Hierarchical Matrix Techniques Based on Matrix Decomposition Algorithm for the Fast Analysis of Planar Layered Structures

The matrix decomposition algorithm (MDA) provides an efficient matrix-vector product for the iterative solution of the integral equation (IE) by a blockwise compression of the impedance matrix. The MDA with a singular value decomposition (SVD) recompression scheme, i.e., so-called MDA-SVD method, shows strong ability for the analysis of planar layered structures. However, iterative solution faces the problem of convergence rate. An efficient hierarchical (
H
-) LU decomposition algorithm based on the
H
-matrix techniques is proposed to handle this problem. Exploiting the data-sparse representation of the MDA-SVD compressed impedance matrix,
H
-LU decomposition can be efficiently implemented by
H
-matrix arithmetic.
H
-matrix techniques provide a flexible way to control the accuracy of the approximate
H
-LU-factors.
H
-LU decomposition with low accuracy can be used as an efficient preconditioner for the iterative solver due to its low computational cost, while
H
-LU decomposition with high accuracy can be used as a direct solver for dealing with multiple right-hand-side (RHS) vector problems particularly. Numerical examples demonstrate that the proposed method is very robust for the analysis of various planar layered structures.