusion of the air layer, traditional MG methods struggle to converge in three-dimensional (3D) electromagnetic (EM) numerical forward modeling. Served as one major contribution of this review, we provide a complete review on strategies, introduced in recent decades to develop efficient MG algorithms for EM forward modeling. We focus on how these strategies handle the convergence difficulties encountered in EM numerical forward modeling. Another observation is that most state-of-the-art MG solvers have been developed and examined against traditional Krylov subspace iterative solvers, but there is little knowledge on the numerical performance of different strategies. Therefore, another primary contribution of this work is to provide a complete review of the numerical performance of different strategies used in MG solvers for 3D EM forward modeling in geophysical applications. For this purpose, firstly, we briefly introduce on finite difference and finite element numerical discretization of the electrical field partial differential equations to demonstrate why EM forward modeling is challenging to solve. Subsequently, some background information on MG methods is provided to show how they can be implemented in general. Then, different strategies used in different MG methods are introduced in great detail to address the convergence issues encountered in EM forward modeling in geophysical applications, caused by the abundant null solution space and the inclusion of the air layer. Finally, we present four newly developed MG algorithms and compare their overall numerical performance in terms of their parallel ability, stability, efficiency and memory cost by using two increasingly complex models. Since one major motivation for improving the EM forward modeling efficiency is to speed up the inversion process, their perspective of efficiency improvement in EM inversions has been discussed. On this basis, authors and researchers can choose one particular MG solver for their own EM forward modeling problems.