Certified Reduced Basis Methods for Wave Problems

Jan S Hesthaven
Division of Applied Mathematics
Brown University
Providence, RI, USA


Abstract:

The development and application of models of reduced computational complexity is used extensively throughout science and engineering to enable the fast/real-time modeling of complex systems for control, design, or prediction purposes. However, these models, while often successful and of undisputed value, are often heuristic in nature and the validity and accuracy of the output is often unknown.

In this talk we will discuss recent attempts to develop reduced basis methods for which one can develop a rigorous a posteriori theory. The approach aims at formulating reduced models for parameterized linear partial differential equations and we will focus the attention on wave problems, although most of the developments are applicable to other types of equations also.

We will provide a brief overview of standard methods, offer an outline of the theoretical developments of 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 performance of the certified reduced basis model will be illustrated through examples drawn from electromagnetics. We will conclude with a few generalizations and outlook toward future developments.

This is work done in collaboration with Prof A. Patera (MIT), Prof. Y. Maday (Paris), Dr. J. Rodriguez (ENSTA), and Dr. Y. Chen (Brown).