Compact and stable modeling of partial inductance and reluctance matrices

The sparsification of the reluctance matrix L^-1 (where L denotes the usual inductance matrix L) has been widely used in several recent investigations to make the problem of simulation of interconnects tractable. Although these sparsification techniques work well in practice, the stability of these approximations has not been established, i.e., the sparsified reluctance and inductance matrices are not guaranteed to be positive-definite. In this work, we propose a band matching method that enjoys two advantages: First, we exploit the elegant structure of the inverse of banded matrices so as to construct an approximate inductance matrix L^~ whose band entries match the band entries of original L, and whose inverse is a banded matrix. This approach yields a compact representation of both inductance and reluctance matrices. Second, we establish that the compact approximant L^~ is guaranteed to be positive-definite. Simulation results show that our approach enjoys an approximation accuracy that is comparable to that of existing methods.