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Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. (English) Zbl 1288.65157

Summary: The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, C. R., Math., Acad. Sci. Paris 350, No. 9–10, 539–542 (2012; Zbl 1245.65105)]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
76Q05 Hydro- and aero-acoustics
76M15 Boundary element methods applied to problems in fluid mechanics

Citations:

Zbl 1245.65105

Software:

rbMIT
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