Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

Kaintura, Arun; Dhaene, Tom; Spina, Domenico

doi:10.3390/electronics7030030

Cited by 103 publications

(93 citation statements)

References 115 publications

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“…Classical families of orthogonal polynomials are related to a specific form of the kernel function. For instance, Hermite polynomials correspond to a kernel function identical to Gaussian distribution function with zero mean and unit variance [64]:…”

Section: Basics Of Polynomial Chaosmentioning

confidence: 99%

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Tarakanov¹,

Elsheikh²

2019

Journal of Computational Physics

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Surrogate-modelling techniques including Polynomial Chaos Expansion (PCE) is commonly used for statistical estimation (aka. Uncertainty Quantification) of quantities of interests obtained from expensive computational models. PCE is a data-driven regression-based technique that relies on spectral polynomials as basis-functions. In this technique, the outputs of few numerical simulations are used to estimate the PCE coefficients within a regression framework combined with regularization techniques where the regularization parameters are estimated using standard cross-validation as applied in supervised machine learning methods.In the present work, we introduce an efficient method for estimating the PCE coefficients combining Elastic Net regularization with a data-driven feature ranking approach. Our goal is to increase the probability of identifying the most significant PCE components by assigning each of the PCE coefficients a numerical value reflecting the magnitude of the coefficient and its stability with respect to perturbations in the input data. In our evaluations, the proposed approach has shown high convergence rate for high-dimensional problems, where standard feature ranking might be challenging due to the curse of dimensionality.The presented method is implemented within a standard machine learning library (scikit-learn [1]) allowing for easy experimentation with various solvers and regularization techniques (e.g. Tikhonov, LASSO, LARS, Elastic Net) and enabling automatic cross-validation techniques using a widely used and well tested implementation. We present a set of numerical tests on standard analytical functions, a two-phase subsurface flow model and a simulation dataset for CO2 sequestration in a saline aquifer. For all test cases, the proposed approach resulted in a significant increase in PCE convergence rates.

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Section: Basics Of Polynomial Chaosmentioning

confidence: 99%

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Tarakanov¹,

Elsheikh²

2019

Journal of Computational Physics

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“…where w is a vector of weight coefficients and b is an offset parameter. To find the best combination of the parameters (w, b) in (7), the following loss-function should be minimized by: For minimizing (9), the below optimization problem can be solved,…”

Section: Support Vector Machine Regressionmentioning

confidence: 99%

“…Recently some intrusive methods using generalized Polynomial Chaos (PC) have been reported to overcome MC problems [6][7][8]. Intrusive methods generally lead to a coupled system of equations by modification of the system fundamental equations, which is very time consuming to solve with respect to the original problem when the dimension of uncertain parameters is high [9]. Alternatively, there are nonintrusive methods that treat the original system of equations as a black-box, and there is no need to change the fundamental system equations.…”

Section: Introductionmentioning

confidence: 99%

“…Alternatively, there are nonintrusive methods that treat the original system of equations as a black-box, and there is no need to change the fundamental system equations. As a result, they are easier to implement, comparing to intrusive methods [9]. For problems with higher dimensions, non-intrusive hierarchical sparse PC approaches have been used, which the PC expansion terms are reduced based form of conventional full-blown PC expansions in general [10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

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Residue-Pole Methods for Variability Analysis of S-parameters of Microwave Devices with 3D FEM and Mesh Deformation

Rufuie¹,

Lamęcki²,

Sypek³

et al. 2020

RADIOENGINEERING

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This paper presents a new approach for variability analysis of microwave devices with a high dimension of uncertain parameters. The proposed technique is based on modeling an approximation of system by its poles and residues using several modeling methods, including ordinary kriging, Adaptive Polynomial Chaos (APCE), and Support Vector Machine Regression (SVM). The computational cost is compared with the traditional Monte-Carlo method. To improve the efficiency, mesh deformation is applied within 3D FEM framework.

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“…[13][14][15] It has been successfully used in various areas, for example, modeling, fault detection, and optimization,. 1,11,[16][17][18] For example, an intrusive gPC has been applied for optimizing the operating condition of a penicillin manufacturing process, 19 solving the full-wave problems or multiconductor transmission lines, [20][21][22] and quantifying uncertainty in flocking dynamics for optimal control design. 23 However, when the governing equations have complicated forms and when the number of parametric uncertainties is large, UA using the SG-based gPC yields a high-dimensional integration problem.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic surrogate models for uncertainty analysis: Dimension reduction‐based polynomial chaos expansion

Son

2019

Numerical Meth Engineering

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Summary This paper presents an approach for efficient uncertainty analysis (UA) using an intrusive generalized polynomial chaos (gPC) expansion. The key step of the gPC‐based uncertainty quantification (UQ) is the stochastic Galerkin (SG) projection, which can convert a stochastic model into a set of coupled deterministic models. The SG projection generally yields a high‐dimensional integration problem with respect to the number of random variables used to describe the parametric uncertainties in a model. However, when the number of uncertainties is large and when the governing equation of the system is highly nonlinear, the SG approach‐based gPC can be challenging to derive explicit expressions for the gPC coefficients because of the low convergence in the SG projection. To tackle this challenge, we propose to use a bivariate dimension reduction method (BiDRM) in this work to approximate a high‐dimensional integral in SG projection with a few one‐ and two‐dimensional integrations. The efficiency of the proposed method is demonstrated with three different examples, including chemical reactions and cell signaling. As compared to other UA methods, such as the Monte Carlo simulations and nonintrusive stochastic collocation (SC), the proposed method shows its superior performance in terms of computational efficiency and UA accuracy.

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Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

Cited by 103 publications

References 115 publications

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Residue-Pole Methods for Variability Analysis of S-parameters of Microwave Devices with 3D FEM and Mesh Deformation

Probabilistic surrogate models for uncertainty analysis: Dimension reduction‐based polynomial chaos expansion

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