| Circuit simulation is an integral parts of the VLSI design process. Complex models have been developed to mimic the various phenomena that occur at the physical level; sophisticated numerical methods have also been simultaneously designed to handle the complexity of the mathematical models. As a result, large realistic models can be simulated accurately and efficiently. The SPICE simulation tool, with an extensive model library and extremely optimized numerical methods, is the current industry standard for circuit simulation.;Recently though, due to the rapid reduction in feature sizes, values assumed for some of the parameters within the model during the design phase cannot be reproduced exactly during the fabrication phase. These aleatoric uncertainties in the model parameters induce non-determinism in the rest of the system variables. This has transformed the traditional circuit simulation problem into one of Statistical Performance Estimation (SPE). Statistical distributions are used to represent parameters and Monte Carlo (MC) type methods are used for analysis. While this approach is robust and easy to implement, it suffers from long analysis times due to its repetitive nature and, more importantly, the curse of dimensionality. The focus of this dissertation is to develop MC alternate methods for SPE: at the top level, we have developed two different methodologies using (a) interval arithmetic and (b) polynomial chaos expansions, with which we have developed intrusive methods to generate a system of equations that amenable to efficient SPE.;The first approach uses interval valued variables to represent the uncertainties. Interval arithmetic follows special computation rules which allows for guaranteed enclosures to be produced. Since the computations are inherently pessimistic and prone to interval blowup, some transformations are necessary to contain these effects. We present a graph theoretic method to transform the DAE modeling the circuit into an ODE. We then use Taylor series expansion to produce a time marching method, this results in reliable guaranteed enclosures without repetitive runs of the deterministic simulation engine. Interval arithmetic, however, is incapable of producing statistical distributions which a MC type analysis can provide. In our second approach, we use polynomial chaos expansions to represent the the inherent and induced uncertainties in the system of equations. Galerkin conditions are used to project system to a finite dimensional basis gives us an extended deterministic DAE, the solution of which allows reintroduction of nondeterminism at a much cheaper cost. While such methods have been been developed for ODEs and PDEs, we have extended the theory to be able to analyze DAEs. In particular, we have shown that an extended form of MNA exists which allows for automatic equation extraction, and that the DAE index does not increase in the extended system. Finally, we have shown that nonlinear terms can also be accommodated in the method through sub-expansions. Experimental results show that the methods is accurate and efficient as compared to the MC method, and is also more immune to the curse of dimensionality. |
