LANCZOS迭代算法及其在三维地电场数值模拟计算中的应用研究

LANCZOS METHOD AND ITS APPLICATION RESEARCH INTHE NUMERICAL MODELING OF 3D GEOELECTRIC FIELD

在三维电阻率的正演计算中往往涉及到快速、准确求解大型线性方程组Ax=b的问题。通过采用有限差分法来构造出求解点电源三维地电场的大型稀疏对称线性方程组,并引入Lanczos迭代技术,构造出三对角阵方程组,然后采用正交分解法进行求解,它是Krylov子空间方法中的一种。与传统迭代算法相比,它占用内存少,收敛速度快且稳定。针对大型稀疏矩阵及MATLAB语言的特点,采用简单记录矩阵的非零元素值及其所在行、列值的方法存储大型稀疏矩阵,可大大节省机器内存,提高运算速度。理论分析和计算实例显示,此算法是地电三维正演计算的有效方法,为下一步的反演计算打好基础。

For 3-D geoelectric field forward modelling, it is successful key that large linear equations Ax=b is solved rapidly and precisely. In this paper, a large sparse symmetric linear equations of solving 3D geoelectric field of point source is formed by finite difference method and simultaneously the theory and alternative course of Lanczos algorithm which is suitable to solving such equations are expounded, which is a Krylov subspace methods and has the advantages of fast convergence, stability and less memory. Aimed at the features of large sparse matrix and MATLAB language, the paper saves large sparse matrix by the method of only recording values of nonzero element and its row and column information, so that the both of the memory and speed of the computing are reduced greatly. Theory analysis and computation examples of 3D geoelectric field show that Lanczos algorithm is a effictive method of 3-D geoelectric field forward modeling, which makes a firm foundation for the following 3D resistivity inversion.