The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data.

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This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients 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 and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method ... continued below

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56 p.

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Webster, Clayton; Tempone, Raul (Florida State University, Tallahassee, FL) & Nobile, Fabio (Politecnico di Milano, Italy) December 1, 2007.

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Description

This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients 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 and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo.

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56 p.

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  • Report No.: SAND2007-8093
  • Grant Number: AC04-94AL85000
  • DOI: 10.2172/934852 | External Link
  • Office of Scientific & Technical Information Report Number: 934852
  • Archival Resource Key: ark:/67531/metadc893089

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  • December 1, 2007

Added to The UNT Digital Library

  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 22, 2016, 4:35 p.m.

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Webster, Clayton; Tempone, Raul (Florida State University, Tallahassee, FL) & Nobile, Fabio (Politecnico di Milano, Italy). The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data., report, December 1, 2007; United States. (digital.library.unt.edu/ark:/67531/metadc893089/: accessed April 26, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.