The progress of integral-equation based fast direct solvers for electromagnetic scattering analysis

This paper reviews a series of integral-equation based fast direct solvers for electromagnetic scattering analysis. The main contribution of fast direct solver is to alleviate the slow-convergence issue of iterative solver or even avoid it. This would provide a fast and stable numerical solution for the electromagnetic integral equations. The paper introduces the advantages and applications of fast direct solvers and the research review of the past few years in detail. The main part of the paper focuses on two different methods, namely hierarchical matrix and hierarchical off-diagonal low-rank matrix, including the construction of the compressed impedance matrix and the factorization of the inverse matrix. The advantages and disadvantages of each method are discussed in further. The effectiveness and accuracy are demonstrated through several numerical results which contain the time and memory complexities and scattering of complex airplane model. Finally, a concise discussion and explorations are drawn to the present challenges and possible remedy strategy for electromagnetic integral-equation based fast direct solvers.