Variation-aware interconnect extraction using statistical moment preserving model order reduction

Abstract

1 In this paper we present a stochastic model order reduction technique for interconnect extraction in the presence of process variabilities, i.e. variation-aware extraction. It is becoming increasingly evident that sampling based methods for variation-aware extraction are more efficient than more computationally complex techniques such as stochastic Galerkin method or the Neumann expansion. However, one of the remaining computational challenges of sampling based methods is how to simultaneously and efficiently solve the large number of linear systems corresponding to each different sample point. In this paper, we present a stochastic model reduction technique that exploits the similarity among the different solves to reduce the computational complexity of subsequent solves. We first suggest how to build a projection matrix such that the statistical moments and/or the coefficients of the projection of the stochastic vector on some orthogonal polynomials are preserved. We further introduce a proximity measure, which we use to determine apriori if a given system needs to be solved, or if it is instead properly represented using the currently available basis. Finally, in order to reduce the time required for the system assembly, we use the multivariate Hermite expansion to represent the system matrix. We verify our method by solving a variety of variation-aware capacitance extraction problems ranging from on-chip capacitance extraction in the presence of width and thickness variations, to off-chip capacitance extraction in the presence of surface roughness. We further solve very large scale problems that cannot be handled by any other state of the art technique.

Stochastic extraction for SoC and SiP interconnect with variability

“…Different strategies have been proposed in the literature to address such challenge, such as parallelization, Krylov subspace recycling [12], [13], stochastic model order reduction [14], [15], [11], [16] and path recycling of floating random walk (PR-FRW) [3] (exclusively for capacitance and resistance extraction).…”

“…The reduced order solution is then computed as y r ( η) = c T Ux r ( η). In [16], the projection matrix U is chosen such that the statistical moments of y r ( η) match those of y( η),…”

Model order reduction of fully parameterized systems by recursive least square optimization

“…Many PMOR methods have been proposed to reduce systems with non-parameterized input and output matrices, such as moment matching [1]- [4], [6], [7], [9], [11], positive-real balanced truncation [13], [14] and the sampling based variational PMTBR [15].…”

An ME-PC Enhanced HDMR Method for Efficient Statistical Analysis of Multiconductor Transmission Line Networks

“…Similar deficiencies are oftentimes encountered in macromodeling [33] and model-order reduction techniques (see the references in [34]) as well. To address this limitation, semi-intrusive stochastic macromodeling techniques [35], [36] and a nonintrusive dimension adaptive sparse grid method [37] have been used in the context of stochastic MTL analysis; implementation of the former hinges on access to the deterministic simulator's inner workings while the latter is easily coupled with any deterministic simulator. In parallel, ME-PC methods have recently been hybridized with high-dimensional model representation (HDMR) techniques [31].…”