Efficient Solution of Ordinary Differential Equations with High-Dimensional Parametrized Uncertainty
Zhen Gao 1, Jan S. Hesthaven 2*1 Research Center for Applied Mathematics, Ocean University of China, Qingdao 266071, Shandong, China; and Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.
2 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.
Received 9 January 2010; Accepted (in revised version) 8 September 2010
Available online 27 April 2011
The important task of evaluating the impact of random parameters on the output of stochastic ordinary differential equations (SODE) can be computationally very demanding, in particular for problems with a high-dimensional parameter space. In this work we consider this problem in some detail and demonstrate that by combining several techniques one can dramatically reduce the overall cost without impacting the predictive accuracy of the output of interests. We discuss how the combination of ANOVA expansions, different sparse grid techniques, and the total sensitivity index (TSI) as a pre-selective mechanism enables the modeling of problems with hundred of parameters. We demonstrate the accuracy and efficiency of this approach on a number of challenging test cases drawn from engineering and science.AMS subject classifications: 62F12, 65C20, 65C30
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Key words: Sparse grids, stochastic collocation method, ANOVA expansion, total sensitivity index.
Email: email@example.com (Z. Gao), Jan.Hesthaven@Brown.edu (J. S. Hesthaven)