Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations
Author(s)Prud'homme, C.; Rovas, D.V.; Veroy, K.; Machiels, L.; Maday, Y.; Patera, Anthony T.; Turinici, G.; ... Show more Show less
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.
High Performance Computation for Engineered Systems (HPCES);
reduced-basis, a posteriori error estimation, output bounds, partial differential equations