Model order reduction (MOR) has long been a mainstream strategy to accelerate large scale transient circuit simulation. Exponential integrator (EI) based on Krylov subspace approximation methods, on the other hand, are more recently developed for a similar goal. This article aims to examine in-depth the underlying relationship between model order reduction (MOR) and exponential integrator (EI) that are commonly seen as two separate methods. The main finding is that EI can be viewed as a moment-matching MOR in the time-domain. Specifically, EI, under certain conditions, is equivalent to performing moment-matching MOR based on rational Krylov subspace projection at each time step with a single input vector and a selected expansion point, then advancing the reduced system one step in the time-domain. The equivalence is mathematically proved under different settings and numerically verified in the experiments. Their differences in the transient circuit analysis context are also elaborated from various perspectives. It is hoped that these new insights would benefit the future development of this classical EDA topic.

Exponential integrator (EI) method has been proved to be an effective technique to accelerate large-scale transient power/ground network analysis. However, EI requires the inputs to be piece-wise linear (PWL) in one step, which greatly limits the step size when the inputs are poorly aligned. To address this issue, in this work we first elucidate with mathematical proof that EI, when used together with the rational Krylov subspace, is equivalent to performing a moment-matching model order reduction (MOR) with single input in each time step, then advancing the reduced system using EI in the same step. Based on this equivalence, we next devise a hybrid method, EI-MOR, to combine the usage of EI and MOR in the same transient simulation. A majority group of well-aligned inputs are still treated by EI as usual, while a few misaligned inputs are selected to be handled by a MOR process producing a reduced model that works for arbitrary inputs. Therefore the step size limitation imposed by the misaligned inputs can be largely alleviated. Numerical experiments are conducted to demonstrate the efficacy of the proposed method.