On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations
type: Item
instance of: scholarly article
title: On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations
zbMATH Open document ID: 1387.65078
author: Paolo Manfredi
author: Dries Vande Ginste
author: Daniël De Zutter
published in: Journal of Engineering Mathematics
publication date: 2018-03-07T00:00:00Z
full work available at URL: https://biblio.ugent.be/publication/8550876
review text: The authors consider the stochastic linear differential algebraic equation \[ \mathbf{C}(\xi)\mathbf{x}'(t,\xi)+\mathbf{G}(\xi)\mathbf{x}(t,\xi)=\mathbf{u}(t,\xi), \] with a suitable initial condition \(\mathbf{x}(t=t_0)\), where the matrices \(\mathbf{C}, \mathbf{G}\in \mathbb R^{N\times N}\) and the source term \(\mathbf{u}\in \mathbb R^N\) depend on one non-deterministic and real valued parameter \(\xi\in \mathbb R\) with a given probability density function \(\omega(\xi)\). To solve it numerically, the strategy, under appropriate assumptions, relies on the fact of considering that the solution \(\mathbf{x}(t,\xi)\) and the matrices \(\mathbf{C}(\xi)\) and \(\mathbf{G}(\xi)\) are expanded into (truncated) series of suitable orthogonal polynomials, the coefficients of the matrices expansions being known. To determine the unknown coefficients for the solution, two methods can be applied: either the stochastic Galerkin method (SGM) or the pseudo-spectral collocation method (PSCM). After describing briefly both the SGM and PSCM methods, the authors are devoted to determine and assess the relationship between the methods for the above linear differential algebraic equation. According to the authors, this is done by means of an approximate decomposition matrix, stemming from the general properties of orthogonal polynomials and decoupling the SGM problem into the PSCM one. It can be stressed that their result contains as a special case the analysis developed in [\textit{R. Pulch}, J. Comput. Appl. Math. 262, 281--291 (2014; Zbl 1301.65090)]. Moreover, their study enables them to generalize and extend an \textit{ad hoc} similar decomposition given for the specific case of Hermite's chaos in [\textit{T. A. Pham} et al., IEEE Trans. Compon. Packag Manuf. Technol. 4, 1634--1647 (2014; \url{doi:10.1109/TCPMT.2014.2340815})], because their factorization is valid for arbitrary polynomial basis functions. In the last part of the paper under review, the authors illustrate their …
Mathematics Subject Classification ID: 65L80
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65C30
zbMATH DE Number: 6847667
zbMATH Keywords: linear differential algebraic equations
zbMATH Keywords: matrix factorization
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: polynomial chaos
zbMATH Keywords: stochastic collocation method
zbMATH Keywords: stochastic Galerkin method
zbMATH Keywords: adaptive Gauss-Kronrod quadrature
zbMATH Keywords: Legendre polynomials
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: RLC-circuit
zbMATH Keywords: multiconductor transmission line
reviewed by: Antonio Linero Bas
describes a project that uses: FERUM
MaRDI profile type: Publication
OpenAlex ID: W2619716728
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DOI: 10.1007/S10665-017-9909-7
description: scientific article; zbMATH DE number 6847667