Division of Applied Mathematics
"Reduced Order Models You Can Trust"
Models of reduced computational complexity are 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 validity and accuracy of the output is often unknown. This limits the predictive value of such models.
We discuss recent and ongoing efforts to develop reduced methods endowed with a rigorous a posteriori theory, hence certifying the accuracy of the model and giving it a true predictive value. The focus is reduced models for parameterized linear partial differential equations such as equations for acoustics or electromagnetics although many others problems are equally treatable. We outline the theoretical 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 as time permits.
The performance of the certified reduced basis model will be illustrated through a number of examples to highlight the major advantages of the proposed approach and, 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), Z. Shun (Brown), Y. Maday (Paris VI), and J. Rodriguez (University of Santiago).