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An overview of blackbox reduced-basis output bound methods for elliptic partial differential equations

Rovas Dimitrios

Πλήρης Εγγραφή

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Περίληψη

We present a two-stage off-line/online blackbox reduced-basis output bound method for the prediction of outputs of interest associated with elliptic partial differential equations with ane parameter dependence. The method is characterized by (i) Galerkin projection onto a reduced-basis space comprising solutions at selected points in parameter space, and (ii) a rigorous output error bound based on the dual norm of the resulting residual. The computational complexity of the on-line stage of the procedure scales only with the dimension of the reduced-basis space and the parametric complexity of the partial di erential operator. The method is thus both ecient and certain: thanks to the a posteriori error bounds, we may safely retain only the minimal number of modes necessary to achieve the prescribed accuracy in the output of interest. The technique is particularly appropriate for applications such as design, optimization, and control, in which repeated and rapid evaluation of the output is required; in the limit of many evaluations, the method can be several orders of magnitude faster than standard (finite element) approximation. To illustrate the method, we consider the design of a thermal fin.

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