fast/real-time fashion. These include simulation-based design, parameter optimization, optimal control, multi-model/scale analysis, uncertainty quantification etc. Thanks to an
offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced
collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RB/RC solution is maintained through a rigorous a posteriori error estimator whose efficient
development is critical.
In this talk, I will give a brief introduction of the RBM and discuss recent and ongoing efforts to develop RCM, and the accompanying parametric analytical preconditioning techniques
which are capable of improving the quality of the error estimation uniformly on the parameter domain, and speeding up the convergence of the reduced solution to the truth approximation significantly.
New ways of effectively bounding the stability constants for the error estimation may also be discussed. These results are critical in certifying the accuracy of the reduced model and giving it a reliable predictive value.