Solution of large-scale 3D controlled-source electromagnetic modeling problem using efficient iterative solvers

Abstract

With geophysical surveys evolving from traditional 2D to 3D models, the large volume of data adds challenges to inversion, especially when aiming to resolve complex 3D structures. An iterative forward solver for a controlled-source electromagnetic method (CSEM) requires less memory than that for a direct solver; however, it is not easy to iteratively solve an ill-conditioned linear system of equations arising from finite-element discretization of Maxwell’s equations. To solve this problem, we have developed efficient and robust iterative solvers for frequency- and time-domain CSEM modeling problems. For the frequency-domain problem, we first transform the linear system into its equivalent real-number format, and then introduce an optimal block-diagonal preconditioner. Because the condition number of the preconditioned linear equation system has an upper bound of √2, we can achieve fast solution convergence when applying a flexible generalized minimum residual solver. Applying the block preconditioner further results in solving two smaller linear systems with the same coefficient matrix. For the time-domain problem, we first discretize the partial differential equation for the electric field in time using an unconditionally stable backward Euler scheme. We then solve the resulting linear equation system iteratively at each time step. After the spatial discretization in the frequency domain, or space-time discretization in the time domain, we exploit the conjugate-gradient solver with auxiliary-space preconditioners derived from the Hiptmair-Xu decomposition to solve these real linear systems. Finally, we check the efficiency and effectiveness of our iterative methods by simulating complex CSEM models. The most significant advantage of our approach is that the iterative solvers that we adopt have almost the same accuracy and robustness as direct solvers but require much less memory, rendering them more suitable for large-scale 3D CSEM forward modeling and inversion.

Iterative solution methods for 3D controlled-source electromagnetic forward modelling of geophysical exploration scenarios

“…We emphasise that PRESB, although resembling the block-diagonal preconditioner presented in some other related work (e.g. [25,44,45,69]), is different. In particular, applying these preconditioner results in slightly different computational procedures although the core of both schemes consists in solving two linear systems.…”

“…This preconditioning framework introduced in [30] has also been adopted to EM modelling problems (cf e.g. [25,44,45,69]).…”

“…We also stress that we use a spectral element discretisation method, whereas previous works use finite element methods. In contrast to [44,45] and [69], we do not disregard displacement currents which complicates the system due to the potential negative shift in matrix elements resulting from the displacement current term in some parts of the modelling domain [25].…”

Three-dimensional modelling of controlled source electro-magnetic surveys using non-conforming finite element methods

“…Iterative solvers have also been used in EM modeling, particularly in 3-D CSEM (e.g. Um et al, 2012;Puzyrev et al, 2013;Sommer et al, 2013;Dunham et al, 2018;Castillo-Reyes et al, 2019;Qiu et al, 2021), due to their relatively low storage requirements. Furthermore, iterative solvers are much easier to implement efficiently on computers with parallel-distributed memory (Puzyrev et al, 2013).…”

“…Efforts to overcome these difficulties have led to numerous strategies such as the formulation and application of novel preconditioning techniques (e.g. Li et al, 2019;Qiu et al, 2021;Peng et al, 2020), the addition of a static divergence correction to the iterative process (e.g. Liu & Yin, 2013;Tang et al, 2021), the modification of the original curl-curl equations with the divergence correction equation (e.g.…”