An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations. (English. Abridged French version) Zbl 1061.65118

Summary: We present an efficient reduced-basis discretization procedure for partial differential equations with nonaffine parameter dependence. The method replaces nonaffine coefficient functions with a collateral reduced-basis expansion which then permits an (effectively affine) offline-online computational decomposition. The essential components of the approach are (i) a good collateral reduced-basis approximation space, (ii) a stable and inexpensive interpolation procedure, and (iii) an effective a posteriori estimator to quantify the newly introduced errors. Theoretical and numerical results respectively anticipate and confirm the good behavior of the technique.


65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35R30 Inverse problems for PDEs
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[1] Almroth, B.O.; Stern, P.; Brogan, F.A., Automatic choice of global shape functions in structural analysis, Aiaa j., 16, 525-528, (1978)
[2] Fink, J.P.; Rheinboldt, W.C., On the error behavior of the reduced basis technique for nonlinear finite element approximations, Z. angew. math. mech., 63, 21-28, (1983) · Zbl 0533.73071
[3] Machiels, L.; Maday, Y.; Oliveira, I.B.; Patera, A.T.; Rovas, D.V., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. acad. sci. Paris, ser. I, 331, 2, 153-158, (2000) · Zbl 0960.65063
[4] Maday, Y.; Patera, A.T.; Turinici, G., Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations, C. R. acad. sci. Paris, ser. I, 335, 3, 289-294, (2002) · Zbl 1009.65066
[5] Noor, A.K.; Peters, J.M., Reduced basis technique for nonlinear analysis of structures, Aiaa j., 18, 4, 455-462, (1980)
[6] Prud’homme, C.; Rovas, D.; Veroy, K.; Maday, Y.; Patera, A.T.; Turinici, G., Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, J. fluids engrg., 124, 1, 70-80, (2002)
[7] Quarteroni, A.; Sacco, R.; Saleri, F., Numer. math., Texts appl. math., vol. 37, (1991), Springer New York
[8] Veroy, K.; Prud’homme, C.; Rovas, D.V.; Patera, A.T., A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, (), AIAA Paper 2003-3847
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