Abstract
AbstractReduction 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 rigorousa posteriorierror 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.
Highlights
Introduction and motivation the increasing computer power makes the numerical solution problems of very large dimensions that model complex phenomena essential, a computational reduction is still determinant whenever interested in real-time simulations and/or repeated output evaluations for different values of some inputs of interest
In this work we review the reduced basis (RB) approximation and a posteriori error estimation methods for the rapid and reliable evaluation of engineering outputs associated with elliptic and parabolic parametrized partial differential equations (PDEs)
We consider a output of interest s(μ) ∈ R expressed as a functional of a field variable u(μ) that is the solution of a partial differential equation, parametrized with respect to the input parameter p-vector μ; the input parameter domain - that is, the set of all possible inputs - is a subset D of Rp
Summary
The increasing computer power makes the numerical solution problems of very large dimensions that model complex phenomena essential, a computational reduction is still determinant whenever interested in real-time simulations and/or repeated output evaluations for different values of some inputs of interest. The reduced basis methodology we recall in this paper is motivated by, and applied within two particular contexts: the real-time context (for example, in-the-field robust parameter-estimation, or nondestructive evaluation); and the many-query context (for example, design or shape optimization, optimal control or multi-model/scale simulation) Both are crucial in view of more widespread application of numerical methods for PDEs in engineering practice and more specific industrial processes. In this paper we shall focus on the case of linear functional outputs of affinely parametrized linear elliptic and parabolic coercive partial differential equations This kind of problems - relatively simple, yet relevant to many important applications in transport (for example, steady/unsteady conduction, convection-diffusion), mass transfer, and more generally in continuum mechanics - proves a convenient expository vehicle for the methodology, with the aim of stressing on the potential impact on possible industrial applications, dealing with optimization for devices and/or processes, diagnosis, control. This paper focuses only on the affine linear elliptic and parabolic coercive cases - in order to allow to catch all the main ingredients - the reduced basis approximation and associated a posteriori error estimation methodology is much more general; many problems can successfully be faced in the even simplest affine case
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