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A stochastic collocation method for elliptic partial differential equations with random input data
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2007 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 45, no 3, 1005-1034 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms ( input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuska, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 ( 2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

Place, publisher, year, edition, pages
2007. Vol. 45, no 3, 1005-1034 p.
Keyword [en]
collocation method, stochastic partial differential equations, finite elements, uncertainty quantification, exponential convergence, finite-element-method, polynomial chaos, uncertainty, coefficients, interpolation, algorithms
National Category
Computer and Information Science
URN: urn:nbn:se:kth:diva-16747ISI: 000247647700006OAI: diva2:334790
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2012-02-07Bibliographically approved

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