Exponential integrator (EI) method based on Krylov subspace approximation is a promising method for large-scale transient circuit simulation. However, it suffers from the singularity problem and consumes large subspace dimensions for stiff circuits when using the ordinary Krylov subspace. Restarting schemes are commonly applied to reduce the subspace dimension, but they also slow down the convergence and degrade the overall computational efficiency. In this paper, we first devise an implicit and sparsity-preserving regularization technique to tackle the singularity problem facing EI in the ordinary Krylov subspace framework. Next, we analyze the root cause of the slow convergence of the ordinary Krylov subspace methods when applied to stiff circuits. Based on the analysis, we propose a deflated restarting scheme, compatible with the above regularization technique, to accelerate the convergence of restarted Krylov subspace approximation for EI methods. Numerical experiments demonstrate the effectiveness of the proposed regularization technique, and up to $50\%$ convergence improvements for Krylov subspace approximation compared to the non-deflated version.

中文翻译：

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含隐式正则化的指数积分器方法的紧缩重启，可实现高效的瞬态电路仿真

基于Krylov子空间逼近的指数积分器（EI）方法是大规模瞬态电路仿真的一种有前途的方法。但是，当使用普通的Krylov子空间时，它会遇到奇异性问题，并且在刚性电路上会占用较大的子空间尺寸。重新启动方案通常用于减小子空间维，但是它们也会减慢收敛速度并降低总体计算效率。在本文中，我们首先设计了一种隐式且稀疏的正则化技术来解决普通Krylov子空间框架中EI所面临的奇异性问题。接下来，我们分析了应用于刚性电路时普通Krylov子空间方法收敛缓慢的根本原因。在分析的基础上，我们提出了一种紧缩的重启方案，与上述正则化技术兼容，以加快针对EI方法的重启Krylov子空间逼近的收敛。数值实验证明了所提出的正则化技术的有效性，直至$50％$ 与非缩小版本相比，Krylov子空间逼近的收敛性改进。