Accurate and online-efficient evaluation of thea posteriorierror bound in the reduced basis method

Abstract

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, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539-542.]. 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.Mathematics Subject Classification. 65N15, 65D05, 68W25, 76Q05.Article published by EDP Sciences c EDP Sciences, SMAI 2014 208 F. CASENAVE ET AL.As described in Section 2, the RB method consists in replacing the sequence P μHere, P denotes the parameter set, E μ : μ → u μ the model problem,Ê μ : μ →û μ its lower-dimensional approximation, Q(u μ ) the quantity of interest, andQ(û μ ) its RB approximation. More specifically, the RB method consists in two steps: (i) A so-called offline stage, where solutions to E μ for well-chosen values of the parameter μ are computed. During this stage,N problems of size N are solved (withN N ), and some quantities related to theN solutions are stored, and (ii) a so-called online stage, where the precomputed quantities are used to solveÊ μ for many values of μ. In this stage, a certification of the approximation is possible by means of an a posteriori error bound. An important feature in the RB method is the use of an online-efficient error bound. The notion of online-efficiency is defined in Section 2.4. Moreover, the error bound must be as sharp as possible to faithfully represent the error. However, as noticed for example in ([34], pp. 148-149), the error bound is subject to round-off errors, especially for the computation of accurate solutions. This difficulty can be encountered in complex industrial applications in the following two cases. First and most importantly, when the stability constant of the underlying bilinear (or sesquilinear) form is very small, the classical formula for the error bound fails to certify, even at a relatively crude error level, as illustrated in Section 5 where the stability constant is about 10 −6 and the classical error bound stagnates at about 10 −4 . Second, in some industrial codes, the single-precision format is used to speed up computat...

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