基于几何多重网格预条件技术的三维大地电磁高效正演模拟

几何多重网格法(GMG)将细网格上的大型稀疏矩阵的求解转化为较粗网格上的更容易求解的问题, 从而快速求解大型稀疏方程组.但是由于大地电磁法(MT)正演模拟中涉及双旋度算子, 传统GMG无法有效平滑高频误差导致其收敛慢甚至发散.为此, 我们引入了四色分块高斯赛德尔法(GS)作为平滑算法, 该算法局部满足电流散度为零的条件, 无需额外的散度校正且具有高度并行性, 可以显著提高GMG的收敛效率.但是随着系数矩阵各向异性(比如电导率的强烈变化等)增加, GMG收敛速度会变慢.Krylov子空间求解法如稳定双共轭梯度法(BiCGstab)可以改善这种收敛变慢的问题.因此, 在本文中针对交错网格有限差分法(FDM)提出了一种结合四色分块GS平滑算法GMG和BiCGstab的MT高效正演模拟方法.在该方法中, 将四色分块GS平滑算法GMG作为BiCGstab求解器的预条件技术, 从而显著提高正演效率.我们设计了一个层状电阻率模型, 通过与其解析解对比验证本文所提算法的正确性.然后设计了一个双异常体电阻率模型和一个Dublin模型1(DTM1), 基于BiCGstab, 对比了GMG预条件技术与其他传统预条件技术的数值表现, 如超松弛预条件技术(SSOR)、分块不完全LU分解预条件技术(block ILU)和高斯赛德尔预条件技术(GS).结果显示本文提出的算法在迭代次数, 计算时间和稳定性方面都远远优于传统预条件技术.对于所有例子, GMG预条件技术均能在10次以内达到收敛, 计算时间比传统预条件技术减少70%以上, 显示了本方法的稳定性和高效性.

Efficient three-dimensional magnetotelluric forward modeling based on a geometric multigrid preconditioner

The geometric multigrid (GMG) method approximates the fine grid problems with the large sparse matrix on a coarser grid, which is easier to be solved. However, due to the double curl operator involved in magnetotelluric (MT) forward modeling, the high-frequency error cannot be smoothed effectively, resulting in an algorithm converging slowly or even diverging. For this reason, we introduce a four-color cell-block Gauss-Seidel (GS) method as smoother to improve the convergence efficiency. The algorithm locally satisfies the current divergence-free condition with high parallelism, and no additional divergence correction is needed which improves the efficiency of GMG significantly. However, with the anisotropy of the coefficient matrix (such as sharp change of conductivity, etc.), the efficiency of the GMG method can deteriorate. Krylov subspace methods such as BiCGstab can improve the problem. Therefore, in this paper, an efficient MT forward modeling method based on the staggered grid finite difference method (FDM) is proposed by combining GMG based on the four-color cell-block GS smoother and BiCGstab. In this method, GMG based on the four-color cell-block GS smoother is used as a preconditioner for the BiCGstab solver to significantly improve the forward modeling efficiency. We designed a layered resistivity model to verify the accuracy of the proposed algorithm by comparing with its analytical solution. Then, a double anomaly body resistivity model and a Dublin Model 1 (DTM1) were designed. Based on BiCGstab, the numerical performance of the GMG preconditioner was compared with other traditional preconditioners, such as the symmetric successive over relaxation (SSOR), block incomplete LU (ILU) factorization and the GS. The results show that the algorithm proposed in this paper is much better than the traditional preconditioners in terms of iteration number, compute time and stability. In all examples, the GMG preconditioner achieves convergence within 10 times, and the compute time is reduced by more than 70% compared with the traditional preconditioners, indicating the stability and high efficiency of the proposed method.