Machiels, L; Maday, Y; Patera, AT; Rovas, DV; (2001) Blackbox reduced-basis output bound methods for shape optimisation. In: Chan, Tony and Kako, Takashi and Kawarada, Hideo and Pironneau, Olivier, (eds.) Proceedings of 12th International Conference on Domain Decomposition Methods. (pp. pp. 429-436). ddm.org: Chiba, Japan.

Text C03.pdf - Published Version Access restricted to UCL open access staff Download (1MB)

Abstract

We present a two-stage off-line/on-line blackbox reduced-basis output bound method for the prediction of outputs of coercive partial differential equations with affine parameter dependence. 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 differential operator. The method is both efficient and certain: thanks to rigorous a posteriori error bounds, we may 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 and optimization, in which repeated and rapid evaluation of the output is required. Reduced-basis methods [ASB78, Nag79, NP80] — projection onto low-order approximation spaces comprising solutions of the problem of interest at selected points in the parameter/design space — are efficient techniques for the prediction of linear functional outputs. These methods enjoy an optimality property which ensures rapid convergence even in high-dimensional parameter spaces; good accuracy is obtained even for very few modes (basis functions), and thus the computational cost is typically very small. It is often the case that the parameter enters affinely in the differential operator. This allows us to separate the computational steps into two stages:(i) the off-line stage, in which the reduced-basis space is constructed; and (ii) the on-line/real time stage, in which for each new parameter value the reduced-basis approximation for the output of interest is calculated. The on-line stage is “blackbox” in the sense that there is no longer any reference to the original problem formulation:the computational complexity of this stage scales only with the dimension of the reduced-basis space and the parametric complexity of the partial differential operator. Although a priori theory [FR83, Por85] suggests the optimality of the reducedbasis space approximation, for a particular choice of the reduced-basis space the error in the output of interest is typically not known, and hence the minimal number of basis functions required to satisfy the desired error tolerance can not be ascertained. As a result, either too many or too few basis functions are retained; the former results in computational inefficiency, the latter in uncertainty and unacceptably inaccurate predictions. In this paper we develop blackbox a posteriori methods that address these shortcomings. We consider here equilibrium solutions of coercive problems within the context of shape optimization; see also [MPR00] for treatment of noncoercive equilibrium problems and [MMO+00] for symmetric eigenvalue problems.

Download activity - last month

Export as CSVXMLJSON

Download activity - last 12 months

Export as CSVXMLJSON

Downloads by country - last 12 months

Export as CSVXMLJSON

Archive Staff Only

View Item