Prud'homme, C; Rovas, DV; Veroy, K; Machiels, L; Maday, Y; Patera, AT; Turinici, G; (2002) Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. Journal of Fluids Engineering, Transactions of the ASME , 124 (1) pp. 70-80. 10.1115/1.1448332.

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Abstract

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 WN 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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