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Reduced-basis Output Bound Methods for Parametrised Partial Differential Equations

Prud'homme, C; Rovas, DV; Veroy, K; Machiels, L; Maday, Y; Patera, AT; Turinici, G; (2002) Reduced-basis Output Bound Methods for Parametrised Partial Differential Equations. In: Proceedings of Proceedings of the Singapore MIT Alliance Symposium 2002. Massachusetts Institute of Technology: Saarbrücken, Germany. Green open access

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Abstract

An efficient and reliable method for the prediction of outputs of interest of partial differential equations with affine parameter dependence is presented. To achieve efficiency we employ the reduced-basis method: a weighted residual Galerkin-type method, where the solution is projected onto low-dimensional spaces with certain problem-specific approximation properties. Reliability is obtained by a posteriori error estimation methods - relaxations of the standard error-residual equation that provide inexpensive but sharp and rigorous bounds for the error in outputs of interest. Special affine parameter dependence of the differential operator is exploited to develop a two-stage off-line/on-line blackbox computational procedure. In the on-line stage, for every new parameter value, we calculate the output of interest and an associated error bound. 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 thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. The theory and corroborating numerical results are presented for: symmetric coercive problems (e.g. problems in conduction heat transfer), parabolic problems (e.g. unsteady heat transfer), noncoercive problems (e.g. the reduced-wave, or Helmholtz, equation), the Stokes problem (e.g flow of highly viscous fluids), and certain nonlinear equations (e.g. eigenvalue problems).

Type: Proceedings paper
Title: Reduced-basis Output Bound Methods for Parametrised Partial Differential Equations
Event: Singapore MIT Alliance Symposium 2002
Location: Saarbrücken, Germany
Open access status: An open access version is available from UCL Discovery
Publisher version: https://dspace.mit.edu/handle/1721.1/16956
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Mechanical Engineering.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of the Built Environment
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of the Built Environment > Bartlett School Env, Energy and Resources
URI: https://discovery.ucl.ac.uk/id/eprint/10062917
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