基于块状有理Krylov方法的大地电磁三维模型降阶正演

三维大地电磁正演需要求解两个极化源在不同频率下的电磁场分布，计算量巨大。基于块状有理Krylov方法，文中实现了大地电磁三维模型降阶快速正演计算。所用算法的创新点包括：①将大地电磁的源项显式表示为平面电流源，将随频率变化的电场响应表示为一个传递函数与电流源常矢量的乘积，从而可通过构建有理Krylov子空间实现所有频率电场响应的快速求解，避免多次求解不同频率的大型稀疏线性方程组；②采用块状Krylov技术，将TE和TM极化源表示为块状源矢量，将求解两个极化源的正演响应简化为构建一个块状有理Krylov子空间。引入渐近收敛公式得到了块状有理Krylov方法的最优化单个重复极点，结合直接求解器，将大地电磁三维正演的计算量降为一次系数矩阵分解和几十次矩阵回代。该算法在保证正演精度的同时，极大地提高了正演速度。半空间模型和三维DTM1模型的正演数值结果表明，相比常规的逐个频率的正演求解方法，块状有理Krylov方法可显著提高正演速度。

Three-dimensional magnetotelluric forward modeling by order reduction based on block rational Krylov method

Three-dimensional (3D) magnetotelluric (MT) forward modeling requires solutions to electromagnetic field distribution of two polarization sources at several frequencies, thus leading to enormous computational costs. This paper accele-rates 3D MT forward modeling by order reduction based on block rational Krylov method. The novelty of this algorithm lies in the following aspects. First, the source-term explicit expression of MT is represented as planar current source, and the frequency-dependent electric field response is a produ-ct of a transfer function and a constant vector of the current source. As a result, the rapid solution of electric field response at all frequencies is reali-zed through the construction of a rational Krylov subspace, which avoids repeated solutions of large sparse linear equations with different frequencies. Second, the paper adopts the block Krylov technique to express polarizations of TE and TM as block source vectors and simplifies forward mode-ling response of the two polarizations into the construction of a block rational Krylov subspace. Additionally, an asymptotic convergence formula is introduced to obtain the optimal single repeated polarization of the Krylov method. Combined with direct solver, the forward modeling computational cost of 3D MT is reduced to a coefficient matrix decomposition and dozens of matrix back substitutions. This algorithm ensures the forward mode-ling accuracy and significantly improves the forward modeling speed. Numerical results of forward modeling in half space model and 3D DTM1 model show that compared with the conventional frequency-dependent forward modeling solutions, block rational Krylov can notably increase the modeling speed.