Hyperspherical Clustering and Sampling for Rare Event Analysis with Multiple Failure Region Coverage

Statistical circuit simulation is exhibiting increasing importance for circuit design under process variations. It has been widely used throughout the design of standard cell circuits (SRAM, Flip-Flop, etc.) to maximize yield, i.e. to minimize the failure probability. Existing approaches cannot effectively analyze the failure probability when failed samples are distributed in multiple disjoint regions, nor handle the circuits with a large number of variations. To tackle these challenges, the proposed hyperspherical clustering and sampling (HSCS) approach first identifies multiple failure regions through a reweighted spherical k-means algorithm, which clusters failed samples on a set of hyperspheres, rather than the high dimensional open space. Next, a modified mixture importance sampling is designed to draw samples at those clusters to achieve multiple failure region coverage. The proposed HSCS is evaluated using both mathematical and circuit-based examples. It achieves about 3-order speedup over Monte Carlo with the same level of accuracy, while other importance sampling based approaches either fail to converge or converge to wrong results. Furthermore, HSCS demonstrates excellent robustness by generating consistent results in multiple replications.