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

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.