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A sparse grid stochastic collocation method for partial differential equations with random input data
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2008 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 5, 2309-2345 p.Article in journal (Refereed) Published
Abstract [en]

This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coeffcients and forcing terms ( input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using L-q norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem ( number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.

Place, publisher, year, edition, pages
2008. Vol. 46, no 5, 2309-2345 p.
Keyword [en]
collocation techniques, stochastic PDEs, finite elements, uncertainty, quantification, sparse grids, Smolyak approximation, multivariate, polynomial approximation, polynomial chaos, uncertainty, quadrature, interpolation, coefficients
National Category
Computer and Information Science
Identifiers
URN: urn:nbn:se:kth:diva-17702DOI: 10.1137/060663660ISI: 000257746600005Scopus ID: 2-s2.0-52749083629OAI: oai:DiVA.org:kth-17702DiVA: diva2:335747
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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CiteExportLink to record
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  • apa
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