Application of Randomized Matrix Approximation in 3-D Volume Integral Equation Domain Decomposition Method for Electromagnetic Scattering in Layered Media With Complex Objects

Electromagnetic (EM) scattering of complex 3-D distributions in a layered-medium background is crucial in many remote sensing and geophysical applications, including ground-penetrating radar, microwave tomography, and atmospheric EM analysis. The Volume Integral Equation (VIE) method offers distinct advantages in simulating such problems. In particular, the recently proposed VIE combined with the Domain Decomposition Method (DDM) and accelerated by the Fast Fourier Transform (VIE-DDM-FFT) has shown promise. However, limitations in handling dense meshes along the z-axis led to the development of a low-rank approximation (LRA) algorithm, specifically the Adaptive Cross Approximation (ACA), for mutual matrix-vector multiplications (VIE-DDM-ACA). Despite reducing computational complexity, ACA suffers from premature, unstable, and unpredictable convergence, particularly when dealing with closely spaced objects, resulting in inaccuracies and inefficiencies. In this paper, we propose utilizing the Randomized Matrix Approximation (RMA) method for LRA. RMA provides superior low-rank factorization by requiring fewer ranks and exhibiting more stable and predictable convergence, making it particularly suitable for reconstructing mutual matrices in remote sensing scenarios involving closely spaced scatterers, such as subsurface imaging and layered-medium inversion problems. While RMA involves multiple matrix-vector multiplications (MVMs) during the preparation stage, these MVMs are independent and can be parallelized, mitigating computational costs. Additionally, the iterative MVMs in the equation-solving procedure retain linear complexity with a smaller coefficient compared to ACA due to the reduced rank requirements. Our results demonstrate that the RMA method outperforms ACA in terms of low-rank factorization and accuracy, making VIE-DDM-RMA more efficient than VIE-DDM-FFT and more accurate than VIE-DDM-ACA for EM scattering in remote sensing and geoscience applications involving close objects and complex scattering environments in layered media.