Fast VIE-DDM for 3-D Electromagnetic Scattering of Complicated Anomalies in Layered Media Using Adaptive Cross Approximation

Simulations of 3-D electromagnetic (EM) scattering from complicated anomalies in a layered medium are important in geophysical applications. The volume integral equation (VIE) method, specifically its fast solvers such as fast Fourier transform (FFT)-based methods, has been widely applied to tackle these problems. Recently, we proposed to combine the domain decomposition method (DDM) with VIE to solve the problems with multiple objects in a layered medium (VIE-DDM). This method leveraged 3-D and 2-D FFTs to accelerate the self- and mutual-coupling matrix-vector multiplications (MVMs) in the stabilized-biconjugate gradient (BCGS) method when solving the linear system derived from the VIE-DDM. Though significantly extending the applicability and improving the efficiency of VIE in some scenarios, the BCGS-FFT-DDM (or VIE-DDM-FFT) has the limitations of requiring conformal and uniform meshes in the
$xy$
plane and not allowing dense meshes in the
$z$
-direction. It turned out both the limitations come from the mutual-coupling matrices. In this article, we propose to exploit the rank-deficient property of the mutual-coupling matrices relating well-separated objects, apply the adaptive cross approximation (ACA) method to find the low-rank factors of these matrices, and then achieve the MVMs using the low-rank factors. The ACA is purely algebraic and only requires a few rows and columns from the original matrix, which can be easily obtained by computing Green’s functions, to find the low-rank factor matrices. With this new method, both the storage and computation complexities become linear. These linear complexities successfully avoid the bottleneck of dense
$z$
-direction meshes in the BCGS-FFT-DDM. Furthermore, the new method does not strictly require conformal and uniform meshes in any direction for different objects, making it more flexible in treating multiscale problems. We tested the performance of this method in two typical scenarios and compared the results with the BCGS-FFT-DDM and commercial platforms. The results show that the new method achieves significant improvement in memory and CPU time cost and has more flexibility in treating multiscale problems.