Exploiting correlation kernels for efficient handling of intra-die spatial correlation, with application to statistical timing

Abstract

Intra-die manufacturing variations are unavoidable in nanoscale processes. These variations often exhibit strong spatial correlation. Standard grid-based models assume model parameters (grid-size, regularity) in an ad hoc manner and can have high measurement cost [1]. The random field model [1][2] overcomes these issues. However, no general algorithm has been proposed for the practical use of this model in statistical CAD tools. In this paper, we propose a robust and efficient numerical method, based on the Galerkin technique [3] and Karhunen Loéve Expansion [4], that enables effective use of the model. We test the effectiveness of the technique using a Monte Carlo-based Statistical Static Timing Analysis algorithm, and see errors less than 2 8%, while reducing the number of random variables from thousands to 25, resulting in speedups of up to 10x.

Practical, fast Monte Carlo statistical static timing analysis: Why and how

“…We anticipate the correlation to drop off monotonically as we move away from any given point x on the chip. A valid covariance kernel on a domain D × D must be non-negative definite [7]. [3] shows how to extract valid kernels from measurement data.…”

“…KLE does not place restrictions on the distribution of the stochastic process. [7] shows that the stochastic behavior of the thousands to millions of gates on a chip can be represented using a much reduced set of r uncorrelated RVs by truncating (1) at the first r eigenpairs. For the ISCAS89 circuits, [7] showed that using r = 25 eigenpairs for each statistical parameter yielded errors of only 0.35% on average.…”

“…The KLE of a random field model of intra-die statistical variation can take an extremely large model, with many random variables (RVs) to represent gate-level statistics, and reduce these to few tens of uncorrelated RVs. [7] showed that this reduction already improves the performance of conventional Monte Carlo samples. But, in this paper, we show that one can do even better, in particular, we show that the KLE model is precisely what is needed to make ultra-fast QMC sampling scale to practical SSTA problems.…”

“…We synthesize two different, recently introduced ideas: reduced sampling via deterministic quasiMonte Carlo (QMC) methods [5], and dimensionality reduction via a Karhunen-Loéve expansion (KLE) of a random field model of intra-die variation [4] [7].…”

“…In the case of SSTA, we have the twin obstacles that the inherent dimensionality is extremely large (thousands to millions), and neither designer expertise nor rank correlation are tractable for measuring statistical importance of each variable. This is where a recent advance in KLE-based spatial correlation modeling [4], made practical by [7], is essential. The KLE of a random field model of intra-die statistical variation can take an extremely large model, with many random variables (RVs) to represent gate-level statistics, and reduce these to few tens of uncorrelated RVs.…”