Certified reduced basis methods and reduced collocation methods

20 February 2015
12:00 pm
Certified reduced basis methods and reduced collocation methods
Yanlai Chen
Associate Professor
Dept. of Mathematics
University of Massachusetts, Dartmouth
Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized partial differential equation are desired in a 
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.