255 research outputs found
Stochastic macromodeling of nonlinear systems via polynomial chaos expansion and transfer function trajectories
A novel approach is presented to perform stochastic variability analysis of nonlinear systems. The versatility of the method makes it suitable for the analysis of complex nonlinear electronic systems. The proposed technique is a variation-aware extension of the Transfer Function Trajectory method by means of the Polynomial Chaos expansion. The accuracy with respect to the classical Monte Carlo analysis is verified by means of a relevant numerical example showing a simulation speedup of 1777 X
Quantification of airfoil geometry-induced aerodynamic uncertainties - comparison of approaches
Uncertainty quantification in aerodynamic simulations calls for efficient
numerical methods since it is computationally expensive, especially for the
uncertainties caused by random geometry variations which involve a large number
of variables. This paper compares five methods, including quasi-Monte Carlo
quadrature, polynomial chaos with coefficients determined by sparse quadrature
and gradient-enhanced version of Kriging, radial basis functions and point
collocation polynomial chaos, in their efficiency in estimating statistics of
aerodynamic performance upon random perturbation to the airfoil geometry which
is parameterized by 9 independent Gaussian variables. The results show that
gradient-enhanced surrogate methods achieve better accuracy than direct
integration methods with the same computational cost
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
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
B-Spline based uncertainty quantification for stochastic analysis
The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification.
At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails.
Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it.
Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided
- …