基于GPU的CSAMT三维正演的并行外推多网格法研究

可控源音频大地电磁法(CSAMT)是近年发展起来的、很有前景的地球物理勘探方法之一，在矿产及油气勘探、地热勘查和水文工程地质勘查等领域得到了广泛应用。多重网格法以其高效率正逐步应用到电磁场数值模拟中。本项目拟开展基于GPU的CSAMT三维正演的并行外推瀑布式多网格法(EXCMG)研究。对Coulomb规范下磁矢势和电标势满足的耦合方程组，导出其线性有限元解的渐近展开式，由此构造不同嵌套网格下的Richardson外推公式，结合局部加密等级网格和双共轭梯度法等技巧，将EXCMG法应用到具有奇性(电偶源)的三维、复数域CSAMT正演问题。以此为基础，研究该方法的GPU并行方案，在高性能计算服务器上实现上亿自由度的三维CSAMT法快速、高精度正演。通过典型地电模型测试并改进算法，为方法的进一步应用奠定基础。本项目的研究对促进勘探电磁法正反演的快速发展，提高资料解释水平，具有重要的理论和实际意义。

Controlled source audio magnetotelluric (CSAMT) method is a promising geophysical method developed in recent years, which is commonly used for mineral and petroleum exploration, geothermal exploration and other hydrological, engineering geological exploration. Because of its high-efficiency, multigrid method has been gradually applied to the numerical simulation of electromagnetic fields. The project will focus on a GPU-based parallel extrapolation cascadic multigrid (EXCMG) method which is used for three-dimensional CSAMT forward modeling. First, we will derive asymptotic expansions of linear finite element solutions to the coupled equations in terms of the magnetic vector potential and electric scalar potential of the Coulomb gauge. Next we will construct Richardson extrapolation formulas for different nested grids, then combine with the locally refined meth and bi-conjugate gradient method and other techniques. We finally apply EXCMG to three-dimensional CSAMT forward problems with singularities (electric dipole) in complex field. Based on the above results, a GPU-based parallel implementation of the EXCMG algorithm will be developed. High-precision numerical simulation of CSAMT data with hundreds of millions of degrees of freedom can also be implemented rapidly on high performance computing server. Through a large number of numerical simulations, we can analyze and improve the algorithm, which lays the foundation for the further application of the method. The expected results of the project have an important theoretical and practical value for promoting the rapid development of forward and inverse electromagnetic modeling and improving the level of data interpretation for geophysical exploration.