Author:
Quarteroni Alfio,Rozza Gianluigi,Manzoni Andrea
Abstract
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 real-time 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 results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.
Publisher
Springer Science and Business Media LLC
Reference87 articles.
1. Rozza G, Huynh P, Patera A: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 2008, 15: 229–275. 10.1007/s11831-008-9019-9
2. Patera, A., Rozza, G.: Reduced Basis Approximation and a posteriori Error Estimation for Parametrized Partial Differential Equations. Version 1.0, MIT. http://augustine.mit.edu (2006)
3. Prud’homme C, Rovas D, Veroy K, Maday Y, Patera A, Turinici G: Reliable real-time solution of parametrized partial differential equations: reduced-basis output bounds methods. J. Fluids Eng. 2002, 124: 70–80. 10.1115/1.1448332
4. Porsching TA: Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comput. 1985,45(172):487–496. 10.1090/S0025-5718-1985-0804937-0
5. Ito K, Ravindran S: A reduced-order method for simulation and control of fluid flow. J. Comput. Phys. 1998,143(2):403–425. 10.1006/jcph.1998.5943
Cited by
202 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献