Reduced order modeling for parametrized PDEs

4:30PM at LWSN 1142
Prof. Jan S Hesthaven, EPFL, Lausanne, Switzerland
Reduced order modeling for parametrized PDEs

The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.

However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large scale applications. In this talk, we discuss recent developments of reduced basis methods that enables the formulation of rigorous error estimates to certify the results obtained with the reduced model. The efficiency and accuracy of this shall be demonstrated by a number of 2D and 3D examples.

We subsequently discuss the extension of such ideas to time-dependent problems and problems with substantial nonlinear behavior. For nonlinear time-dependent problems we pay particular attention to reduced models for Hamiltonian problems but shall also consider more general problems. For general nonlinear problems such as the Navier-Stokes equations, a direct development of a reduced model is complex. To overcome this we discuss the use of machine learning, combining reduced basis methods and neural networks. This opens the doors for the development of fast reduced models for very general parameterized nonlinear problems, yet many questions remains open.

Throughout the talk we illustrate the different ideas with examples and also point out some of the many remaining open problems, in particular the application and understanding of reduced basis methods for general nonlinear problems.

A reception with refreshments will follow immediately after the lecture outside of LWSN 1142