SSD - Hesthaven Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Jan Hesthaven Ph.D.- New Directions in Reduced Order Modeling

Chair of Computational Mathematics and Simulation Science, Ecole Polytechnique Fédérale de Lausanne, Switzerland

Abstract

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 particular for nonlinear and/or timedependent problems. After giving a brief general introduction to reduced order models, we discuss developments in two different directions. In the first part, we discuss recent developments of reduced methods that conserve chosen invariants for nonlinear time-dependent problems. We pay particular attention to the development of reduced models for Hamiltonian problems and propose a greedy approach to build the basis. As we shall demonstrate, attention to the construction of the basis must be paid not only to ensure accuracy but also to ensure stability of the reduced model. As alternative approach we shall also, time permitting, discuss the importance of using a skew-symmetric form to ensure stability of the reduced models for more general conservation laws.

The second part of the talk discusses the combination of reduced order modeling for nonlinear problems with the use of neural networks to overcome known problems of online efficiency for general nonlinear problems. We discuss the general idea in which training of the neural network becomes part of the offline part and demonstrate its potential through a number of examples, including for the incompressible Navier-Stokes equations with geometric variations and a prototype combustion problem.

This work has been done with in collaboration with B.F.  Afkram (EPFL, CH), N. Ripamonti EPFL, CH) and S. Ubbiali (USI, CH), Q. Wang (EPFL, CH).