Abstract:

We consider (hierarchical, Lagrange) reduced basis approximation and a posteriori
error estimation for linear functional outputs of affinely parametrized  partial differential equations. 
The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional
space associated with a smooth ``parametric manifold'' --- dimension reduction;
efficient and effective greedy sampling methods for identification of optimal and numerically
stable approximations --- rapid convergence; a posteriori error estimation procedures ---
rigorous and sharp bounds for the linear-functional outputs of interest; and Offline-Online
computational decomposition strategies --- minimum marginal cost for high performance
in the real-time/embedded (e.g., parameter-estimation, control) and many-query
(e.g., design optimization,  multiscale) contexts. We present illustrative results
(worked problems)  for heat conduction and convection-diffusion, inviscid potential flows,
viscous flows and linear elasticity by a software library available at the
web address: http://augustine.mit.edu.