Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input {\it a-priori}. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201

Uncertainty quantification for integrated circuits: Stochastic spectral methods

Due to significant manufacturing process variations, the performance of integrated circuits (ICs) has become increasingly uncertain. Such uncertainties must be carefully quantified with efficient stochastic circuit simulators. This paper discusses the recent advances of stochastic spectral circuit simulators based on generalized polynomial chaos (gPC). Such techniques can handle both Gaussian and non-Gaussian random parameters, showing remarkable speedup over Monte Carlo for circuits with a small or medium number of parameters. We focus on the recently developed stochastic testing and the application of conventional stochastic Galerkin and stochastic collocation schemes to nonlinear circuit problems. The uncertainty quantification algorithms for static, transient and periodic steady-state simulations are presented along with some practical simulation results. Some open problems in this field are discussed.MIT Masdar Program (196F/002/707/102f/70/9374

Clustering-based collocation for uncertainty propagation with multivariate correlated inputs

In this article, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation (SC). The key idea is to use cluster centers as the nodes for collocation. In this way, we can extend the use of collocation methods to uncertainty propagation with multivariate, correlated input. The approach is particularly useful in situations where the probability distribution of the input is unknown, and only a sample from the input distribution is available. We examine several clustering methods and assess their suitability for stochastic collocation numerically using the Genz test functions as benchmark. The proposed methods work well, most notably for the challenging case of nonlinearly correlated inputs in higher dimensions. Tests with input dimension up to 16 are included. Furthermore, the clustering-based collocation methods are compared to regular SC with tensor grids of Gaussian quadrature nodes. For 2-dimensional uncorrelated inputs, regular SC performs better, as should be expected, however the clustering-based methods also give only small relative errors. For correlated 2-dimensional inputs, clustering-based collocation outperforms a simple adapted version of regular SC, where the weights are adjusted to account for input correlatio