Models of reduced computational complexity is used extensively throughout science and engineering to enable the fast/real-time/subscale modeling of complex systems for control, design, multi-scale analysis, uncertainty quantification etc While of undisputed value these reduced models are, however, often heuristic in nature and the accuracy of the output is often unknown, hence limiting the predictive value.
We discuss ongoing efforts to develop reduced methods endowed with a rigorous a posteriori theory to certify the accuracy of the model. The focus will be on reduced models for parameterized linear partial differential equations. We outline the basic ideas behind certified reduced basis methods, discuss an offline-online approach to ensure computational efficiency, and emphasize how the error estimator can be exploited to construct an efficient basis at minimal computational off-line cost.
The discussion will draw on examples based both on differential and integral equations formulations. Time permitting, we shall discuss the challenges and some ideas to enable the development of reduced models for high-dimensional parametric problems.
This is work done in collaboration with Y. Chen (UMass Dartmouth), B. Stamm (Paris VI), S. Zhang (Brown), and Y. Maday (Paris VI).