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On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations

Paolo Manfredi (UGent) , Daniël De Zutter (UGent) and Dries Vande Ginste (UGent)
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Abstract
Polynomial chaos-based methods have been extensively applied in electrical and other engineering problems for the stochastic simulation of systems with uncertain parameters. Most of the implementations are based on either the intrusive stochastic Galerkin method or on non-intrusive collocation approaches, of which a very common example is the pseudo-spectral method based on Gaussian quadrature rules. This paper shows that, for the important class of linear differential algebraic equations, the latter can be cast as an approximate factorization of the stochastic Galerkin approach, thus generalizing recent discussions in literature in this regard. Consistently with this literature, we show that the factorization turns out to be exact for first-order random inputs, and hence the two methods coincide under this assumption. Further, the presented results also generalize recent work in the field of electrical circuit simulation, in which a similar decomposition was derived ad hoc, via error minimization, for the case of Hermite chaos. We demonstrate that the factorization stems from the general properties of orthogonal polynomials and the error introduced by the approximation-or in other terms, the error of the stochastic collocation method in comparison with the stochastic Galerkin method-is carefully quantified and assessed. An illustrative example concerning the stochastic analysis of an RLC circuit is used to illustrate the main findings of this paper. In addition, a more complex and real-life example allows emphasizing the generality of the achieved results.
Keywords
IBCN, GENERALIZED POLYNOMIAL CHAOS, UNCERTAINTY QUANTIFICATION, Linear differential algebraic equations, Matrix factorization, Orthogonal polynomials, Polynomial chaos, Stochastic collocation method, Stochastic Galerkin method

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Citation

Please use this url to cite or link to this publication:

Chicago
Manfredi, Paolo, Daniël De Zutter, and Dries Vande Ginste. 2018. “On the Relationship Between the Stochastic Galerkin Method and the Pseudo-spectral Collocation Method for Linear Differential Algebraic Equations.” Journal of Engineering Mathematics 108 (1): 73–90.
APA
Manfredi, P., De Zutter, D., & Vande Ginste, D. (2018). On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations. JOURNAL OF ENGINEERING MATHEMATICS, 108(1), 73–90.
Vancouver
1.
Manfredi P, De Zutter D, Vande Ginste D. On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations. JOURNAL OF ENGINEERING MATHEMATICS. Dordrecht: Springer; 2018;108(1):73–90.
MLA
Manfredi, Paolo, Daniël De Zutter, and Dries Vande Ginste. “On the Relationship Between the Stochastic Galerkin Method and the Pseudo-spectral Collocation Method for Linear Differential Algebraic Equations.” JOURNAL OF ENGINEERING MATHEMATICS 108.1 (2018): 73–90. Print.
@article{8550876,
  abstract     = {Polynomial chaos-based methods have been extensively applied in electrical and other engineering problems for the stochastic simulation of systems with uncertain parameters. Most of the implementations are based on either the intrusive stochastic Galerkin method or on non-intrusive collocation approaches, of which a very common example is the pseudo-spectral method based on Gaussian quadrature rules. This paper shows that, for the important class of linear differential algebraic equations, the latter can be cast as an approximate factorization of the stochastic Galerkin approach, thus generalizing recent discussions in literature in this regard. Consistently with this literature, we show that the factorization turns out to be exact for first-order random inputs, and hence the two methods coincide under this assumption. Further, the presented results also generalize recent work in the field of electrical circuit simulation, in which a similar decomposition was derived ad hoc, via error minimization, for the case of Hermite chaos. We demonstrate that the factorization stems from the general properties of orthogonal polynomials and the error introduced by the approximation-or in other terms, the error of the stochastic collocation method in comparison with the stochastic Galerkin method-is carefully quantified and assessed. An illustrative example concerning the stochastic analysis of an RLC circuit is used to illustrate the main findings of this paper. In addition, a more complex and real-life example allows emphasizing the generality of the achieved results.},
  author       = {Manfredi, Paolo and De Zutter, Dani{\"e}l and Vande Ginste, Dries},
  issn         = {0022-0833},
  journal      = {JOURNAL OF ENGINEERING MATHEMATICS},
  keyword      = {IBCN,GENERALIZED POLYNOMIAL CHAOS,UNCERTAINTY QUANTIFICATION,Linear differential algebraic equations,Matrix factorization,Orthogonal polynomials,Polynomial chaos,Stochastic collocation method,Stochastic Galerkin method},
  language     = {eng},
  number       = {1},
  pages        = {73--90},
  publisher    = {Springer},
  title        = {On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations},
  url          = {http://dx.doi.org/10.1007/s10665-017-9909-7},
  volume       = {108},
  year         = {2018},
}

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