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 the controlled-source electromagnetic (CSEM) method 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 $\sqrt{2}$, 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 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.

中文翻译：

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使用高效迭代求解器解决大规模 3D 受控源电磁建模问题

随着地球物理调查从传统的 2D 模型发展到 3D 模型，大量数据给反演增加了挑战，尤其是在旨在解析复杂的 3D 结构时。用于受控源电磁 (CSEM) 方法的迭代正向求解器比直接求解器需要更少的内存；然而，迭代求解由麦克斯韦方程的有限元离散化产生的病态线性方程组并不容易。为了解决这个问题，我们为频域和时域 CSEM 建模问题开发了高效且稳健的迭代求解器。对于频域问题，我们首先将线性系统转换为其等效的实数格式，然后引入最优块对角预处理器。$\sqrt{2}$，当应用灵活的广义最小残差求解器时，我们可以实现快速解收敛。应用块预处理器进一步导致求解具有相同系数矩阵的两个较小的线性系统。对于时域问题，我们首先使用无条件稳定的后向欧拉格式对电场的偏微分方程进行时间离散化。然后我们在每个时间步迭代地求解得到的线性方程组。在频域中的空间离散化或时域中的时空离散化之后，我们利用从 Hiptmair-Xu 分解导出的带有辅助空间预处理器的共轭梯度求解器来求解这些实线性系统。最后，我们通过模拟复杂的 CSEM 模型来检查我们的迭代方法的效率和有效性。