Novel methods to quantify aleatory and epistemic uncertainty in high speed networks

2017 Summer.Includes bibliographical references.With the sustained miniaturization of integrated circuits to sub-45 nm regime and the increasing packaging density, random process variations have been found to result in unpredictability in circuit performance. In existing literature, this unpredictability has been modeled by creating polynomial expansions of random variables. But the existing methods prove inefficient because as the number of random variables within a system increase, the time and computational cost increases in a near-polynomial fashion. In order to mitigate this poor scalability of conventional approaches, several techniques are presented, in this dissertation, to sparsify the polynomial expansion. The sparser polynomial expansion is created, by identifying the contribution of each random variable on the total response of the system. This sparsification is performed primarily using two different methods. It translates to immense savings, in the time required, and the memory cost of computing the expansion. One of the two methods presented is applied to aleatory variability problems while the second method is applied to problems involving epistemic uncertainty. The accuracy of the proposed approaches is validated through multiple numerical examples

The relationship between Galerkin and collocation methods in statistical transmission line analysis

This paper discusses the relationship between two standard methods for the stochastic analysis of linear circuits, namely the stochastic Galerkin method (SGM) and the stochastic collocation method (SCM), based on a multidimensional Gaussian quadrature. It is established that the SCM corresponds to an approximate factorization of the SGM, involving matrix polynomials sharing the same coefficients as the pertinent polynomial chaos basis functions. Under certain assumptions, the two methods coincide. These findings are illustrated by means of a frequency-domain simulation of a transmission line circuit

Generation of stochastic interconnect responses via gaussian process latent variable models

We introduce a novel generative model for stochastic device responses using limited available data. This model is oblivious to any varying design parameters or their distribution and only requires a small set of "training" responses. Using this model, new responses are efficiently generated whose distribution closely matches that of the real data, e.g., for use in Monte-Carlo-like analyses. The modeling methodology consists of a vector fitting step, where device responses are represented by a rational model, followed by the optimization of a Gaussian process latent variable model. Passivity is guaranteed by a posteriori discarding of nonpassive responses. The novel model is shown to considerably outperform a previous generative model, as evidenced by comparing accuracies of distribution estimation for the case of differential-to-common mode conversion in two coupled microstrip lines

A Data Compression Strategy for the Efficient Uncertainty Quantification of Time-Domain Circuit Responses

This paper presents an innovative modeling strategy for the construction of efficient and compact surrogate models for the uncertainty quantification of time-domain responses of digital links. The proposed approach relies on a two-step methodology. First, the initial dataset of available training responses is compressed via principal component analysis (PCA). Then, the compressed dataset is used to train compact surrogate models for the reduced PCA variables using advanced techniques for uncertainty quantification and parametric macromodeling. Specifically, in this work sparse polynomial chaos expansion and least-square support-vector machine regression are used, although the proposed methodology is general and applicable to any surrogate modeling strategy. The preliminary compression allows limiting the number and complexity of the surrogate models, thus leading to a substantial improvement in the efficiency. The feasibility and performance of the proposed approach are investigated by means of two digital link designs with 54 and 115 uncertain parameters, respectively

Review of polynomial chaos-based methods for uncertainty quantification in modern integrated circuits

Advances in manufacturing process technology are key ensembles for the production of integrated circuits in the sub-micrometer region. It is of paramount importance to assess the effects of tolerances in the manufacturing process on the performance of modern integrated circuits. The polynomial chaos expansion has emerged as a suitable alternative to standardMonte Carlo-based methods that are accurate, but computationally cumbersome. This paper provides an overview of the most recent developments and challenges in the application of polynomial chaos-based techniques for uncertainty quantification in integrated circuits, with particular focus on high-dimensional problems

Uncertainty quantification of cable inductances and capacitances via mixed-fidelity models

In this paper, we investigate a mixed-fidelity approach for the uncertainty quantification of the per-unit-length (p.u.l.) capacitance and inductance of cables with random geometrical and material parameters. Polynomial chaos expansion is used to model uncertainty, whereas a numerical discretization technique is used to calculate p.u.l. inductances and capacitances. However, instead of using a model with high fidelity in both features, the results are obtained as a combination of two complementary models with mixed fidelity in each feature. Numerical examples concerning the statistical assessment of the p.u.l. inductance and capacitance matrices of two shielded cables show that similar accuracy is attained at a fraction of the computational cost compared to conventional approaches

Machine Learning Regression Techniques for the Modeling of Complex Systems: An Overview

Recently, machine learning (ML) techniques have gained widespread diffusion, since they have been successfully applied in several research fields. This paper investigates the effectiveness of advanced ML regressions in two EMC applications. Specifically, support vector machine, least-squares support vector machine and Gaussian process regressions are adopted to construct accurate and fast-to-evaluate surrogate models able to predict the output variable of interest as a function of the system parameters. The resulting surrogates, built from a limited set of training samples, can be suitably adopted for both uncertainty quantification and optimization purposes. The accuracy and the key features of each of the considered machine learning techniques are investigated by comparing their predictions with the ones provided by either circuital simulations or measurements

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

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