On the relationship between the stochastic Galerkin method and the pseudo-spectral collocation method for linear differential algebraic equations
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 analysis with an RLC-circuit with stochastic elements and with a stochastic multiconductor transmission line.
scientific article; zbMATH DE number 6847667
scientific article