Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations
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HPCES017.pdf
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Author(s)
Prud'homme, C.
Rovas, D.V.
Veroy, K.
Machiels, L.
Maday, Y.
Patera, Anthony T.
Turinici, G.
Date Issued
January 2002
Series/Report no.
High Performance Computation for Engineered Systems (HPCES);
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.
Subjects
reduced-basis
a posteriori error estimation
output bounds
partial differential equations
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