Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations

Abstract

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with (approximately) 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 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 (approximate) 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 the case in which the parametrization of the partial differential equation is exact; in this paper, we address the situation in which our mathematical model is not “complete.” In particular, we permit error in the data that define our partial differential operator: this error may be introduced, for example, by imperfect specification, measurement, calculation, or parametric expansion of a coefficient function. We develop both accurate predictions for the outputs of interest and associated rigorous a posteriori error bounds; and the latter incorporate both numerical discretization and “model truncation” effects. Numerical results are presented for a particular instantiation in which the model error originates in the (approximately) prescribed velocity field associated with a three-dimensional convection-diffusion problem.