基于并行化直接解法的频率域可控源电磁三维正演

电磁法的三维数值模拟是一个对数值算法和计算机硬件要求都非常高的问题.对常用的微分类方法如有限单元法和有限差分法而言,求解最后所得的大型线性方程组是至关重要的一步,直接影响到正演算法的实用性.如何高效、稳定且准确地解线性方程长期以来一直是被探讨的问题.本文实现了基于线性系统直接求解技术的频率域可控源电磁(CSEM)三维正演.使用交错网格有限体积法(FV)来离散化关于二次电场的Helmholtz方程;使用直接解法取代传统的迭代解法来求解离散线性系统,即对系统矩阵进行完全LU分解,具体通过调用大规模并行矩阵直接求解器(MUMPS)来实现.基于理论模型做了一系列数值实验,首先证明了直接解法的高精度和稳定性,并考察了其内存需求、计算时间和并行可伸缩性等主要计算性能,最后检验了所开发的算法快速模拟多场源CSEM问题的能力以及对常规海洋和陆地CSEM模拟的有效性.

3-D frequency-domain CSEM modeling using a parallel direct solver

Three-dimensional modeling of electromagnetic data is a computationally demanding problem. For frequently-used numerical techniques such as finite-element and finite-difference methods, solving the large linear systems arising from the discretization of Maxwell's equations is a key step which has a major impact on the applicability of the solution, and it has always been a research topic to solve the linear equations efficiently, robustly and accurately. A 3D modeling scheme based on direct solutions of the linear system is presented for frequency domain controlled-source electromagnetic(CSEM)surveys. The Helmholtz equation in terms of secondary electric fields is discretized using a finite-volume(FV)method over a staggered grid. Taking advantage of recent developments in numerical algorithms and the availability of computational resources, the resulting linear system of FV equations is solved directly using the massively parallel solver, namely MUMPS, instead of the most commonly used linear solvers, i.e. Krylov subspace iterative techniques. The direct solver carries out an LU(and possibly LDL^T)decomposition of the system matrix and then computes solutions efficiently by applying forward and backward substitutions.To evaluate the computational performance of the direct solver, a series of numerical tests based on synthetic 1D models were conducted, and the results indicate that(1)Normalized residuals of solutions are almost independent of the conductivity value assigned to air layers but increase rapidly as the frequency value decreases. Nevertheless, the order of magnitude of the largest normalized residual is as small as 10^-11. At the same time, although the matrix factorization time varies as either the air conductivity or the frequency changes, the variation is only a fraction of the total run time.(2)Both the execution time and required memory increase rapidly(more than linearly)with increasing grid sizes.(3)By executing MUMPS in parallel over multiple processors, not only the total run time but also the average memory used per processor can be reduced a lot. However, the total memory requirement increases with the number of processes. The scalability of MUMPS is limited. Additional numerical experiments considering specific survey settings were done to demonstrate the reliability and effectiveness of the code, and the results are as follows:(1)The FV numerical solutions show excellent agreement with semi-analytic solutions for the 1D models.(2)The computation time of a multitransmitter problem is comparable to that of a single-transmitter problem.(3)Reasonable modeling results of 3D models can be obtained for both typical land and marine survey scenarios.In summary, compared with iterative linear solvers, the direct solvers generally benefit CSEM modeling in three aspects. The first is they often provide more accurate solutions. The second is that direct solvers are much more stable for ill-conditioned linear systems, which are almost inevitable because of large electrical conductivity contrasts and/or non-uniform grid. The last is in multitransmitter problems, only a single matrix factorization is necessary, and multiple solutions can be achieved very easily by reusing the factors. The presented 3D CSEM modeling scheme, which employs the MUMPS direct solver, possesses all these advantages. In addition, solving linear systems can be executed in parallel to speed up the computation and to reduce the average memory used per node, although the parallel scalability of MUMPS is limited. In spite of the fact that matrix factorizations for large models can entail tremendous computational cost, it can be anticipated that direct solvers will be used more and more widely as the development of both numerical algorithms and computers.