Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations

We discuss reduced basis approximation and associated a posteriori error estimation for reliable and rapid solution of parametrized partial differential equations.

The crucial ingredients are rapidly convergent Galerkin approximations over a space spanned by snapshots on the parametrically induced solution manifold; efficient POD (in time)/Greedy (in parameter) selection of quasi-optimal samples;
effective Successive Constraint Method construction of stability-constant lower bounds; rigorous and sharp a posteriori error estimators for the norms and outputs/quantities of interest; and Offline-Online computational procedures for very rapid response in the real-time and many-query contexts.

In this talk we first describe the technical ingredients in the context of linear parabolic equations. We then present illustrative
results for a broad range of elliptic, parabolic, and hyperbolic examples drawn from heat transfer and mechanics.