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Volume 2, Issue 2
Efficient Collocational Approach for Parametric Uncertainty Analysis

D. Xiu

Commun. Comput. Phys., 2 (2007), pp. 293-309.

Published online: 2007-02

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  • Abstract

A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudo-spectral” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations. 

  • Keywords

Collocation methods, pseudo-spectral methods, stochastic inputs, random differential equations, uncertainty quantification.

  • AMS Subject Headings

65C20, 65C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-2-293, author = {D. Xiu}, title = {Efficient Collocational Approach for Parametric Uncertainty Analysis}, journal = {Communications in Computational Physics}, year = {2007}, volume = {2}, number = {2}, pages = {293--309}, abstract = {

A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudo-spectral” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations. 

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7907.html} }
TY - JOUR T1 - Efficient Collocational Approach for Parametric Uncertainty Analysis AU - D. Xiu JO - Communications in Computational Physics VL - 2 SP - 293 EP - 309 PY - 2007 DA - 2007/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7907.html KW - Collocation methods, pseudo-spectral methods, stochastic inputs, random differential equations, uncertainty quantification. AB -

A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudo-spectral” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations. 

D. Xiu. (2007). Efficient Collocational Approach for Parametric Uncertainty Analysis. Communications in Computational Physics. 2 (2). 293-309. doi:
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