A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners

Veroy, Karen; Rovas, Dimitrios V.; Patera, Anthony T.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 1007-1028.

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations - Galerkin projection onto a space $W_N$ 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 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. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity, is close to (but always greater than) unity. However, in other cases, the necessary “bound conditioners” - in essence, operator preconditioners that (i) satisfy an additional spectral “bound” requirement, and (ii) admit the reduced-basis off-line/on-line computational stratagem - either can not be found, or yield unacceptably large effectivities. In this paper we introduce a new class of improved bound conditioners: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); we thereby accommodate higher-order (e.g., piecewise linear) effectivity constructions while simultaneously preserving on-line efficiency. Simple convex analysis and elementary approximation theory suffice to prove the necessary bounding and convergence properties.

Classification : 35J50, 65N15
Mots clés : elliptic partial differential equations, reduced-basis methods, output bounds, Galerkin approximation, a posteriori error estimation, convex analysis

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