J Sci Comput (2014) 59:187–216 DOI 10.1007/s10915-013-9764-2 Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems Peng Chen · Alfio Quarteroni · Gianluigi Rozza Received: 28 January 2013 / Revised: 28 May 2013 / Accepted: 27 July 2013 / Published online: 9 August 2013 © Springer Science+Business Media New York 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 Com- put 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 Com- put 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), computa- tional 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 colloca- P. Chen · A. Quarteroni Modelling and Scientific Computing, CMCS, Mathematics Institute of Computational Science and Engineering, MATHICSE, Ecole Polytechnique Fédérale de Lausanne, EPFL, Station 8, 1015 Lausanne, Switzerland e-mail: peng.chen@epfl.ch A. Quarteroni Modellistica e Calcolo Scientifico, MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy e-mail: alfio.quarteroni@epfl.ch G. Rozza (B ) SISSA MathLab, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy e-mail: gianluigi.rozza@sissa.it 123