Statistical analysis of on-chip power grid networks by variational extended truncated balanced realization method

In this paper, we present a novel statistical analysis approach for large power grid network analysis under process variations. The new algorithm is very efficient and scalable for huge networks with a large number of variational variables. This approach, called varETBR for variational extended truncated balanced realization, is based on model order reduction techniques to reduce the circuit matrices before the variational simulation. It performs the parameterized reduction on the original system using variation-bearing subspaces. varETBR calculates variational response Gramians by Monte-Carlo based numerical integration considering both system and input source variations for generating the projection subspace. varETBR is very scalable for the number of variables and is flexible for different variational distributions and ranges as demonstrated in experimental results. After the reduction, Monte-Carlo based statistical simulation is performed on the reduced system and the statistical responses of the original system are obtained thereafter. Experimental results, on a number of IBM benchmark circuits up to 1.6 million nodes, show that the varETBR can be 4500X faster than the Monte-Carlo method and is much more scalable than one of the recently proposed approaches.