Efficient modeling and analysis for large scale circuits

Hong Li, Purdue University

Abstract

As semiconductor technology progresses, there has been a significant increase in the size of problems resulting from the modeling of VLSI circuits. Analysis and simulation of these large-sized problems have become problems of pivotal importance in modern VLSI Design. We have explored some of these problems in this thesis: interconnect modeling and simulation, power grid analysis, and timing analysis. Models underlying all these three problems have a similar sparsity structure: either matrices parameterizing these problems or the inverses of these matrices are "multi-banded." We propose efficient techniques for modeling and analyzing these problems by exploiting the sparsity and structure of the underlying matrices. Recent successful techniques for the efficient modeling and simulation of large-scale interconnect models rely on the sparsification of the inverse of the inductance matrix L. While there are several techniques for sparsification, 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 novel band matching method that enjoys several advantages: First the resulting sparse approximation is guaranteed to be positive definite. Second, the approximation is proved to be optimal, in a certain well-defined sense. Third, owing to its computational efficiency and numerical stability, the algorithm is applicable for very large problem sizes. Finally, our approach yields a compact representation of both inductance and reluctance matrices for general cases. For analysis of on-chip power delivery networks, we reveal a compact Cholesky factorization for the coefficient matrix of the system of linear equations encountered in power grid analysis problems. By exploiting this compact structure, we obtain techniques for fast matrix inversion and matrix-vector multiplication. This new method takes full advantage of the special structure of power grids and the runtime for analysis is several times to a few hundred times faster than existing methods. The impact of parameter variations on timing has become significant in recent years. Existing statistical timing analysis (STA) tools suffer from either high computational complexity or significant errors when timing variables become correlated. In this work, we present an efficient algorithm to predict the probability distribution of the circuit delay while accounting for spatial correlations. We exploit the structure of the covariance matrix to decouple the correlated variables to independent ones in linear-time, as opposed to a cubic time-complexity that a conventional technique would require. Experiments show that the proposed method is both accurate and efficient.

Degree

Ph.D.

Advisors

Koh, Purdue University.

Subject Area

Electrical engineering

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