Parallel iterative methods for dense linear systems in inductance extraction

Abstract

Accurate estimation of the inductive coupling between interconnect segments of a VLSI circuit is critical to the design of high-end microprocessors. This paper presents a class of parallel iterative methods for solving the linear systems of equations that arise in the inductance extraction process. The coefficient matrices are made up of dense and sparse submatrices where the dense structure is due to the inductive coupling between current filaments and the sparse structure is due to Kirchoff’s constraints on current. By using a solenoidal basis technique to represent current, the problem is transformed to an unconstrained one that is subsequently solved by an iterative method. A dense preconditioner resembling the inductive coupling matrix is used to increase the rate of convergence of the iterative method. Multipole-based hierarchical approximations are used to compute products with the dense coefficient matrix as well as the preconditioner. A parallel formulation of the preconditoned iterative solver is outlined along with parallelization schemes for the hierarchical approximations. A variety of experiments is presented to show the parallel efficiency of the algorithms on shared-memory multiprocessors.