基于几何多重网格和无矩阵方法的三维大地电磁法自适应有限元正演研究

Magnetotelluric is one of the most widely used frequency domain electromagnetic methods in practice. Improving the speed and accuracy of three-dimensional forward modeling method can effectively improve the speed and effect of three-dimensional inversion, thus enhancing the application ability of the method in practice. In three-dimensional adaptive finite element forward modeling of the magnetotelluric method, the linear equations are difficult to solve and the calculation speed of the matrix-vector product is limited by the memory bandwidth. To resolve these limitations, we study the iterative solver of the linear equations based on the geometric multigrid preconditioner. At the same time, the matrix-vector production is accelerated by the matrix-free technique. The algorithm is parallelized by the distributed mesh partition technique. On this basis, the efficiency and robustness of the iterative algorithm and preconditioner and the scalability of the parallel algorithm are analyzed. Finally, the response of the complex model and the topographic influence of the magnetotelluric method are discussed. The research on the above problems can effectively improve the accuracy, speed, and adaptability of the three-dimensional forward modeling of the magnetotelluric method, and lay a foundation for further practical application of three-dimensional inversion.

大地电磁法是实际中应用最为广泛的频率域电磁勘探方法之一，提高大地电磁法三维正演的计算速度和精度可有效提高三维反演的速度和效果，从而增强方法在实际中的应用能力。本项目针对大地电磁法三维自适应有限元正演中存在的线性方程组求解困难和矩阵向量乘积计算速度受限于内存带宽等问题，研究了基于几何多重网格预条件算法的线性方程组的迭代解法，同时通过无矩阵技术加速迭代算法计算过程中矩阵向量乘积的计算速度，并基于分布式网格分区技术实现了算法的并行化。在此基础之上，重点分析迭代算法和预条件的效率、鲁棒性以及并行算法的可扩展性，并对复杂模型的响应和大地电磁法的地形影响进行讨论。对上述问题的研究可有效提高大地电磁法三维正演的计算精度、速度和适应性，为进一步实现基于有限元正演的三维反演实用化奠定基础。

大地电磁法是实际中使用最为广泛的频率域电磁勘探方法之一，三维正反演巨大的计算量是限制其在实际中应用的主要因素之一。本项目围绕高精度和高性能的大地电磁法三维自适应正演算法开展研究。针对大地电磁法三维正演中复杂模型模拟中网格设置较为困难的问题，研究了面向目标的自适应网格加密算法。针对线性方程组求解困难的问题，研究了两种迭代求解算法：分块对角预条件和几何多重网格预条件。在并行算法方面，基于分布式网格分区技术实现了正演算法的并行化，同时针对矩阵向量乘积计算速度受限于内存带宽等问题，通过无矩阵技术加速迭代算法计算过程中矩阵向量乘积的计算速度。通过数值算例对上述算法的性能进行了测试，模拟结果表明：面向目标自适应正演算法具有较强的适应性，对于非常复杂的地电模型也能有效地进行局部网格加密，提高正演计算的精度和效率；两种预条件算法均能在很少的迭代次数内收敛，且迭代次数与电阻率、频率和问题规模无关；并行算法能够大大加快正演的计算速度，达到接近线性加速的效果。本项目研究的正演算法已经应用在三维反演中，有望进一步应用于与实际的电磁勘探中。