A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations |
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Authors: | Yvon Maday Anthony T. Patera Gabriel Turinici |
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Affiliation: | (1) Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris, France;(2) Department of Mechanical Engineering, M.I.T., 77 Mass. Ave., Cambridge, Massachusetts, 02139;(3) ASCI-CNRS Orsay, and INRIA Rocquencourt M3N, B.P. 105, 78153 Le Chesnay, France |
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Abstract: | We consider Lagrangian reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions. |
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Keywords: | reduced basis method interpolation methods exponential convergence |
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