Oct-tree-based multilevel low-rank decomposition algorithm for rapid 3-D parasitic extraction

Fast parasitic extraction is an integral part of high-speed microelectronic simulation at the package and on-chip level. Integral equation methods and related fast solvers for the iterative solution of the resulting dense matrix systems have enabled linear time complexity and memory usage. However, these methods tend to have large disparities between setup and matrix-vector product times that affect their efficiency when applied to multiple excitation problems, i.e., problems with a large number of nets. For example, FastCap, which is based on the fast multipole method, has a significantly faster setup time than the multilevel QR decomposition-based IES³, but relatively slow matrix-vector products. In this paper, we present a novel oct-tree-based QR compression technique for fast iterative solution. The regular cube structure of the fast multipole method and the QR compression scheme for interaction submatrices as in IES³ are combined to achieve a predetermined compressible matrix-block structure and, consequently, superior memory, setup, and solve time efficiencies.