In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem - we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe - Galerkin projection onto a space ${W}_{N}$ spanned by solutions of the governing partial differential equation at $N$ selected points in parameter-time space - and develop a new greedy adaptive procedure to “optimally” construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve a priori to construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline-online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on $N$ (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.

Classification: 35K15, 65M15

Keywords: parabolic partial differential equations, diffusion equation, parameter-dependent systems, reduced-basis methods, output bounds, Galerkin approximation, a posteriori error estimation

@article{M2AN_2005__39_1_157_0, author = {Grepl, Martin A. and Patera, Anthony T.}, title = {A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {39}, number = {1}, year = {2005}, pages = {157-181}, doi = {10.1051/m2an:2005006}, zbl = {1079.65096}, mrnumber = {2136204}, language = {en}, url = {http://www.numdam.org/item/M2AN_2005__39_1_157_0} }

Grepl, Martin A.; Patera, Anthony T. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 1, pp. 157-181. doi : 10.1051/m2an:2005006. http://www.numdam.org/item/M2AN_2005__39_1_157_0/

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