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.

Introduction

A key component in the design of high-end microprocessors is the estimation of signal delay of the VLSI circuit. The signal delay depends on several factors including parasitic resistance, capacitance, and inductance due to the on-chip interconnect. At higher frequencies, it is critical to estimate the effect of the inductive coupling between the interconnect segments quickly and accurately.

This paper presents a class of parallel algorithms for solving the linear systems of equations that arise in inductance extraction of VLSI circuits. To obtain the parasitic impedance at a particular frequency, the interconnect segments are discretized by a uniform mesh of filaments. Each filament carries an unknown amount of current which depends on the potential difference between the end nodes of the filament. The net flow of current into any node must be zero. These conditions give rise to linear systems with large coefficient matrices that have both sparse and dense submatrices. The sparse submatrices represent the constraints on current flow at each node. The dense submatrices represent the potential drop across each filament due to the inductive effect of current in all other filaments as well as due to its own resistance.

For accurate impedance estimation, discretizations with millions of filaments may be necessary. Since the computation and storage of the associated dense submatrix is not feasible, one must use iterative methods that rely on the ability to compute matrix–vector products with the coefficient matrix. The nature of the inductive coupling between filaments permits use of hierarchical multipole-based techniques such as the fast multipole method (FMM) [3], [8] to compute approximations to the matrix–vector products. The non-availability of the coefficient matrix makes it difficult to construct preconditioners that can be used to increase the rate of convergence of the iterative methods. The constraints imposed on the flow of current leads to an indefinite coefficient matrix, further complicating efforts to design robust preconditioners.

The approach presented in the paper uses a solenoidal basis method to restrict the current to the subspace where Kirchoff’s current law is satisfied, and solves the resulting reduced system by a preconditioned iterative method. The preconditioners are constructed from the inductive coupling of solenoidal functions of the basis, and the preconditioning step is implemented using hierarchical multipole-based approximations. The description of the inductance problem and the solution methodology in 2 Inductance extraction problem, 3 A preconditioned iterative solver follows the presentation in [7]. Section 4 describes a parallelization scheme for the hierarchical approximation algorithms used to compute matrix–vector products with the dense coefficient submatrix as well as the preconditioner. Benchmark experiments presented in Section 5 demonstrate that the algorithm can achieve high parallel efficiency on shared-memory multiprocessors. Concluding remarks are presented in Section 6.