An efficient multigrid solver based on a four-color cell-block Gauss-Seidel smoother for 3D magnetotelluric forward modeling
An efficient multigrid solver based on a four-color cell-block Gauss-Seidel smoother for 3D magnetotelluric forward modeling
复制标题
基于四色单元块高斯-赛德尔平滑器的高效多重网格求解器，用于 3D 大地电磁正演建模
DOI：
10.1190/geo2021-0275.1
复制
发表时间：
2022
期刊：
影响因子：
3.3
通讯作者：
Hang Chen
中科院分区：
文献类型：
--
作者：
Rongwen Guo;Yongfei Wang;Gary D. Egbert;Jianxin Liu;Rong Liu;Kejia Pan;Jian Li;Hang Chen
三维大地电磁反演的实际应用需要对地球电磁场进行有效的正演模拟。为了求解真实的三维结构，需要大量的计算域和极大的线性方程组。由于旋旋算子具有丰富的零空间，因此几乎专门用于求解这些系统的迭代求解器可能效率低下。多网格（MG）求解器被认为是解决此类问题的潜在有效技术。然而，由于存在丰富的零解空间和空气层，MG解仍然会缓慢收敛甚至发散。我们开发了一种有效的有限差分频域电磁求解器。该算法利用高效的四色单元块高斯-塞德尔（GS）的平滑特性有效地去除近程误差，并利用插值算子和延长算子处理远程误差。它们作为一个整体，显著加快了我们算法的收敛速度。由于四色单元块GS的所有节点都被分组为四种颜色，并且每种颜色中附加到不同节点的边缘分量完全解耦，因此可以用于开发高度矢量化或并行化的算法。另一个重要的性质是我们的算法是局部电流散度自由的，有效地消除了旋旋算子零空间中的伪解。通过将MG求解器所得到的数值解与基于综合模型和三维反演模型的不同预条件的双共轭梯度稳定求解器所得到的数值解进行比较，验证了算法的准确性和效率。在迭代次数和计算时间方面的比较表明，我们的算法相对于其他求解器是非常稳定和高效的。我们的MG算法也将适用于大规模并行计算。
Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.