模型降阶与指数积分混合方法研究

长期以来模型降阶是一种被视为有效加速大规模电路瞬态仿真的主流算法。而指数积分则是近年来被提出的一种与模型降阶相似的加速方法，之前很多工作已经证明了这种指数积分方法在加速电源地网络分析是有效的。然而由于指数积分要求输入波形在一个步长内满足分段线性的条件，因此当输入波形不对齐的情况下会产生大量的断点从而严重限制仿真步长大小使得效率降低。为了解决这一问题，本文对于指数积分与模型降阶这两种长期被视为完全独立的方法进行了研究并且揭示出它们之间存在着一定的等价性。具体来说，在一定条件下，指数积分相当于在每个时间步内进行了一次基于有理Krylov子空间投影的矩匹配模型降阶，然后将降阶后的系统在时域中向前求解一步。此外，本文对于它们在瞬态电路分析情景中的表现也从不同的角度进行了一定深度的阐述。希望这些新见解将推动这一经典课题的发展。除此之外，基于本文所提出两种方法的等价性，本文设计出一种指数积分与模型降阶的混合方法，即在瞬态仿真中结合使用指数积分和模型降阶。大多数对齐良好的输入仍然像往常一样由指数积分处理，而一些未对齐的输入被选择由模型降阶过程处理，生成适用于任意输入的降阶模型。这种做法可以大大放宽由未对齐输入带来的步长限制，本文还进行了数值实验以证明所提出方法的有效性。

Model order reduction has long been a mainstream strategy to accelerate large-scale transient circuit simulation. Exponential integrator methods, on the other hand, are more recently developed for a similar goal. The exponential integrator method has been proved to be an effective technique to accelerate large-scale transient power/ground network analysis. However, exponential integrator requires the inputs to be piece-wise linear in one step, which greatly limits the step size when the inputs are poorly aligned. To address this issue, in this work we first examine in-depth the underlying relationship between model order reduction and exponential integrator that are commonly seen as two separate methods. The main finding is that exponential integrator can be viewed as a moment-matching model order reduction in the time domain. Specifically, exponential integrator, under certain conditions, is equivalent to performing moment-matching model order reduction 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. Their differences in the transient circuit analysis context are also elaborated from various perspectives. It is hoped that these new insights would benefit the development of this classical topic. Based on this equivalence and their differences, we next devise a hybrid method, to combine the usage of exponential integrator and model order reduction in the same transient simulation. A majority group of well-aligned inputs are still treated by exponential integrator as usual, while a few misaligned inputs are selected to be handled by a model order reduction 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.

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