Fast ${cal H}$ -Matrix-Based Direct Integral Equation Solver With Reduced Computational Cost for Large-Scale Interconnect Extraction

摘要

In this paper, we propose a fast ${cal H}$-matrix-based direct solution with a significantly reduced computational cost for an integral-equation-based capacitance extraction of large-scale 3-D interconnects in multiple dielectrics. We reduce the computational cost of an ${cal H}$ -matrix-based computation by simultaneously optimizing the ${cal H}$ -matrix partition to minimize the number of matrix blocks and minimizing the rank of each matrix block based on a prescribed accuracy. With the proposed cost-reduction method, we develop a fast LU-based direct solver. This solver possesses a complexity of $kC_{rm sp}O (N{bf log}N)$ in storage, a complexity of $k^{2}C_{{rm sp}}^{2}O(N{log}^{2}{rm N})$ in LU factorization, and a complexity of $kC_{{rm sp}}O(Nlog {rm N})$ in LU solution, where $k$ is the maximal rank, $C_{{rm sp}}$ is a constant dependent on matrix partition, and the constant $kC_{{rm sp}}$ is minimized based on accuracy by the proposed cost-reduction method. The proposed solver successfully factorizes dense matrices that involve millions of unknowns in fast CPU time and modest memory consumption, and with the prescribed accuracy satisfied. As an algebraic method, the underlying fast technique is kernel independent.