大规模三维大地电磁各向异性正演的非均匀网格外推多重网格法

大地电磁正演线性方程组求解的主要加速手段有并行技术和多重网格技术.传统的几何多重网格（GMG）方法依赖于均匀嵌套的正交网格，处理带跃变系数的问题时存在一定缺陷，且对各向异性问题需采用半粗化、线/面磨光等策略特殊处理，限制了GMG方法的应用.本文提出一种基于非均匀网格的外推瀑布式多重网格方法（EXCMG），快速求解三维大地电磁各向异性有限元正演形成的大型复线性方程组.首先，对库仑规范下电磁耦合势满足的向量Helmholtz方程，由Galerkin加权余量法推导有限元离散系统，得到大规模、稀疏、复线性方程组.然后，从最密的非均匀正交网格出发，逐层粗化得到一系列嵌套网格；借助Richardson外推及Lagrange二次插值技术，设计非均匀正交网格下全新的多网格延拓算子；利用前两层网格上的数值解，构造下层密网上有限元解的高精度逼近，作为多网格磨光算子——复数域不完全LU分解预处理稳定双共轭梯度算法（BiCGStab）的迭代初值，加速收敛.最后，通过典型地电模型验证算法的精度和效率.数值实验表明，本文提出的算法适用于较宽频带，加速效果明显，相较于预条件BiCGStab方法，求解效率提升数十倍.本文算法能够求解上亿自由度的超大规模、强各向异性问题，且求解问题的规模越大，算法效率优势越明显.EXCMG算法有望在其他地球物理领域得到应用.

Large-scale three-dimensional magnetotelluric forward modeling in anisotropic media using an extrapolation multigrid algorithm on non-uniform grids

In solving large-scale linear equations arising from three-dimensional (3-D) magnetotelluric (MT) forward modeling, parallelization techniques and multigrid methods are two of the widely-used acceleration approaches. Traditional geometric multigrid methods (GMG) depend on nested orthogonal grids and exhibit disadvantages when solving problems with jumping coefficients. For anisotropic problems, some special strategies like semi-coarsening and line/face smoothing are required, which limits the applications of such methods. Therefore, in this paper, an extrapolation cascadic multi-grid method (EXCMG) based on non-uniform hexahedral grids is proposed, and applied to accelerate the solution of large complex linear system sarising from 3-D MT anisotropic finite-element (FE) forward modeling. First, for the vector Helmholtz equation using Coulomb-gauged electromagnetic potentials a large sparse complex linear system can be derived with Galerkin weighted residual method. Then, a series of nested grids is generated through coarsening a fine nonuniform orthogonal grid level by level. Next, Richardson extrapolation and Lagrange's quadratic interpolation are employed to construct a new prolongation operator for nonuniform orthogonal grids. In the implementation of EXCMG, the highly accurate approximation to the FE solution on the next finer grid is constructed with the numerical solutions of the two coarse grids, and used as good initial guess for the multigrid smoother: Stable bi-conjugate gradient method (BiCGStab) with incomplete LU (ILU) preconditioner, to accelerate its convergence. Finally, the accuracy and efficiency of the algorithm are validated through several typical geoelectric models. Numerical experiments demonstrate that the proposed algorithm is appropriate for a wide frequency range, and the acceleration effect is obvious. Compared with preconditioned BiCGStab, the solving efficiency can be improved by dozens of times. It has been validated that our algorithm is capable of handling problems with strong anisotropy, as well as large-scale problems with more than 0.1 billion unknowns. The advantage of this algorithm is more evident for larger scale problems, and it is promising to apply EXCMG to other geophysical problems.