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Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

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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.

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Acknowledgments

We acknowledge the use of the Matlab packages rbMIT developed by the group of Prof. Anthony Patera in MIT http://augustine.mit.edu/methodology/methodology_rbMIT_System.htm, MLife previously developed by Prof. Fausto Saleri from MOX, Politecnico di Milano and spinterp by Dr. Andreas Klimke from Universität Stuttgart http://www.ians.uni-stuttgart.de/spinterp/. The authors thank Prof. Fabio Nobile for several helpful insights and the numerical verification by the package sparse grid toolkit http://www2.mate.polimi.it:8080/NUMQUES/codes co-developed with Dr. Lorenzo Tamellini and the anonymous referees for the improvement of the presentation of the work. This work is partially supported by Swiss National Science Foundation under Grant No. \(200021\_141034\).

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Authors and Affiliations

  1. Modelling and Scientific Computing, CMCS, Mathematics Institute of Computational Science and Engineering, MATHICSE, Ecole Polytechnique Fédérale de Lausanne, EPFL, Station 8, 1015 , Lausanne, Switzerland

    Peng Chen & Alfio Quarteroni

  2. Modellistica e Calcolo Scientifico, MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 , Milano, Italy

    Alfio Quarteroni

  3. SISSA MathLab, International School for Advanced Studies, via Bonomea 265, 34136 , Trieste, Italy

    Gianluigi Rozza

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  1. Peng Chen
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Correspondence to Gianluigi Rozza.

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Chen, P., Quarteroni, A. & Rozza, G. Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems. J Sci Comput 59, 187–216 (2014). https://doi.org/10.1007/s10915-013-9764-2

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