A Parallel and Incremental Extraction of Variational Capacitance With Stochastic Geometric Moments

This paper presents a parallel and incremental solver for stochastic capacitance extraction. The random geometrical variation is described by stochastic geometrical moments, which lead to a densely augmented system equation. To efficiently extract the capacitance and solve the system equation, a parallel fast-multipole-method (FMM) is developed in the framework of stochastic geometrical moments. This can efficiently estimate the stochastic potential interaction and its matrix-vector product (MVP) with charge. Moreover, a generalized minimal residual (GMRES) method with incremental update is developed to calculate both the nominal value and the variance. Our overall extraction show is called piCAP. A number of experiments show that piCAP efficiently handles a large-scale on-chip capacitance extraction with variations. Specifically, a parallel MVP in piCAP is up 3 × to faster than a serial MVP, and an incremental GMRES in piCAP is up to 15× faster than non-incremental GMRES methods.