Abstract
Hierarchical (H-) matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE-) based computational electromagnetics,H-matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solveH-matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure ofH-matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI) preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solvingH-matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.
Highlights
Integral equation (IE) method [1] is widely used in electromagnetic analysis and simulation
With the efforts for years, various fast algorithms had developed focusing on reducing the computational complexity for IE method such as adaptive integral method (AIM) [4], fast multipole method (FMM) [5], IEFast Fourier transform (IE-FFT) [6], and fast low-rank compression methods [7], Though these fast algorithms are based on different theories, they use the same idea to reduce CPU time and memory usage complexity, which is to compute and store the major entries of the dense system matrix indirectly, and employ iterative methods instead of direct methods to solve the system
The hierarchical matrice methods presented in this paper is embedded in electromagnetic IE method
Summary
Integral equation (IE) method [1] is widely used in electromagnetic analysis and simulation. Iterative methods cannot always guarantee a reasonable solution with high precision; in some complex cases, we expect to employ powerful preconditioners to obtain visible acceleration of convergence or even use direct methods to avoid this problem entirely To implement these ideas in traditional fast algorithms encounters some difficulties that are, once a fast algorithm is applied, the major entries of the system matrix are computed and stored indirectly, and only a few entries can be accessed to form the sparse pattern which is essential to construct the preconditioners. A novel sparse approximate inverse (SAI) preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving H matrices system in electromagnetic applications, which is mentioned in [10], and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases.
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