Deflated Restarting of Exponential Integrator Method with an Implicit Regularization for Efficient Transient Circuit Simulation

Meng Zhang,Chengcheng Yang,Jiaxin Li,Quan Chen

doi:10.3390/electronics10091124

Abstract

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.

Highlights

High performance and full-accuracy transient circuit simulation has always been one of the major demands in modern IC design industry
Besides IC design, SPICE-type simulators have been widely used in some other scientific domains, such as electric/thermal analysis [1], electric/magnetic analysis [2], and advection-diffusion analysis [3], where the systems of interest can be described by differential algebraic equations (DAEs)
We aim to address the above two issues limiting the performance of Exponential integrator (EI) for large stiff circuit simulation

Summary

Introduction

High performance and full-accuracy transient circuit simulation has always been one of the major demands in modern IC design industry. Besides IC design, SPICE-type simulators have been widely used in some other scientific domains, such as electric/thermal analysis [1], electric/magnetic analysis [2], and advection-diffusion analysis [3], where the systems of interest can be described by differential algebraic equations (DAEs). The essence of transient circuit simulation is to numerically solve a system of DAEs, normally derived from Modified Nodal Analysis (MNA), by an explicit or implicit integration method. Being a category of implicit methods, Backward differentiation formula (BDF) is widely adopted by existing SPICE simulators as it offers larger time steps and better stability than explicit methods. It is more suitable for handling stiff DAEs with time constants differing by several orders of magnitude.

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