Three term recurrence for the evaluation of multivariate orthogonal polynomials

Barrio, Roberto; Peña, J. M.; Sauer, Tomas

doi:10.1016/j.jat.2009.07.005

Cited by 17 publications

(13 citation statements)

References 17 publications

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“…However, the algorithm does not converge well due to the numerical instability of the basis functions, and designers cannot easily extract statistical information (e.g., mean value and variance) from the obtained solution. In the applied math community, multivariate orthogonal polynomials may be constructed via the multivariate three-term recurrence [54], [55]. However, the theories in [54], [55] either are hard to implement or can only guarantee weak orthogonality.…”

Section: Non-gaussian Correlated Casesmentioning

confidence: 99%

Stochastic Collocation With Non-Gaussian Correlated Process Variations: Theory, Algorithms, and Applications

Cui

Zhang

2019

IEEE Trans. Compon., Packag. Manufact. Technol.

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Stochastic spectral methods have achieved great success in the uncertainty quantification of many engineering problems, including electronic and photonic integrated circuits influenced by fabrication process variations. Existing techniques employ a generalized polynomial-chaos expansion, and they almost always assume that all random parameters are mutually independent or Gaussian correlated. However, this assumption is rarely true in real applications. How to handle non-Gaussian correlated random parameters is a long-standing and fundamental challenge. A main bottleneck is the lack of theory and computational methods to perform a projection step in a correlated uncertain parameter space. This paper presents an optimization-based approach to automatically determinate the quadrature nodes and weights required in a projection step, and develops an efficient stochastic collocation algorithm for systems with non-Gaussian correlated parameters. We also provide some theoretical proofs for the complexity and error bound of our proposed method. Numerical experiments on synthetic, electronic and photonic integrated circuit examples show the nearly exponential convergence rate and excellent efficiency of our proposed approach. Many other challenging uncertainty-related problems can be further solved based on this work.

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Section: Non-gaussian Correlated Casesmentioning

confidence: 99%

Stochastic Collocation With Non-Gaussian Correlated Process Variations: Theory, Algorithms, and Applications

Cui

Zhang

2019

IEEE Trans. Compon., Packag. Manufact. Technol.

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“…In general, one may construct multivariate orthogonal polynomials via the three-term recurrence in [45] or [46]. However, their theories either are hard to implement or can only guarantee weak orthogonality [45], [46]. Inspired by [47], we present a simple yet efficient method for computing a set of multivariate orthonormal polynomial basis functions.…”

Section: A Multivariate Basis Functionsmentioning

confidence: 99%

High-Dimensional Uncertainty Quantification of Electronic and Photonic IC With Non-Gaussian Correlated Process Variations

Cui

Zhang

2020

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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Uncertainty quantification based on generalized polynomial chaos has been used in many applications. It has also achieved great success in variation-aware design automation. However, almost all existing techniques assume that the parameters are mutually independent or Gaussian correlated, which is rarely true in real applications. For instance, in chip manufacturing, many process variations are actually correlated. Recently, some techniques have been developed to handle non-Gaussian correlated random parameters, but they are timeconsuming for high-dimensional problems. We present a new framework to solve uncertainty quantification problems with many non-Gaussian correlated uncertainties. Firstly, we propose a set of smooth basis functions to well capture the impact of non-Gaussian correlated process variations. We develop a tensor approach to compute these basis functions in a highdimension setting. Secondly, we investigate the theoretical aspect and practical implementation of a sparse solver to compute the coefficients of all basis functions. We provide some theoretical analysis for the exact recovery condition and error bound of this sparse solver in the context of uncertainty quantification. We present three adaptive sampling approaches to improve the performance of the sparse solver. Finally, we validate our methods by synthetic and practical electronic/photonic ICs with 19 to 57 non-Gaussian correlated variation parameters. Our approach outperforms Monte Carlo by thousands of times in terms of efficiency. It can also accurately predict the output density functions with multiple peaks caused by non-Gaussian correlations, which are hard to capture by existing methods.Index Terms-High dimensionality, uncertainty quantification, electronic and photonic IC, non-Gaussian correlation, process variations, tensor, sparse solver, adaptive sampling.

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“…Three term relations. The construction follows [12]. For more information on orthogonal polynomials of several variables see also [16], and [33].…”

Section: Theorem 32 the Moments Smentioning

confidence: 99%

Generalized Fock space and moments

Alpay¹,

Cerejeiras²,

Kähler³

2020

Preprint

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In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-Itô expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of stochastic processes.

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Three term recurrence for the evaluation of multivariate orthogonal polynomials

Abstract: In this paper we obtain some explicit three term recurrence relations for the determination of multivariate orthogonal polynomials. These formulas allow us to obtain evaluation algorithms of finite series of these polynomials.

Cited by 17 publications

References 17 publications

Stochastic Collocation With Non-Gaussian Correlated Process Variations: Theory, Algorithms, and Applications

Stochastic Collocation With Non-Gaussian Correlated Process Variations: Theory, Algorithms, and Applications

High-Dimensional Uncertainty Quantification of Electronic and Photonic IC With Non-Gaussian Correlated Process Variations

Generalized Fock space and moments

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