Certified reduced basis approximation for parametrized partial differential equations and applications

By Alfio Maria Quarteroni, Gianluigi Rozza, Andrea Manzoni. 2011

300 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/462b73450e9ea13d83f476529a1a416ff867d09e

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In Computational Mechanics

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Citations in the corpus

Digital Twin: Values, Challenges and Enablers From a Modeling Perspective (2019, unknown): “[202] A.”
Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “2 For a complete presentation of the reduced basis method we refer to [25, 29], now we just recall its main features and we introduce some notations.”