We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations—Galerkin projection onto a space spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation—relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; 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.
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March 2002
Technical Papers
Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods
C. Prud’homme,
C. Prud’homme
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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D. V. Rovas,
D. V. Rovas
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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K. Veroy,
K. Veroy
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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L. Machiels,
L. Machiels
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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Y. Maday,
Y. Maday
Laboratoire d’Analyse Nume´rique, Universite´ Pierre et Marie Curie, Boı^te courrier 187, 75252 Paris, Cedex 05, France
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A. T. Patera,
A. T. Patera
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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G. Turinici
G. Turinici
ASCI-CNRS Orsay, and INRA Rocquencourt M3N, B.P. 105, 78153 LeChesnay Cedex France
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C. Prud’homme
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
D. V. Rovas
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
K. Veroy
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
L. Machiels
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
Y. Maday
Laboratoire d’Analyse Nume´rique, Universite´ Pierre et Marie Curie, Boı^te courrier 187, 75252 Paris, Cedex 05, France
A. T. Patera
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
G. Turinici
ASCI-CNRS Orsay, and INRA Rocquencourt M3N, B.P. 105, 78153 LeChesnay Cedex France
Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering Division September 13, 2001; revised manuscript received November 2, 2001. Associate Editor. G. Karmadakis.
J. Fluids Eng. Mar 2002, 124(1): 70-80 (11 pages)
Published Online: November 2, 2001
Article history
Received:
September 13, 2001
Revised:
November 2, 2001
Citation
Prud’homme , C., Rovas , D. V., Veroy , K., Machiels, L., Maday, Y., Patera, A. T., and Turinici, G. (November 2, 2001). "Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods ." ASME. J. Fluids Eng. March 2002; 124(1): 70–80. https://doi.org/10.1115/1.1448332
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