Journal of Mathematics in Industry (2011) 1:3 DOI RESEARCH Open Access Certified reduced basis approximation for parametrized partial differential equations and applications Alfio Quarteroni · Gianluigi Rozza · Andrea Manzoni Received: 2 March 2011 / Accepted: 3 June 2011 / Published online: 3 June 2011 © 2011 Quarteroni et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Abstract Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods (built. | Journal of Mathematics in Industry 2011 1 3 DOI 2190-5983-1-3 Journal of Mathematics in Industry a SpringerOpen Journal RESEARCH Open Access Certified reduced basis approximation for parametrized partial differential equations and applications Alfio Quarteroni Gianluigi Rozza Andrea Manzoni Received 2 March 2011 Accepted 3 June 2011 Published online 3 June 2011 2011 Quarteroni et al. licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Abstract Reduction strategies such as model order reduction MOR or reduced basis RB methods in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods built upon a high-fidelity truth finite element approximation for a rapid and reliable approximation of parametrized partial differential equations and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction ideally suited for realtime simulation and many-query contexts for example optimization control or parameter identification . After a brief excursus on the methodology we focus on linear elliptic and parabolic problems discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some re- A Quarteroni G Rozza A Manzoni Modelling and Scientific Computing CMCS Mathematics Institute of Computational Science and Engineering MATHICSE Ecole .
