Variation-Aware Stochastic Extraction with Large Parameter Dimensionality: Review and Comparison of State of the Art Intrusive and Non-intrusive Techniques (Invited)

T. El-Moselhy and L. Daniel
Massachusetts Institute of Technology


Uncertainties in integrated circuit manufacturing processes typically result in structures, whose dimensions, shapes and material properties deviate significantly from the original design intent. Variation-aware stochastic solvers extract and propagate statistical descriptions of the physical geometrical variations to statistical descriptions of electrical parameters (e.g. capacitance, resistance, inductance, threshold voltage etc...), and generate variation-aware stochastic macromodels to enable system level optimization and synthesis of sensitive circuit blocks. The vast majority of variation-aware stochastic extraction tools can be classified into two different categories of algorithms, namely, "intrusive" and "non-intrusive". "Non-intrusive" algorithms are those that rely on sampling the parameter space (e.g. Montecarlo, stochastic collocation, adapting sampling, importance sampling, etc...), and then use any standard deterministic solver to compute a deterministic solution at the different sampling points. The main challenge facing such methods is the need to simultaneously and efficiently solve a very large number of similar linear systems corresponding to each different sample point possibly recycling some form of computation (e.g. Krylov subspace recycling, or floating random walk path recycling, etc...). The term "intrusive" refers to the fact that a stochastic representation of the solution is instead directly computed using specialized stochastic algorithms (e.g. the stochastic Galerkin method, the Neumann expansion, the Combined Neumann-Hermite Expansion, or the Stochastic Dominant Singular Vector methods). In this talk I will review a variety of state of the art intrusive and non-intrusive algorithms, recently proposed for IC variation-aware extraction. I will then describe in more details the best performing non-intrusive and the best performing intrusive algorithms up to date, showing results and comparisons on examples ranging from specific small to very large practical problems.