三维电阻率正演计算中的Lanczos迭代算法

Lanczos's iteration glgorithm in 3-D forward calculations of resistivity

在三维电阻率的正反演计算中,快速、准确的正演计算是反演的关键。而正演计算往往涉及到求解大型线性方程组Ax=b的问题,通过Lanczos迭代构造出对称三对角阵方程组,并采用正交分解法进行求解,与传统算法相比,此算法占用内存少、收敛速度快、且稳定;针对大型稀疏矩阵的特点,采用简单地记录矩阵的非零元素值及其所在行、列值的方法,来存储大型稀疏矩阵,可大大节省机器内存,提高运算速度。通过理论分析和点电源三维地电场计算实例,阐述该法是地电三维正演计算的有效方法。

For 3-D resistivity field forward modeling and inversion , it is successful key that large linear equations Ax=b is solved rapidly and exactly. In this paper, a symmetric tridiagonal system of equations is formed by Lanczos algorithm, simultaneously solving such equations is explained by using orthogonal decomposition method, this algorithm has the advantage of fast convergence, stability and less memory. Aimed at the character of large sparse matrice, the paper saves large sparse matrice by the method of only recording values of nonzero elment and its row and column, thereby it requires less memory of computer, and bumps operation velocity up. Theory analyses and computation examples of 3-D geoelectric field show Lanczos algorithm is a efficacious method of 3-D geoelectric field forward modeling.