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

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 appro...

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Veröffentlicht in:	Journal of engineering mathematics Jg. 108; H. 1; S. 73 - 90
Hauptverfasser:	Manfredi, Paolo, De Zutter, Daniël, Vande Ginste, Dries
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Dordrecht Springer Netherlands 01.02.2018 Springer Nature B.V
Schlagworte:	Algebra Applications of Mathematics Collocation methods Computational Mathematics and Numerical Analysis Differential equations Factorization Galerkin method Mathematical analysis Mathematical and Computational Engineering Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Parameter uncertainty Polynomials Spectral methods Theoretical and Applied Mechanics Stochastic Galerkin method Stochastic collocation method Matrix factorization Polynomial chaos Linear differential algebraic equations Orthogonal polynomials
ISSN:	0022-0833, 1573-2703
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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.
AbstractList	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	Vande Ginste, Dries De Zutter, Daniël Manfredi, Paolo
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Cites_doi	10.1016/S0021-9991(03)00092-5 10.1137/S1064827501387826 10.1109/TCPMT.2014.2340815 10.1007/s10665-016-9848-8 10.1109/TCPMT.2013.2285877 10.1007/978-1-4612-3094-6 10.1109/TCAD.2013.2263039 10.1109/TCPMT.2013.2259295 10.1016/j.cam.2013.10.046 10.1142/p032 10.1007/978-1-4757-2272-7 10.1098/rsta.1929.0004 10.1109/TCSI.2014.2304667 10.1145/1391978.1391981 10.1090/S0025-5718-69-99647-1 10.1016/S0266-8920(02)00031-0 10.1109/TCS.1975.1084079
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A spectral approach1991New YorkSpringer10.1007/978-1-4612-3094-60722.73080 – reference: GolubGHWelschJHCalculation of Gauss quadrature rulesMath Comput19692322123024520110.1090/S0025-5718-69-99647-10179.21901 – reference: Bigoni D, Engsig-Karup AP, Eskilsson C (2016) Efficient uncertainty quantification of a fully nonlinear and dispersive water wave model with random inputs. J Eng Math 101:87–113 – reference: GradshteynISRyzhikIMTable of integrals, series, and products20077San DiegoAcademic Press1208.65001 – reference: ManfrediPVande GinsteDDe ZutterDCanaveroFGStochastic modeling of nonlinear circuits via SPICE-compatible spectral equivalentsIEEE Trans Circuits Syst I Reg Pap20146172057206510.1109/TCSI.2014.2304667 – reference: PaulCRAnalysis of multiconductor transmission lines1994New YorkWiley – volume: 187 start-page: 137 year: 2003 ident: 9909_CR10 publication-title: J Comput Phys doi: 10.1016/S0021-9991(03)00092-5 – volume: 24 start-page: 619 issue: 2 year: 2002 ident: 9909_CR2 publication-title: SIAM J Sci Comput doi: 10.1137/S1064827501387826 – volume: 4 start-page: 1634 issue: 10 year: 2014 ident: 9909_CR15 publication-title: IEEE Trans Compon Packag Manuf Techol doi: 10.1109/TCPMT.2014.2340815 – ident: 9909_CR12 doi: 10.1007/s10665-016-9848-8 – volume: 4 start-page: 673 issue: 4 year: 2014 ident: 9909_CR7 publication-title: IEEE Trans Compon Packag Manuf Techol doi: 10.1109/TCPMT.2013.2285877 – volume-title: Stochastic finite elements. 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SubjectTerms	Algebra Applications of Mathematics Collocation methods Computational Mathematics and Numerical Analysis Differential equations Factorization Galerkin method Mathematical analysis Mathematical and Computational Engineering Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Parameter uncertainty Polynomials Spectral methods Theoretical and Applied Mechanics
Title	On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations
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