摘要
区域分解算法作为求解大规模科学与工程问题的一种有效计算手段,已经在地球物理电磁法领域取得了一定的应用,但影响区域分解算法计算效率的因素复杂,前人的研究缺乏对计算效率影响因素的系统讨论和研究。将目前应用广泛的平衡区域分解算法引入到直流电阻率三维正演中,首先对三维模型进行有限差分离散得到线性方程组,然后将求解区域分解为多个不重叠的子域,使用Schur补偿算法将线性方程解耦为子域和共享边界的方程,最后对边界方程进行平衡预处理,并对子域和共享边界的方程进行迭代求解,实现了直流电阻率三维正演,通过与2层水平介质模型的解析解对比验证了算法的准确性和可行性。着重对影响平衡区域分解算法计算效率的因素进行了讨论分析,结果表明子域数目、子域问题和边界方程的解法以及网格大小都会对计算效率产生不同程度的影响。平衡区域分解算法的计算速度随子域数目先减小后增大,随网格增大呈指数增加。采用的3种子域问题和边界方程的解法中,预处理共轭梯度法效率最高,稳定双共轭梯度法次之,最速下降法效率最低。
As a powerful means to solve large-scale scientific and engineering problems, domain decomposition method has been applied in the field of geophysical electromagnetic to some extent. However, the factors affecting the computational efficiency of domain decomposition method are complex, and previous studies lack of discussion on the factors affecting the computational efficiency. In this paper, the widely used balanced domain decomposition method was introduced into the three-dimensional forward modeling of DC resistivity. Firstly, the three-dimensional model was discretized by finite difference method to obtain a set of linear equations. Then, the solution domain was decomposed into several non-overlapping sub-domains. The Schur compensation algorithm was used to decouple the linear equations into sub-domains and shared boundary equations. Finally, the boundary equation was balanced preconditioned, the equations of sub-domains and shared boundary were solved iteratively, and the three-dimensional forward modeling of DC resistivity is realized. The accuracy and feasibility of the proposed algorithm were verified by comparing with the analytical solutions of the two-layer horizontal medium model. In this paper, the factors affecting the computational efficiency of the balanced domain decomposition method were discussed and analyzed. The results show that the number of subdomains, the solution of subdomain and boundary problems, and the mesh sizes will have different effects on the computational efficiency. The computational speed of the balanced domain decomposition method decreases first and then increases with the number of sub-domains, and increases exponentially with the increase of meshes. Among three kinds of iteration methods for subdomain and boundary problems adopted in this paper, PCG has the highest efficiency, BICG takes the second place, Steepest Descent method has the lowest efficiency.
作者
赵航
ZHAO Hang(School of Geophysics and Mesurement-Control Technology,East China University of Technology, 330013, Nanchang, PRC)
出处
《江西科学》
2019年第2期170-174,245,共6页
Jiangxi Science
关键词
直流电阻率法
平衡区域分解
Schur补偿算法
正演模拟
direct current resistivity
balanced domain decomposition method
Schur complement of matrix
forward modeling
