An efficient modified hierarchical domain decomposition for two-dimensional magnetotelluric forward modelling

Abstract

We use 2-D magnetotelluric (MT) problems as a feasibility study to demonstrate that 3-D MT problems can be solved with a direct solver, even on a standard single processor PC. The scheme used is a hierarchical domain decomposition (HDD) method in which a global computational domain is uniformly split into many smaller non-overlapping subdomains. To make it more efficient, two modifications are made to the standard HDD method. Instead of three levels as in the standard HDD method, we classify the unknowns into four classes: the interiors, the horizontal interfaces, the vertical interfaces and the intersections. Four sets of smaller systems of equations are successively solved with a direct method (an LU factorization). The separation significantly reduces the large memory requirements of a direct solver. It also reduces the CPU time to almost half that of the standard HDD method although it is still slower than the conventional finite difference (FD) method. To further enhance the speed of the code, a red-black ordering is applied to solve the horizontal and vertical interface reduced systems. Numerical experiments on a 2-D MT problem of a given size running on a single processor machine shows that CPU time and memory used are almost constant for any resistivity models, frequencies and modes. This is a clear advantage of our algorithm and is of particular importance if the method is applied to 3-D problems. We show that our new method results in reductions in both memory usage and CPU time for large enough domains when compared to the standard FD and HDD schemes. In addition, we also introduce a ‘memory minimization map’, a graphical tool we can use instead of trial-and-error to pre-select the optimal size of subdomains, which yield the best performance in both CPU time and memory even running on a serial machine.