A Perturbative Stochastic Galerkin Method for the Uncertainty Quantification of Linear Circuits

Abstract

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.

Propagation algorithm for hybrid uncertainty parameters based on polynomial chaos expansion

“…39 As the core of constructing the PCE model, the methods to obtain the PCE coefficients are the Galerkin projection method and the stochastic response surface method (regression method). [40][41][42] However, inevitably, as the dimensionality of the uncertain parameters of the system and the accuracy of the solution increase, the number of orders required for the PCE model will become higher so that the computational cost of both methods increases (i.e., dimensional curse). To overcome this drawback, some methods have been proposed to filter and discard terms in the Galerkin projection that have less impact on the results, for example, the adaptive method, 43 the sparse grid numerical integration method (SGNIM).…”

“…As research progressed, a PCE based on Askey's law was proposed to solve the uncertainty propagation problem for uncertain variables with various types of distributions 39 . As the core of constructing the PCE model, the methods to obtain the PCE coefficients are the Galerkin projection method and the stochastic response surface method (regression method) 40‐42 . However, inevitably, as the dimensionality of the uncertain parameters of the system and the accuracy of the solution increase, the number of orders required for the PCE model will become higher so that the computational cost of both methods increases (i.e., dimensional curse).…”

Failure Mechanism Analysis of the Stochastic Galerkin Method in EMC Simulation Considering Geometric Randomness

“…Since 2013, the stochastic Galerkin method (SGM) [8][9][10][11] and the stochastic collocation method (SCM) [12][13][14] have always been research hotspots and have been widely applied in EMC field till now due to their calculation accuracy and efficiency. Both are based on generalized polynomial chaos theory.…”

Fast Stochastic Surrogate Modeling via Rational Polynomial Chaos Expansions and Principal Component Analysis

“…The PCE framework became widely popular also in the field of electrical engineering [2], e.g., to investigate the impact of process variations in large-scale integration circuits [3]- [17]. The available techniques can be subdivided into two classes: intrusive ones [3]- [8], chiefly…”