In this paper we discuss the parallel implementation of the 3-D electromagnetic forward modeling algorithm, based on the contraction integral equation (CIE) method. We use a Linux Beowulf Class Cluster available at the Center for High Performance Computing (CHPC) at the University of Utah. We have developed a working version of the 3-D CIE code based on the Complex Generalized Minimal Residual (CGMRM) method of solving the system of linear equations. This new parallel implementation of the CGMRM algorithm ensures the convergence of the corresponding matrix inversion and allows us to consider large-scale forward modeling problems. We have investigated the solution of a large-scale forward problem with different discretizations on a PC cluster of up to 32 nodes.
There are several examples of successful realization of EM forward modeling methods using parallel computing (Alumbaugh and Newman, 1997; Newman and Alumbaugh, 1999; Wilson et al., 1999; Xie et al., 2000). However, until recently mostly the finite difference (FD) and the finite element (FE) algorithms were implemented using parallel computing. This can be explained by the fact that the parallelization of the FD and FE solutions are relatively straightforward and it is a well developed area of computer science. The integral equation (IE) forward modeling method represents a more challenging problem than FD and FE modeling even in the case of using a single PC. The parallelization of the IE codes is an extremely difficult task. At the same time, we should notice that the IE approach has one principal advantage over FD and FE methods: the IE method requires the discretization of the anomalous domain only, while the FD and FE methods need a huge grid covering the entire modeling domain. Hursan and Zhdanov, 2002 developed an efficient IE modeling software, INTEM3D, based on the Contraction Integral Equation (CIE) method. This is a powerful code which allows us to use a fine discretization grid to describe the 3-D inhomogeneous conductivity model. However, due to the standard PC memory limitations, the INTEM3D code can operate with a discretization grid whose number of cells does not exceed 100,000. We have developed a parallel IE solution using a PC cluster system available at the Center for High Performance Computing (CHPC) at the University of Utah which provides high-end computer services to advanced research programs in computational sciences and simulations. We use the Complex Generalized Minimal Residual (CGMRM) method (Zhdanov, 2002) as a solver of the system of linear equations arising in the CIE method. This new parallel implementation of the CGMRM algorithm ensures the convergence of the corresponding matrix inversion for arbitrary conductivity models. As an example, we present a solution of a large-scale forward problem with different discretizations up to a quarter million cells on a PC cluster of up to 32 nodes.
In order to test the new parallel version of the CIE forward modeling code, we used the complex synthetic geoelectrical model obtained as a result of inversion of 3-D MT data set.
