This paper presents an iterative and decoupled perturbative stochastic Galerkin (SG) method for the variability analysis of stochastic linear circuits with a large number of uncertain parameters. State-of-the-art implementations of polynomial chaos expansion and SG projection produce a large deterministic circuit that is fully coupled, thus becoming cumbersome to implement and inefficient to solve when the number of random parameters is large. In a perturbative approach, component variability is interpreted as a perturbation of its nominal value. The relaxation of the resulting equations and the application of a SG method lead to a decoupled system of equations, corresponding to a modified equivalent circuit in which each stochastic component is replaced by the nominal element equipped with a parallel current source accounting for the effect of variability. The solution of the perturbation problem is carried out in an iterative manner by suitably updating the equivalent current sources by means of Jacobi- or Gauss-Seidel strategies, until convergence is reached. A sparse implementation allows avoiding the refinement of negligible coefficients, yielding further efficiency improvement. Moreover, for time-invariant circuits, the iterations are effectively performed in post-processing after characterizing the circuit in time or frequency domain by means of a limited number of simulations. Several application examples are used to illustrate the proposed technique and highlight its performance and computational advantages.

中文翻译：

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线性电路不确定度量化的微扰随机伽辽金方法

本文提出了一种迭代解耦微扰随机伽辽金（SG）方法，用于具有大量不确定参数的随机线性电路的可变性分析。多项式混沌展开和 SG 投影的最新实现产生了一个完全耦合的大型确定性电路，因此当随机参数的数量很大时，实现起来很麻烦且求解效率低下。在微扰方法中，组件可变性被解释为其标称值的微扰。结果方程的松弛和 SG 方法的应用导致方程解耦系统，对应于修改后的等效电路，其中每个随机分量被标称元件替换，标称元件配备有考虑可变性影响的并联电流源。扰动问题的求解以迭代方式通过使用 Jacobi-或 Gauss-Seidel 策略适当地更新等效电流源来执行，直到达到收敛。稀疏实现允许避免对可忽略系数的细化，从而进一步提高效率。此外，对于时不变电路，在通过有限数量的模拟在时域或频域中表征电路之后，在后处理中有效地执行迭代。