NDLI: Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

Content Provider	Springer Nature Link
Author	Chen, Peng Quarteroni, Alfio Rozza, Gianluigi
Copyright Year	2013
Abstract	The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005–1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411–2442, 2008a; SIAM J Numer Anal 46(5):2309–2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118–1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289–294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu , 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229–275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41–44):3187–3206, 2009; Arch Comput Methods Eng 17:435–454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $$O(1)$$ to moderate dimensions $$O(10)$$ and to high dimensions $$O(100)$$ . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.
Starting Page	187
Ending Page	216
Page Count	30
File Format	PDF
ISSN	08857474
Journal	Journal of Scientific Computing
Volume Number	59
Issue Number	1
e-ISSN	15737691
Language	English
Publisher	Springer US
Publisher Date	2013-08-09
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Stochastic elliptic problem Reduced basis method Stochastic collocation method Sparse grid Greedy algorithm Offline–online computational decomposition Convergence analysis Algorithms Computational Mathematics and Numerical Analysis ApplicationMathematics/Computational Methods of Engineering Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Engineering Computational Theory and Mathematics Computational Mathematics Numerical Analysis Software

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